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Eureka Math
Precalculus
Module 1
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Mathematics at Louisiana State University for their support in the development of
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10 9 8 7 6 5 4 3 2 1
ISBN 978-1-64054-386-7
G12-M1-UTE-1.3.0-05.2018
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Board of Trustees
Lynne Munson, President and Executive Director of Great Minds
Nell McAnelly, Chairman, Co-Director Emeritus of the Gordon A. Cain Center for STEM Literacy at Louisiana
State University
William Kelly, Treasurer, Co-Founder and CEO at ReelDx
Jason Griffiths, Secretary, Director of Programs at the National Academy of Advanced Teacher Education
Pascal Forgione, Former Executive Director of the Center on K-12 Assessment and Performance Management
at ETS
Lorraine Griffith, Title I Reading Specialist at West Buncombe Elementary School in Asheville, North Carolina
Bill Honig, President of the Consortium on Reading Excellence (CORE)
Richard Kessler, Executive Dean of Mannes College the New School for Music
Chi Kim, Former Superintendent, Ross School District
Karen LeFever, Executive Vice President and Chief Development Officer at ChanceLight Behavioral Health and
Education
Maria Neira, Former Vice President, New York State United Teachers
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DĂƚŚĞŵĂƚŝĐƐ CƵƌƌŝĐƵůƵŵ
PRECALCULUS AND ADVANCED TOPICS ͻ DODULE 1
Table of Contents1
CŽŵƉůĞdž NƵŵďĞƌƐ ĂŶĚ TƌĂŶƐĨŽƌŵĂƚŝŽŶƐ
DŽĚƵůĞ OǀĞƌǀŝĞǁ .............................................................................................................................. .................... 3
Topic A: A Question of Linearity͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ ....................................................................................... 17
Lessons 1–2: Wishful Thinking—Does Linearity Hold? ........................................................................... 19
Lesson 3: Which Real Number Functions Define a Linear Transformation? .......................................... 34
Lessons 4–5: An Appearance of Complex Numbers ............................................................................... 47
Lesson 6: Complex Numbers as Vectors ................................................................................................. 73
Lessons 7–8: Complex Number Division................................................................................................. 85
Topic B: Complex Number Operations as Transformations ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘......... 104
Lessons 9–10: The Geometric Effect of Some Complex Arithmetic ..................................................... 106
Lessons 11–12: Distance and Complex Numbers ................................................................................. 126
Lesson 13: Trigonometry and Complex Numbers ................................................................................ 145
Lesson 14: Discovering the Geometric Effect of Complex Multiplication ............................................ 170
Lesson 15: Justifying the Geometric Effect of Complex Multiplication ................................................ 182
Lesson 16: Representing Reflections with Transformations ................................................................ 202
Lesson 17: The Geometric Effect of Multiplying by a Reciprocal ......................................................... 212
DŝĚ-DŽĚƵůĞ AƐƐĞƐƐŵĞŶƚ ĂŶĚ RƵďƌŝĐ ................................................................................................................ 226
Topics A through B (assessment 1 day, return 1 day, remediation or further applications 3 days)
Topic C: The Power of the Right Notation .. .......................................................................................................
... 242
Lessons 18–19: Exploiting the Connection to Trigonometry................................................................ 244
Lesson 20: Exploiting the Connection to Cartesian Coordinates.......................................................... 271
Lesson 21: The Hunt for Better Notation ............................................................................................. 281
Lessons 22–23: Modeling Video Game Motion with Matrices ............................................................ 293
Lesson 24: Matrix Notation Encompasses New Transformations! ...................................................... 325
1Each lesson is ONE day, and ONE day is considered a 45-minute period.
DŽĚƵůĞ 1: Complex Numbers and Transformations 1
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A STORY OF FUNCTIONS DŽĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
Lesson 25: Matrix Multiplication and Addition .................................................................................... 342
Lessons 26–27: Getting a Handle on New Transformations ................................................................ 353
Lessons 28–30: When Can We Reverse a Transformation? ................................................................. 380
EŶĚ-ŽĨ-DŽĚƵůĞ AƐƐĞƐƐŵĞŶƚ ĂŶĚ RƵďƌŝĐ ............................................................................................................ 411
Topics A through C (assessment 1 day, return 1 day, remediation or further applications 4 days)
2 DŽĚƵůĞ 1: Complex Numbers and Transformations
©201ϴ Great Minds ®. eureka-math.org
A STORY OF FUNCTIONS DŽĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
Precalculus and Advanced Topics ͻ Module 1
CŽŵƉůĞdž NƵŵďĞƌƐ ĂŶĚ TƌĂŶƐĨŽƌŵĂƚŝŽŶƐ
OVERVIEW
Module 1 sets the stage for expanding students’ understanding of transformations by first exploring the
notion of linearity in an algebraic context (“Which familiar algebraic functions are linear?”). This quickly leads
to a return to the study of complex numbers and a study of linear transformations in the complex plane.
Thus, Module 1 builds on skills introduced in Geometry and Algebra Ϯ.
Topic A opens with a study of common misconceptions by asking questions such as “For which numbers ܽ and
ଵ ଵ ଵ
ܾ does (ܽ + ܾ)ଶ = ܽଶ + ܾ ଶ happen to hold?”; “Are there numbers ܽ and ܾ for which = + ?”; and so
ା
on. This second equation has only complex solutions, which launches a study of quotients of complex
numbers and the use of conjugates to find moduli and quotients. The topic ends by classifying real and
complex functions that satisfy linearity conditions. (A function ܮis linear if, and only if, there is a real or
complex value ݓsuch that ݖݓ = )ݖ(ܮfor all real or complex ݖ.) Complex number multiplication is
emphasized in the last lesson.
In Topic B, students develop an understanding that when complex numbers are considered points in the
complex plane, complex number multiplication has the geometric effect of a rotation followed by a dilation
in the complex plane. This is a concept that has been developed since Algebra II and builds upon ĂŶ
ŝŶƚƌŽĚƵĐtionŽĨĐŽŵƉůĞdž numberƐ ĂŶĚƉĞƌĨŽƌŵŝŶŐĂĚĚŝƚŝŽŶ͕ƐƵďƚƌĂĐƚŝŽŶ͕ĂŶĚŵƵůƚŝƉůŝĐĂƚŝŽŶǁŝƚŚƚŚĞŵ͕
ǁŚŝĐŚǁĞƌĞĂĐĐŽŵƉĂŶŝĞĚǁŝƚŚƚŚĞŽďƐĞƌǀĂƚŝŽŶƚŚĂƚŵƵůƚŝƉůŝĐĂƚŝŽŶďLJ݅ŚĂƐƚŚĞŐĞŽŵĞƚƌŝĐĞĨĨĞĐƚŽĨƌŽƚĂƚŝŶŐ
ĂŐŝǀĞŶĐŽŵƉůĞdžŶƵŵďĞƌ90° about the origin in a counterclockwise direction. The algebraic inverse of a
complex number (its reciprocal) provides the inverse geometric operation. Analysis of the angle of rotation
and the scale of the dilation brings a return to topics in trigonometry first introduced in Geometry and
expanded on in Algebra II. It also reinforces the geometric interpretation of the modulus of a complex
number and introduces the notion of the argument of a complex number.
The theme of Topic C is to highlight the effectiveness of changing notations and the power provided by
certain notations such as matrices. By exploiting the connection to trigonometry, students see how much
complex arithmetic is simplified. By regarding complex numbers as points in the complex plane, students can
begin to write analytic formulas for translations, rotations, and dilations in the plane and revisit the ideas of
high school Geometry in this light. Taking this work one step further, students develop the 2 × 2 matrix
notation for planar transformations represented by complex number arithmetic. This work sheds light on
how geometry software and video games efficiently perform rigid motion calculations. Finally, the flexibility
implied by 2 × 2 matrix notation allows students to study additional matrix transformations (shears, for
example) that do not necessarily arise from our original complex number thinking context.
In Topic C, the study of vectors and matrices is introduced through a coherent connection to transformations
and complex numbers. Students learn to see matrices as representing transformations in the plane and
develop understanding of multiplication of a matrix by a vector as a transformation acting on a point in the
DŽĚƵůĞ 1: Complex Numbers and Transformations 3
©201ϴ Great Minds ®. eureka-math.org
A STORY OF FUNCTIONS DŽĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
plane. While more formal study of multiplication of matrices occurs in Module 2, in Topic C, students are
exposed to initial ideas of multiplying 2 × 2 matrices including a geometric interpretation of matrix
invertibility and the meaning of the zero and identity matrices. Addition, subtraction, and multiplication with
matrices are introduced in a strictly geometric context and is expanded upon more formally in Module 2.
The Mid-Module Assessment follows Topic B. The End-of-Module Assessment follows Topic C.
FŽĐƵƐ SƚĂŶĚĂƌĚƐ
PĞƌĨŽƌŵ ĂƌŝƚŚŵĞƚŝĐ ŽƉĞƌĂƚŝŽŶƐ ǁŝƚŚ ĐŽŵƉůĞdž ŶƵŵďĞƌƐ͘
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of
complex numbers.
RĞƉƌĞƐĞŶƚ ĐŽŵƉůĞdž ŶƵŵďĞƌƐ ĂŶĚ ƚŚĞŝƌ ŽƉĞƌĂƚŝŽŶƐ ŽŶ ƚŚĞ ĐŽŵƉůĞdž ƉůĂŶĞ͘
(+) Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar forms of
a given complex number represent the same number.
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation.
ଷ
For example, ൫െͳ ξ͵݅൯ ൌ ͺ because ൫െͳ ξ͵݅൯ has modulus ʹ and argument ͳʹͲι.
(+) Calculate the distance between numbers in the complex plane as the modulus of the
difference, and the midpoint of a segment as the average of the numbers at its endpoints.
PĞƌĨŽƌŵ ŽƉĞƌĂƚŝŽŶƐ ŽŶ ŵĂƚƌŝĐĞƐ ĂŶĚ ƵƐĞ ŵĂƚƌŝĐĞƐ ŝŶ ĂƉƉůŝĐĂƚŝŽŶƐ͘
(+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of Ͳ and ͳ in the real numbers. The determinant of a
square matrix is nonzero if and only if the matrix has a multiplicative inverse.
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable
dimensions to produce another vector. Work with matrices as transformations of vectors.
(+) Work with ʹ ൈ ʹ matrices as transformations of the plane, and interpret the absolute
value of the determinant in terms of area.
4 DŽĚƵůĞ 1: Complex Numbers and Transformations
©201ϴ Great Minds ®. eureka-math.org
A STORY OF FUNCTIONS DŽĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
FŽƵŶĚĂƚŝŽŶĂů SƚĂŶĚĂƌĚƐ
RĞĂƐŽŶ ƋƵĂŶƚŝƚĂƚŝǀĞůLJ ĂŶĚ ƵƐĞ ƵŶŝƚƐ ƚŽ ƐŽůǀĞ ƉƌŽďůĞŵƐ͘
Define appropriate quantities for the purpose of descriptive modeling.ƾ
PĞƌĨŽƌŵ ĂƌŝƚŚŵĞƚŝĐ ŽƉĞƌĂƚŝŽŶƐ ǁŝƚŚ ĐŽŵƉůĞdž ŶƵŵďĞƌƐ͘
Know there is a complex number ݅ such that ݅ ଶ ൌ െͳ, and every complex number has the
form ܽ ܾ݅ with ܽ and ܾ real.
Use the relation ݅ ଶ ൌ െͳ and the commutative, associative, and distributive properties to
add, subtract, and multiply complex numbers.
UƐĞ ĐŽŵƉůĞdž ŶƵŵďĞƌƐ ŝŶ ƉŽůLJŶŽŵŝĂů ŝĚĞŶƚŝƚŝĞƐ ĂŶĚ ĞƋƵĂƚŝŽŶƐ͘
Solve quadratic equations with real coefficients that have complex solutions.
(+) Extend polynomial identities to the complex numbers. For example, rewrite ݔଶ Ͷ as
ሺ ݔ ʹ݅ሻሺ ݔെ ʹ݅ሻ.
IŶƚĞƌƉƌĞƚ ƚŚĞ ƐƚƌƵĐƚƵƌĞ ŽĨ ĞdžƉƌĞƐƐŝŽŶƐ͘
Interpret expressions that represent a quantity in terms of its context.ƾ
Interpret parts of an expression, such as terms, factors, and coefficients.
Interpret complicated expressions by viewing one or more of their parts as a single
entity. For example, interpret ܲሺͳ ݎሻ as the product of ܲ and a factor not
depending on ܲ.
WƌŝƚĞ ĞdžƉƌĞƐƐŝŽŶƐ ŝŶ ĞƋƵŝǀĂůĞŶƚ ĨŽƌŵƐ ƚŽ ƐŽůǀĞ ƉƌŽďůĞŵƐ͘
Choose and produce an equivalent form of an expression to reveal and explain properties of
ƾ
the quantity represented by the expression.
Factor a quadratic expression to reveal the zeros of the function it defines.
Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
Use the properties of exponents to transform expressions for exponential functions.
ଵଶ௧
For example the expression ͳǤͳͷ௧ can be rewritten as ൫ͳǤͳͷଵΤଵଶ ൯ ൎ ͳǤͲͳʹଵଶ௧ to
reveal the approximate equivalent monthly interest rate if the annual rate is ͳͷΨ.
CƌĞĂƚĞ ĞƋƵĂƚŝŽŶƐ ƚŚĂƚ ĚĞƐĐƌŝďĞ ŶƵŵďĞƌƐ Žƌ ƌĞůĂƚŝŽŶƐŚŝƉƐ͘ƾ
Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential
functions.
DŽĚƵůĞ 1: Complex Numbers and Transformations 5
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Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law ܸ ൌ ܴܫto highlight resistance ܴ.
UŶĚĞƌƐƚĂŶĚ ƐŽůǀŝŶŐ ĞƋƵĂƚŝŽŶƐ ĂƐ Ă ƉƌŽĐĞƐƐ ŽĨ ƌĞĂƐŽŶŝŶŐ ĂŶĚ ĞdžƉůĂŝŶ ƚŚĞ ƌĞĂƐŽŶŝŶŐ͘
Explain each step in solving a simple equation as following from the equality of numbers
asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
SŽůǀĞ ĞƋƵĂƚŝŽŶƐ ĂŶĚ ŝŶĞƋƵĂůŝƚŝĞƐ ŝŶ ŽŶĞ ǀĂƌŝĂďůĞ͘
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
SŽůǀĞ ƐLJƐƚĞŵƐ ŽĨ ĞƋƵĂƚŝŽŶƐ͘
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
EdžƉĞƌŝŵĞŶƚ ǁŝƚŚ ƚƌĂŶƐĨŽƌŵĂƚŝŽŶƐ ŝŶ ƚŚĞ ƉůĂŶĞ͘
Represent transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs and give other
points as outputs. Compare transformations that preserve distance and angle to those that
do not (e.g., translation versus horizontal stretch).
Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another.
EdžƚĞŶĚ ƚŚĞ ĚŽŵĂŝŶ ŽĨ ƚƌŝŐŽŶŽŵĞƚƌŝĐ ĨƵŶĐƚŝŽŶƐ ƵƐŝŶŐ ƚŚĞ ƵŶŝƚ ĐŝƌĐůĞ͘
Understand radian measure of an angle as the length of the arc on the unit circle subtended
by the angle.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
6 DŽĚƵůĞ 1: Complex Numbers and Transformations
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(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for
ߨȀ͵, ߨȀͶ, and ߨȀ, and use the unit circle to express the values of sine, cosine, and tangent
for ߨ െ ݔ, ߨ ݔ, and ʹߨ െ ݔin terms of their values for ݔ, where ݔis any real number.
PƌŽǀĞ ĂŶĚ ĂƉƉůLJ ƚƌŝŐŽŶŽŵĞƚƌŝĐ ŝĚĞŶƚŝƚŝĞƐ͘
Prove the Pythagorean identity ଶ ሺߠሻ ଶሺߠሻ ൌ ͳ and use it to find ሺߠሻ, ሺߠሻ, or
ሺߠሻ given ሺߠሻ, ሺߠሻ, or ሺߠሻ and the quadrant of the angle.
FŽĐƵƐ SƚĂŶĚĂƌĚƐ ĨŽƌ DĂƚŚĞŵĂƚŝĐĂů PƌĂĐƚŝĐĞ
RĞĂƐŽŶ ĂďƐƚƌĂĐƚůLJ ĂŶĚ ƋƵĂŶƚŝƚĂƚŝǀĞůLJ͘ Students come to recognize that multiplication by a
complex number corresponds to the geometric action of a rotation and dilation from the
origin in the complex plane. Students apply this knowledge to understand that
multiplication by the reciprocal provides the inverse geometric operation to a rotation and
dilation. Much of the module is dedicated to helping students quantify the rotations and
dilations in increasingly abstract ways so they do not depend on the ability to visualize the
transformation. That is, they reach a point where they do not need a specific geometric
model in mind to think about a rotation or dilation. Instead, they can make generalizations
about the rotation or dilation based on the problems they have previously solved.
CŽŶƐƚƌƵĐƚ ǀŝĂďůĞ ĂƌŐƵŵĞŶƚƐ ĂŶĚ ĐƌŝƚŝƋƵĞ ƚŚĞ ƌĞĂƐŽŶŝŶŐ ŽĨ ŽƚŚĞƌƐ͘ Throughout the module,
students study examples of work by algebra students. This work includes a number of
common mistakes that algebra students make, but it is up to students to decide about the
validity of the argument. Deciding on the validity of the argument focuses students on
justification and argumentation as they work to decide when purported algebraic identities
do or do not hold. In cases where they decide that the given student work is incorrect,
students work to develop the correct general algebraic results and justify them by reflecting
on what they perceived as incorrect about the original student solution.
DŽĚĞů ǁŝƚŚ ŵĂƚŚĞŵĂƚŝĐƐ͘ As students work through the module, they become attuned to
the geometric effect that occurs in the context of complex multiplication. However, initially
it is unclear to them why multiplication by complex numbers entails specific geometric
effects. In the module, students create a model of computer animation in the plane. The
focus of the mathematics in the computer animation is such that students come to see
rotating and translating as dependent on matrix operations and the addition of ʹ ൈ ͳ
vectors. Thus, their understanding becomes more formal with the notion of complex
numbers.
DŽĚƵůĞ 1: Complex Numbers and Transformations 7
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TĞƌŵŝŶŽůŽŐLJ
NĞǁ Žƌ RĞĐĞŶƚůLJ IŶƚƌŽĚƵĐĞĚ TĞƌŵƐ
AƌŐƵŵĞŶƚ (The argument of the complex number ݖis the radian (or degree) measure of the
counterclockwise rotation of the complex plane about the origin that maps the initial ray (i.e., the
ray corresponding to the positive real axis) to the ray from the origin through the complex number ݖ
in the complex plane. The argument of ݖis denoted ሺݖሻ.)
ŽƵŶĚ VĞĐƚŽƌ (A bound vector is a directed line segment (an arrow). For example, the directed line
segment ܤܣሬሬሬሬሬԦ is a bound vector whose initial point (or tail) is ܣand terminal point (or tip) is ܤ.
Bound vectors are bound to a particular location in space. A bound vector ܤܣ ሬሬሬሬሬԦ has a magnitude given
by the length of ܤܣand direction given by the ray ܤܣ ሬሬሬሬሬԦ . Many times, only the magnitude and
direction of a bound vector matters, not its position in space. In that case, any translation of that
bound vector is considered to represent the same free vector.)
CŽŵƉůĞdž NƵŵďĞƌ (A complex number is a number that can be represented by a point in the complex
plane. A complex number can be expressed in two forms:
1. The rectangular form of a complex number is ܽ ܾ݅ where ݖcorresponds to the point
ሺܽǡ ܾሻ in the complex plane, and ݅ is the imaginary unit. The number ܽ is called the real part
of ܽ ܾ݅, and the number ܾ is called the imaginary part of ܽ ܾ݅. Note that both the real
and imaginary parts of a complex number are themselves real numbers.
2. For Ͳ ് ݖ, the polar form of a complex number ݖis ݎሺ ሺߠሻ ݅ ሺߠሻሻ where ݎൌ ȁݖȁ and
ߠ ൌ ሺݖሻ, and ݅ is the imaginary unit.)
CŽŵƉůĞdž PůĂŶĞ (The complex plane is a Cartesian plane equipped with addition and multiplication
operators defined on ordered pairs by the following:
à Addition: ሺܽǡ ܾሻ ሺܿǡ ݀ሻ ൌ ሺܽ ܿǡ ܾ ݀ሻ.
When expressed in rectangular form, if ݖൌ ܽ ܾ݅ and ݓൌ ܿ ݀݅, then
ݖ ݓൌ ሺܽ ܿሻ ሺܾ ݀ሻ݅.
à Multiplication: ሺܽǡ ܾሻ ڄሺܿǡ ݀ሻ ൌ ሺܽܿ െ ܾ݀ǡ ܽ݀ ܾܿሻ.
When expressed in rectangular form, if ݖൌ ܽ ܾ݅ and ݓൌ ܿ ݀݅, then
ݓ ڄ ݖൌ ሺܽܿ െ ܾ݀ሻ ሺܽ݀ ܾܿሻ݅. The horizontal axis corresponding to points of the form
ሺݔǡ Ͳሻ is called the real axis, and a vertical axis corresponding to points of the form ሺͲǡ ݕሻ is
called the imaginary axis.)
CŽŶũƵŐĂƚĞ (The conjugate of a complex number of the form ܽ ܾ݅ is ܽ െ ܾ݅. The conjugate of ݖis
denoted ݖ.)
ܽ ܾ
DĞƚĞƌŵŝŶĂŶƚ oĨ ൈ DĂƚƌŝdž (The determinant of the ʹ ൈ ʹ matrix ቂ ቃ is the number computed
ܿ ݀
ܽ ܾ
by evaluating ܽ݀ െ ܾܿ and is denoted by ቀቂ ቃቁ.)
ܿ ݀
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ܽଵଵ ܽଵଶ ܽଵଷ
DĞƚĞƌŵŝŶĂŶƚ oĨ ൈ DĂƚƌŝdž (The determinant of the ͵ ൈ ͵ matrix ܽଶଵ ܽଶଶ ܽଶଷ ൩ is the number
ܽଷଵ ܽଷଶ ܽଷଷ
computed by evaluating the expression,
ܽଶଶ ܽଶଷ ܽଶଵ ܽଶଷ ܽଶଵ ܽଶଶ
ܽଵଵ ቀቂܽ ቃቁ െ ܽ ቀቂ ቃቁ ܽ ቀቂܽଷଵ ܽଷଶ ቃቁ ,
ଷଶ ܽଷଷ ܽଷଵ ܽଷଷ
ଵଶ ଵଷ
ܽଵଵ ܽଵଶ ܽଵଷ
and is denoted by ൭ܽଶଵ ܽଶଶ ܽଶଷ ൩൱.)
ܽଷଵ ܽଷଶ ܽଷଷ
DŝƌĞĐƚĞĚ GƌĂƉŚ (A directed graph is an ordered pair ܦሺܸǡ ܧሻ with
à ܸ a set whose elements are called vertices or nodes, and
à ܧa set of ordered pairs of vertices, called arcs or directed edges.)
DŝƌĞĐƚĞĚ SĞŐŵĞŶƚ (A directed segment ܤܣ ሬሬሬሬሬԦ is the line segment ܤܣtogether with a direction given by
connecting an initial point ܣto a terminal point ܤ.)
FƌĞĞ VĞĐƚoƌ (A free vector is the equivalence class of all directed line segments (arrows) that are
equivalent to each other by translation. For example, scientists often use free vectors to describe
physical quantities that have magnitude and direction only, freely placing an arrow with the given
magnitude and direction anywhere in a diagram where it is needed. For any directed line segment in
the equivalence class defining a free vector, the directed line segment is said to be a representation
of the free vector or is said to represent the free vector.)
IĚĞŶƚŝƚLJ DĂƚƌŝdž (The ݊ ൈ ݊ identity matrix is the matrix whose entry in row ݅ and column ݅ for
ͳ ݅ ݊ is ͳ and whose entries in row ݅ and column ݆ for ͳ ݅ǡ ݆ ݊, and ݅ ് ݆ are all zero. The
identity matrix is denoted by ܫ.)
IŵĂŐŝŶĂƌLJ AdžŝƐ (See complex plane.)
IŵĂŐŝŶĂƌLJ NƵŵďĞƌ (An imaginary number is a complex number that can be expressed in the form ܾ݅
where ܾ is a real number.)
IŵĂŐŝŶĂƌLJ PĂƌƚ (See complex number.)
IŵĂŐŝŶĂƌLJ UŶŝƚ (The imaginary unit, denoted by ݅, is the number corresponding to the point ሺͲǡͳሻ in
the complex plane.)
IŶĐŝĚĞŶĐĞ DĂƚƌŝdž (The incidence matrix of a network diagram is the ݊ ൈ ݊ matrix such that the entry
in row ݅ and column ݆ is the number of edges that start at node ݅ and end at node ݆.)
IŶǀĞƌƐĞ DĂƚƌŝdž (An ݊ ൈ ݊ matrix ܣis invertible if there exists an ݊ ൈ ݊ matrix ܤso that ܤܣൌ ܣܤൌ ܫ,
where ܫis the ݊ ൈ ݊ identity matrix. The matrix ܤ, when it exists, is unique and is called the inverse
of ܣand is denoted by ିܣଵ .)
LŝŶĞĂƌ FƵŶĐƚŝoŶ (A function ݂ǣ Թ ՜ Թ is called a linear function if it is a polynomial function of
degree one, that is, a function with real number domain and range that can be put into the form
݂ሺݔሻ ൌ ݉ ݔ ܾ for real numbers ݉ and ܾ. A linear function of the form ݂ሺݔሻ ൌ ݉ ݔ ܾ is a linear
transformation only if ܾ ൌ Ͳ.)
DoĚƵůĞ 1: Complex Numbers and Transformations 9
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LŝŶĞĂƌ TƌĂŶƐĨoƌŵĂƚŝoŶ (A function ܮǣ Թ ՜ Թ for a positive integer ݊ is a linear transformation if
the following two properties hold:
à ܮሺ ܠ ܡሻ ൌ ܮሺܠሻ ܮሺܡሻ for all ܠǡ א ܡԹ , and
à ܮሺ݇ܠሻ ൌ ݇ ܮ ڄሺܠሻ for all א ܠԹ and ݇ אԹ,
where א ܠԹ means that ܠis a point in Թ .)
LŝŶĞĂƌ TƌĂŶƐĨoƌŵĂƚŝoŶ IŶĚƵĐĞĚ ďLJ DĂƚƌŝdž (Given a ʹ ൈ ʹ matrix ܣ, the linear transformation
ݔ ݔ
induced by matrix ܣis the linear transformation ܮgiven by the formula ܮቀቂݕቃቁ ൌ ڄ ܣቂݕቃ. Given a
͵ ൈ ͵ matrix ܣ, the linear transformation induced by matrix ܣis the linear transformation ܮgiven by
ݔ ݔ
the formula ܮቆቈ ቇ ൌ ڄ ܣቈݕ.)
ݕ
ݖ ݖ
DĂƚƌŝdž (An ݉ ൈ ݊ matrix is an ordered list of ݊݉ real numbers, ܽଵଵ , ܽଵଶ , ǥ, ܽଵ , ܽଶଵ , ܽଶଶ , ǥ, ܽଶ , ǥ,
ܽଵ , ܽଶ , ǥ, ܽ , organized in a rectangular array of ݉ rows and ݊ columns:
ܽଵଵ ܽଵଶ ڮ ܽଵ
ܽଶଵ ܽଶଶ ڮ ܽଶ
൦ ڭ ڭ ڰ ڭ൪. The number ܽ is called the entry in row ݅ and column ݆.)
ܽଵ ܽଶ ڮ ܽ
DĂƚƌŝdž DŝĨĨĞƌĞŶĐĞ (Let ܣbe an ݉ ൈ ݊ matrix whose entry in row ݅ and column ݆ is ܽ , and let ܤbe
an ݉ ൈ ݊ matrix whose entry in row ݅ and column ݆ is ܾ . Then, the matrix difference ܣെ ܤis the
݉ ൈ ݊ matrix whose entry in row ݅ and column ݆ is ܽ െ ܾ .)
DĂƚƌŝdž PƌoĚƵĐƚ (Let ܣbe an ݉ ൈ ݊ matrix whose entry in row ݅ and column ݆ is ܽ , and let ܤbe an
݊ ൈ matrix whose entry in row ݅ and column ݆ is ܾ . Then, the matrix product ܤܣis the ݉ ൈ
matrix whose entry in row ݅ and column ݆ is ܽଵ ܾଵ ܽଶ ܾଶ ڮ ܽ ܾ .)
DĂƚƌŝdž SĐĂůĂƌ DƵůƚŝƉůŝĐĂƚŝoŶ (Let ݇ be a real number, and let ܣbe an ݉ ൈ ݊ matrix whose entry in
row ݅ and column ݆ is ܽ . Then, the scalar product ݇ ܣ ڄis the ݉ ൈ ݊ matrix whose entry in row ݅
and column ݆ is ݇ ܽ ڄ .)
DĂƚƌŝdž SƵŵ (Let ܣbe an ݉ ൈ ݊ matrix whose entry in row ݅ and column ݆ is ܽ , and let ܤbe an
݉ ൈ ݊ matrix whose entry in row ݅ and column ݆ is ܾ . Then, the matrix sum ܣ ܤis the ݉ ൈ ݊
matrix whose entry in row ݅ and column ݆ is ܽ ܾ .)
DoĚƵůƵƐ (The modulus of a complex number ݖ, denoted ȁݖȁ, is the distance from the origin to the
point corresponding to ݖin the complex plane. If ݖൌ ܽ ܾ݅, then ȁݖȁ ൌ ξܽଶ ܾ ଶ.)
NĞƚǁoƌŬ DŝĂŐƌĂŵ (A network diagram is a graphical representation of a directed graph where the ݊
vertices are drawn as circles with each circle labeled by a number ͳ through ݊ and the directed
edges are drawn as segments or arcs with the arrow pointing from the tail vertex to the head
vertex.)
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ሬሬሬሬሬԦ , the opposite vector,
OƉƉoƐŝƚĞ VĞĐƚoƌ (For a vector ݒԦ represented by the directed line segment ܤܣ
ݒଵ
ݒଶ
denoted െݒԦ, is the vector represented by the directed line segment ሬሬሬሬሬԦ
ܣܤ. If ݒԦ ൌ ൦ ڭ൪ in Թ , then
ݒ
െݒଵ
െݒଶ
െݒԦ ൌ ൦ ڭ൪.)
െݒ
PoůĂƌ Foƌŵ oĨ a ComƉůĞdž NƵmďĞƌ (The polar form of a complex number ݖis ݎሺ ሺߠሻ ݅ ሺߠሻሻ
where ݎൌ ȁݖȁ and ߠ ൌ ሺݖሻ.)
ݒଵ
ݒଶ
PoƐŝƚŝoŶ VĞĐƚoƌ (For a point ܲሺݒଵ ǡ ݒଶ ǡ ǥ ǡ ݒ ሻ in Թ , the position vector ݒԦ, denoted by ൦ ڭ൪ or
ݒ
ሬሬሬሬሬԦ
ݒۃଵ ǡ ݒଶ ǡ ǥ ǡ ݒ ۄ, is a free vector ݒԦ that is represented by the directed line segment ܱܲ from the origin
ܱሺͲǡͲǡͲǡ ǥ ǡͲሻ to the point ܲ. The real number ݒ is called the ݅ th component of the vector ݒԦ.)
RĞaů CooƌĚŝŶaƚĞ SƉaĐĞ (For a positive integer ݊, the ݊-dimensional real coordinate space, denoted
Թ , is the set of all ݊-tuple of real numbers equipped with a distance function ݀ that satisfies
݀ሾሺݔଵ ǡ ݔଶ ǡ ǥ ǡ ݔ ሻǡ ሺݕଵ ǡ ݕଶ ǡ ǥ ǡ ݕ ሻሿ ൌ ඥሺݕଵ െ ݔଵ ሻଶ ሺݕଶ െ ݔଵ ሻଶ ڮ ሺݕ െ ݔ ሻଶ
for any two points in the space. One-dimensional real coordinate space is called a number line, and
the two-dimensional real coordinate space is called the Cartesian plane.)
RĞĐƚaŶŐƵůaƌ Foƌm oĨ a ComƉůĞdž NƵmďĞƌ (The rectangular form of a complex number ݖis ܽ ܾ݅
where ݖcorresponds to the point ሺܽǡ ܾሻ in the complex plane and ݅ is the imaginary unit. The
number ܽ is called the real part of ܽ ܾ݅, and the number ܾ is called the imaginary part of ܽ ܾ݅.)
TƌaŶƐůaƚŝoŶ ďLJ a VĞĐƚoƌ ŝŶ RĞaů CooƌĚŝŶaƚĞ SƉaĐĞ (A translation by a vector ݒԦ in Թ is the translation
ݒଵ
ݒ ଶ
transformation ܶ௩ሬԦ ǣ Թ ՜ Թ given by the map that takes ݔԦ հ ݔԦ ݒԦ for all ݔԦ in Թ . If ݒԦ ൌ ൦ ڭ൪ in
ݒ
ݔଵ ݔଵ ݒଵ
ݔ ଶ ݔ ݒ
Թ , then ܶ௩ሬԦ ൮൦ ڭ൪൲ ൌ ൦ ଶ ڭଶ ൪ for all ݔԦ in Թ .)
ݔ ݔ ݒ
DoĚƵůĞ 1: Complex Numbers and Transformations 11
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PRECALCULUS AND ADVANCED TOPICS
ሬሬԦ in Թ , the sum ݒԦ ݓ
VĞĐƚoƌ AĚĚŝƚŝoŶ (For vectors ݒԦ and ݓ ሬሬԦ is the vector whose ݅ th component is the
ݒଵ ݓଵ
ݒଶ ݓଶ
ሬሬԦ for ͳ ݅ ݊. If ݒԦ ൌ ൦ ڭ൪ and ݓ
sum of the ݅ th components of ݒԦ and ݓ ሬሬԦ ൌ ൦ ڭ൪ in Թ , then
ݒ ݓ
ݒଵ ݓଵ
ݒଶ ݓଶ
ሬሬԦ ൌ ൦
ݒԦ ݓ ڭ ൪.)
ݒ ݓ
VĞĐƚoƌ DaŐŶŝƚƵĚĞ (The magnitude or length of a vector ݒԦ, denoted ȁݒԦȁ or ԡݒԦԡ, is the length of any
ݒଵ
ݒଶ
directed line segment that represents the vector. If ݒԦ ൌ ൦ ڭ൪ in Թ , then ȁݒԦȁ ൌ ඥݒଵଶ ݒଶଶ ڮ ݒଶ ,
ݒ
which is the distance from the origin to the associated point ܲሺݒଵ ǡ ݒଶ ǡ ǥ ǡ ݒ ሻ.)
VĞĐƚoƌ RĞƉƌĞƐĞŶƚaƚŝoŶ oĨ a ComƉůĞdž NƵmďĞƌ (The vector representation of a complex number ݖis
the position vector ݖԦ associated to the point ݖin the complex plane. If ݖൌ ܽ ܾ݅ for two real
ܽ
numbers ܽ and ܾ, then ݖԦ ൌ ቂ ቃ.)
ܾ
VĞĐƚoƌ SĐaůaƌ DƵůƚŝƉůŝĐaƚŝon (For a vector ݒԦ in Թ and a real number ݇, the scalar product ݇ ݒ ڄԦ is
the vector whose ݅ th component is the product of ݇ and the ݅ th component of ݒԦ for ͳ ݅ ݊. If ݇ is
ݒଵ ݇ݒଵ
ݒଶ ݇ݒ
a real number and ݒԦ ൌ ൦ ڭ൪ in Թ , then ݇ ݒ ڄԦ ൌ ൦ ଶ ൪.)
ڭ
ݒ ݇ݒ
VĞĐƚoƌ SƵďƚƌaĐƚŝon (For vectors ݒԦ and ݓ ሬሬԦ, the difference ݒԦ െ ݓ ሬሬԦ is the sum of ݒԦ and the opposite of
ݒଵ ݓଵ ݒଵ െ ݓଵ
ݒଶ ݓଶ ݒଶ െ ݓଶ
ݓ
ሬሬԦ; that is, ݒԦ െ ݓ ሬሬԦሻ. If ݒԦ ൌ ൦ ڭ൪ and ݓ
ሬሬԦ ൌ ݒԦ ሺെݓ ሬሬԦ ൌ ൦ ڭ൪ in Թ , then ݒԦ െ ݓሬሬԦ ൌ ൦ ڭ ൪.)
ݒ ݓ ݒ െ ݓ
Ğƌo Daƚƌŝdž (The ݉ ൈ ݊ zero matrix is the ݉ ൈ ݊ matrix in which all entries are equal to zero. For
Ͳ Ͳ Ͳ
Ͳ Ͳ
example, the ʹ ൈ ʹ zero matrix is ቂ ቃ, and the ͵ ൈ ͵ zero matrix is Ͳ Ͳ Ͳ൩.)
Ͳ Ͳ
Ͳ Ͳ Ͳ
Ğƌo VĞĐƚoƌ (The zero vector in Թ is the vector in which each component is equal to zero. For
Ͳ
Ͳ
example, the zero vector in Թଶ is ቂ ቃ, and the zero vector in Թଷ is Ͳ൩.)
Ͳ
Ͳ
12 DoĚƵůĞ 1: Complex Numbers and Transformations
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A STORY OF FUNCTIONS DoĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
FamŝůŝaƌTĞƌmƐ anĚ SLJmďoůƐϮ
Dilation
Rectangular Form
Rotation
Translation
SƵŐŐĞƐƚĞĚ TooůƐ anĚ RĞƉƌĞƐĞnƚaƚŝonƐ
Geometer’s Sketchpad software
Graphing calculator
Wolfram Alpha software
PƌĞƉaƌŝnŐ ƚo TĞaĐŚ a DoĚƵůĞ
Preparation of lessons will be more effective and efficient if there has been an adequate analysis of the
module first. Each module in A Story of Functions can be compared to a chapter in a book. How is the
module moving the plot, the mathematics, forward? What new learning is taking place? How are the topics
and objectives building on one another? The following is a suggested process for preparing to teach a
module.
Step 1: Get a preview of the plot.
A: Read the Table of Contents. At a high level, what is the plot of the module? How does the story
develop across the topics?
B: Preview the module’s Exit Tickets to see the trajectory of the module’s mathematics and the nature
of the work students are expected to be able to do.
ϮThese are terms and symbols students have seen previously.
DoĚƵůĞ 1: Complex Numbers and Transformations 13
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A STORY OF FUNCTIONS DoĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
Note: When studying a PDF file, enter “Exit Ticket” into the search feature to navigate from one Exit
Ticket to the next.
Step 2: Dig into the details.
A: Dig into a careful reading of the Module Overview. While reading the narrative, liberally reference
the lessons and Topic Overviews to clarify the meaning of the text—the lessons demonstrate the
strategies, show how to use the models, clarify vocabulary, and build understanding of concepts.
B: Having thoroughly investigated the Module Overview, read through the Student Outcomes of each
lesson (in order) to further discern the plot of the module. How do the topics flow and tell a
coherent story? How do the outcomes move students to new understandings?
Step 3: Summarize the story.
Complete the Mid- and End-of-Module Assessments. Use the strategies and models presented in the
module to explain the thinking involved. Again, liberally reference the lessons to anticipate how students
who are learning with the curriculum might respond.
PƌĞƉaƌŝnŐ ƚo TĞaĐŚ a LĞƐƐon
A three-step process is suggested to prepare a lesson. It is understood that at times teachers may need to
make adjustments (customizations) to lessons to fit the time constraints and unique needs of their students.
The recommended planning process is outlined below. Note: The ladder of Step 2 is a metaphor for the
teaching sequence. The sequence can be seen not only at the macro level in the role that this lesson plays in
the overall story, but also at the lesson level, where each rung in the ladder represents the next step in
understanding or the next skill needed to reach the objective. To reach the objective, or the top of the
ladder, all students must be able to access the first rung and each successive rung.
14 DoĚƵůĞ 1: Complex Numbers and Transformations
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A STORY OF FUNCTIONS DoĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
Step 1: Discern the plot.
A: Briefly review the module’s Table of Contents, recalling the overall story of the module and analyzing
the role of this lesson in the module.
B: Read the Topic Overview related to the lesson, and then review the Student Outcome(s) and Exit
Ticket of each lesson in the topic.
C: Review the assessment following the topic, keeping in mind that assessments can be found midway
through the module and at the end of the module.
Step 2: Find the ladder.
A: Work through the lesson, answering and completing
each question, example, exercise, and challenge.
B: Analyze and write notes on the new complexities or
new concepts introduced with each question or
problem posed; these notes on the sequence of new
complexities and concepts are the rungs of the ladder.
C: Anticipate where students might struggle, and write a
note about the potential cause of the struggle.
D: Answer the Closing questions, always anticipating how
students will respond.
Step 3: Hone the lesson.
Lessons may need to be customized if the class period is not long enough to do all of what is presented
and/or if students lack prerequisite skills and understanding to move through the entire lesson in the
time allotted. A suggestion for customizing the lesson is to first decide upon and designate each
question, example, exercise, or challenge as either “Must Do” or “Could Do.”
A: Select “Must Do” dialogue, questions, and problems that meet the Student Outcome(s) while still
providing a coherent experience for students; reference the ladder. The expectation should be that
the majority of the class will be able to complete the “Must Do” portions of the lesson within the
allocated time. While choosing the “Must Do” portions of the lesson, keep in mind the need for a
balance of dialogue and conceptual questioning, application problems, and abstract problems, and a
balance between students using pictorial/graphical representations and abstract representations.
Highlight dialogue to be included in the delivery of instruction so that students have a chance to
articulate and consolidate understanding as they move through the lesson.
B: “Must Do” portions might also include remedial work as necessary for the whole class, a small group,
or individual students. Depending on the anticipated difficulties, the remedial work might take on
different forms as suggested in the chart below.
DoĚƵůĞ 1: Complex Numbers and Transformations 15
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A STORY OF FUNCTIONS DoĚƵůĞ OǀĞƌǀŝĞǁ D1
PRECALCULUS AND ADVANCED TOPICS
AnƚŝĐŝƉaƚĞĚ DŝĨĨŝĐƵůƚLJ ͞DƵƐƚ Do͟ RĞmĞĚŝaů PƌoďůĞm SƵŐŐĞƐƚŝon
The first problem of the lesson is Write a short sequence of problems on the board that
too challenging. provides a ladder to Problem 1. Direct students to
complete those first problems to empower them to begin
the lesson.
There is too big of a jump in Provide a problem or set of problems that bridge student
complexity between two problems. understanding from one problem to the next.
Students lack fluency or Before beginning the lesson, do a quick, engaging fluency
foundational skills necessary for exercise 3. Before beginning any fluency activity for the first
the lesson. time, assess that students have conceptual understanding
of the problems in the set and that they are poised for
success with the easiest problem in the set.
More work is needed at the Provide manipulatives or the opportunity to draw solution
concrete or pictorial level. strategies.
More work is needed at the Add a set of abstract problems to be completed toward the
abstract level. end of the lesson.
C: “Could Do” problems are for students who work with greater fluency and understanding and can,
therefore, complete more work within a given time frame.
D: At times, a particularly complex problem might be designated as a “Challenge!” problem to provide
to advanced students. Consider creating the opportunity for students to share their “Challenge!”
solutions with the class at a weekly session or on video.
E: If the lesson is customized, be sure to carefully select Closing questions that reflect such decisions,
and adjust the Exit Ticket if necessary.
AƐƐĞƐƐmĞnƚ SƵmmaƌLJ
AƐƐĞƐƐmĞnƚ TLJƉĞ AĚmŝnŝƐƚĞƌĞĚ Format
Mid-Module
After Topic B Constructed response with rubric
Assessment Task
End-of-Module
After Topic C Constructed response with rubric
Assessment Task
3Look for fluency suggestions at www.eureka-math.org.
16 DoĚƵůĞ 1: Complex Numbers and Transformations
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^dKZzK&&hEd/KE^
DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ
PRECALCULUS AND ADVANCED TOPICS ͻDKh> 1
Topic A
YƵĞƐƚŝŽŶŽĨ>ŝŶĞĂƌŝƚLJ
&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Find the conjugate of a complex number; use conjugates to find moduli and
quotients of complex numbers.
(+) Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
/ŶƐƚƌƵĐƚŝŽŶĂů ĂLJƐ͗ 8
>ĞƐƐŽŶƐϭ–2: Wishful Thinking—Does Linearity Hold? (E, E) 1
>ĞƐƐŽŶϯ: Which Real Number Functions Define a Linear Transformation? (S)
>ĞƐƐŽŶƐ ϰ–5: An Appearance of Complex Numbers (P, P)
>ĞƐƐŽŶϲ͗ Complex Numbers as Vectors (P)
>ĞƐƐŽŶƐϳ–8: Complex Number Division (P, P)
Linear transformations are a unifying theme of Module 1 Topic A. In Lesson 1, students are introduced to the
term linear transformation and its definition. A function is a linear transformation if it satisfies the conditions
݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ and ݂݇ሺݔሻ ൌ ݂ሺ݇ݔሻ. Students contrast this to their previous understanding of a
linear transformation, which was likely a function whose graph is a straight line. This idea of linearity is
revisited as students study complex numbers and their transformations in Topic B and matrices in Topic C.
Lesson 1 begins as students look at common mistakes made in algebra and asks questions such as “For which
numbers ܽ and ܾ does ሺܽ ܾሻଶ ൌ ܽଶ ܾ ଶ happen to hold?” Students discover that these statements are
usually false by substituting real number values for the variables and then exploring values that make the
statements true. Lesson 2 continues this exploration asking, “Are there numbers ܽ and ܾ for which
ଵ ଵ ଵ
ൌ ?” and so on. This exercise has only complex solutions, which launches a study of complex
ା
numbers. Lesson 3 concludes this study of misconceptions by defining a linear function (a function whose
graph is a line) and explaining the difference between a linear function and a linear transformation. The
concept of a linear transformation is developed in the first three lessons and is revisited throughout the
module. Linear transformations are important because they help students link complex numbers and their
1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
dŽƉŝĐ: A Question of Linearity 17
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A STORY OF FUNCTIONS dŽƉŝĐA Dϭ
PRECALCULUS AND ADVANCED TOPICS
transformations to matrices as this module progresses. Linear transformations are also essential in college
mathematics as they are a foundational concept in linear algebra. Lessons 4 and 5 begin the study of complex
numbers defining ݅ geometrically by rotating the number line ͻͲι and thus giving a “number” ݅ with the
property ݅ଶൌെͳ. Students then add, subtract, and multiply complex numbers. Lesson 6 explores complex
numbers as vectors. Lessons 7 and 8 conclude Topic A with the study of quotients of complex numbers and
the use of conjugates to find moduli and quotients. Linearity is revisited when students classify real and
complex functions that satisfy linearity conditions. (A function ܮis linear if and only if there is a real or
complex value ݓsuch that ܮሺݖሻൌ ݖݓfor all real or complex ݖ.) Complex number multiplication is again
emphasized in Lesson 8. This topic focuses on ĐƌŝƚŝƋƵŝŶŐĂŶĚũƵƐƚŝĨLJŝŶŐƌĞĂƐŽŶŝŶŐas students study common
mistakes that algebra students make and determine the validity of the statements.
18 dŽƉŝĐ: A Question of Linearity
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A STORY OF FUNCTIONS Lesson 1 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 1: Wishful Thinking—Does Linearity Hold?
Student Outcomes
Students learn when ideal linearity properties do and do not hold for classes of functions studied in previous
years.
Students develop familiarity with linearity conditions.
Lesson Notes
This is a two-day lesson that introduces a new definition of a linear transformation and looks at common mistakes that
students make when assuming that all linear functions meet the requirements for this new definition. A linear
transformation is not equivalent to a linear function, which is a function whose graph is a line and can be written as
ݕൌ ݉ ݔ ܾ. In this sequence of lessons, a linear transformation is defined as it is in linear algebra courses, which is that
a function is linear if it satisfies two conditions: ݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ and ݂ሺ݇ݔሻ ൌ ݂݇ሺݔሻ. This definition leads to
surprising results when students study the function ݂ሺݔሻ ൌ ͵ ݔ ͳ. Students apply this new definition of linear
transformation to classes of functions learned in previous years and explore why the conditions for linearity sometimes
produce false statements. Students then solve to find specific solutions when the conditions for linearity produce true
statements, giving the appearance that a linear function is a linear transformation when it is not. In Lesson 1, students
explore polynomials and radical equations. Lesson 2 extends this exploration to trigonometric, rational, and logarithmic
functions. Lessons 1 and 2 focus on linearity for real-numbered inputs but lead to the discovery of complex solutions
and launch the study of complex numbers. This study includes operations on complex numbers as well as the use of
conjugates to find moduli and quotients.
Classwork
Exploratory Challenge (13 minutes)
In this Exploratory Challenge, students work individually while discussing the steps as a class. Students complete the
exercises in pairs with the class coming together at the end to present their findings and to watch a video.
Wouldn’t it be great if functions were sensible and behaved the way we expected them to do?
Let ݂ሺݔሻ ൌ ʹ ݔand ݃ሺݔሻ ൌ ͵ ݔ ͳ.
Write down three facts that you know about ݂ሺݔሻ and ݃ሺݔሻ.
à Answers will vary. Both graphs are straight lines. ݂ሺݔሻ has a ݕ-intercept of Ͳ. ݃ሺݔሻ has a ݕ-intercept
of ͳ. The slope of ݂ሺݔሻ is ʹ. The slope of ݃ሺݔሻ is ͵.
Which of these functions is linear?
à Students will probably say both because they are applying a prior definition of a linear function:
ݕൌ ݉ ݔ ܾ.
Introduce the following definition: A function is a linear transformation if ݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ and
݂ሺ݇ݔሻ ൌ ݂݇ሺݔሻ.
Lesson 1: Wishful Thinking—Does Linearity Hold? 19
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A STORY OF FUNCTIONS Lesson 1 M1
PRECALCULUS AND ADVANCED TOPICS
Based on this definition, which function is a linear transformation? Explain how you know.
à ݂ሺݔሻ ൌ ʹ ݔis a linear transformation because ʹሺ ݔ ݕሻ ൌ ʹ ݔ ʹ ݕand ʹሺ݇ݔሻ ൌ ݇ሺʹݔሻ.
à ݃ሺݔሻ ൌ ͵ ݔ ͳ is not a linear transformation because ͵ሺ ݔ ݕሻ ͳ ് ሺ͵ ݔ ͳሻ ሺ͵ ݕ ͳሻ and
͵ሺ݇ݔሻ ͳ ് ݇ሺ͵ ݔ ͳሻ.
Is ݄ሺݔሻ ൌ ʹ ݔെ ͵ a linear transformation? Explain.
à ݄ሺݔሻ ൌ ʹ ݔെ ͵ is not a linear transformation because ʹሺ ݔ ݕሻ െ ͵ ് ሺʹ ݔെ ͵ሻ ሺʹ ݕെ ͵ሻ and
ʹሺ݇ݔሻ െ ͵ ് ݇ሺʹ ݔെ ͵ሻ.
ͳ
Is ሺݔሻ ൌ ݔa linear transformation? Explain.
ʹ
ͳ ଵ ଵ ଵ ଵ ଵ
à ሺݔሻ ൌ ݔis a linear transformation because
ʹ
ሺ ݔ ݕሻ ൌ ݔ ݕand ሺ݇ݔሻ ൌ ݇ ቀ ݔቁ.
ଶ ଶ ଶ ଶ ଶ
ଶ
Let ݃ሺݔሻ ൌ ݔ.
Is ݃ሺݔሻ a linear transformation?
à No. ሺ ݔ ݕሻଶ ് ݔଶ ݕଶ , and ሺܽݔሻଶ ് ܽሺݔሻଶ .
A common mistake made by many math students is saying Scaffolding:
ሺܽ ܾሻଶ ൌ ܽଶ ܾ ଶ . How many of you have made this mistake before? Remind students that ሺܽ ܾሻଶ
ଶ ଶ ଶ
Does ሺܽ ܾሻ ൌ ܽ ܾ ? Justify your claim. means ሺܽ ܾሻሺܽ ܾሻ.
Substitute some values of ܽ and ܾ into this equation to show that this Have students complete the
statement is not generally true. following chart for different values
à Answers will vary, but students could choose ܽ ൌ ͳ and of ܽ and ܾ and the expressions.
ܾ ൌ ͳ. In this case, ሺͳ ͳሻଶ ൌ ͳଶ ͳଶ leads to Ͷ ൌ ʹ, which ܽ ܾ ሺܽ ܾሻଶ ܽଶ ܾଶ ܽଶ ܾ ଶ
we know is not true. There are many other choices. ͳ ʹ ͻ ͳ Ͷ ͷ
Did anyone find values of ܽ and ܾ that made this statement true? ʹ ͵ ʹͷ Ͷ ͻ ͳ͵
à Answers will vary but could include ܽ ൌ Ͳ, ܾ ൌ Ͳ or ܽ ൌ ͳ,
ܾ ൌ Ͳ or ܽ ൌ Ͳ, ܾ ൌ ͳ.
We can find all values of ܽ and ܾ for which this statement is true by solving for one of the variables. I want half
the class to solve this equation for ܽ and the other half to solve for ܾ.
à Expanding the left side and then combining like terms gives
ܽଶ ʹܾܽ ܾ ଶ ൌ ܽଶ ܾ ଶ ʹܾܽ ൌ Ͳ. This leads to ܽ ൌ Ͳ if students are solving for ܽ and ܾ ൌ Ͳ if
students are solving for ܾ.
We have solutions for two different variables. Can you explain this to your neighbor?
à If ܽ ൌ Ͳ and/or ܾ ൌ Ͳ, the statement ሺܽ ܾሻଶ ൌ ܽଶ ܾ ଶ is true.
Take a moment and discuss with your neighbor what we have just shown. What statement is true for all real
values of ܽ and ܾ?
à ሺܽ ܾሻଶ ൌ ܽଶ ܾ ଶ is true for only certain values of ܽ and ܾ, namely, if either or both variables equal
Ͳ. The statement that is true for all real numbers is ሺܽ ܾሻଶ ൌ ܽଶ ʹܾܽ ܾ ଶ .
A function is a linear transformation when the following are true: ݂ሺ݇ݔሻ ൌ ݂݇ሺݔሻ and
݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ. We call this function a linear transformation.
Repeat what I have just said to your neighbor.
à Students repeat.
20 Lesson 1: Wishful Thinking—Does Linearity Hold?
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Look at the functions ݂ሺݔሻ ൌ ʹ ݔand ݃ሺݔሻ ൌ ݔଶ listed above. Which is a linear transformation? Explain.
à ݂ሺݔሻ ൌ ʹ ݔis a linear transformation because ݂ሺܽݔሻ ൌ ݂ܽሺݔሻ and ݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ.
݃ሺݔሻ ൌ ݔଶ is not a linear transformation because ݃ሺܽݔሻ ് ܽ݃ሺݔሻ and ݃ሺ ݔ ݕሻ ് ݃ሺݔሻ ݃ሺݕሻ.
Linear transformations are introduced in Lessons 1 and 2. This leads to the discussion in Lesson 3 on when functions are
linear transformations. In Lesson 3, students discover that a function whose graph is a line may or may not be a linear
transformation.
Exercises (10 minutes)
In the exercises below, instruct students to work in pairs and to go through the Scaffolding:
same steps that they went through in the Exploratory Challenge. Call the class
For advanced learners, assign
back together, and have groups present their results. All groups can be assigned
Exercises 1 and 2 with no leading
both examples, or half the class can be assigned Exercise 1 and the other half
question.
Exercise 2. Exercise 2 is slightly more difficult than Exercise 1.
Monitor group work, and target
Exercises
some groups with more specific
questions to help them with the
Look at these common mistakes that students make, and answer the questions that
algebra needed. For example,
follow.
remind them that
1. If ࢌሺ࢞ሻ ൌ ξ࢞, does ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ, when ࢇ and ࢈ are not negative? ଶ
a. Can we find a counterexample to refute the claim that ൫ξܽ ξܾ൯ ൌ ൫ξܽ ξܾ൯൫ξܽ ξܾ൯.
ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ for all nonnegative values of ࢇ and ࢈? Also, remind them how to multiply
Answers will vary, but students could choose ࢇ ൌ and ࢈ ൌ . In this binomials.
case, ξ ൌ ξ ξ, or ξ ൌ , which we know is not true. There are
many other choices.
b. Find some nonnegative values for ࢇ and ࢈ for which the statement, by coincidence, happens to be true.
Answers will vary but could include ࢇ ൌ , ࢈ ൌ or ࢇ ൌ , ࢈ ൌ or ࢇ ൌ , ࢈ ൌ .
c. Find all values of ࢇ and ࢈ for which the statement is true. Explain your work and the results.
ξࢇ ࢈ ൌ ξࢇ ξ࢈
ሺξࢇ ࢈ሻ ൌ ൫ξࢇ ξ࢈൯
ࢇ ࢈ ൌ ࢇ ξࢇ࢈ ࢈
ξࢇ࢈ ൌ
ࢇ࢈ ൌ ,
which leads to ࢇ ൌ if students are solving for ࢇ and ࢈ ൌ if students are solving for ࢈.
Anytime ࢇ ൌ and/or ࢈ ൌ , then ξࢇ ࢈ ൌ ξࢇ ξ࢈, and the equation is true.
d. Why was it necessary for us to consider only nonnegative values of ࢇ and ࢈?
If either variable is negative, then we would be taking the square root of a negative number, which is not a
real number, and we are only addressing real-numbered inputs and outputs here.
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e. Does ࢌሺ࢞ሻ ൌ ξ࢞ display ideal linear properties? Explain.
No, because ξࢇ ࢈ ് ξࢇ ξ࢈ for all real values of the variables.
2. If ࢌሺ࢞ሻ ൌ ࢞ , does ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ?
a. Substitute in some values of ࢇ and ࢈ to show this statement is not true in general.
Answers will vary, but students could choose ࢇ ൌ and ࢈ ൌ . In this case, ሺ ሻ ൌ or ૡ ൌ ,
which we know is not true. There are many other choices.
b. Find some values for ࢇ and ࢈ for which the statement, by coincidence, happens to work.
Answers will vary but could include ࢇ ൌ , ࢈ ൌ or ࢇ ൌ , ࢈ ൌ or ࢇ ൌ , ࢈ ൌ .
c. Find all values of ࢇ and ࢈ for which the statement is true. Explain your work and the results.
ሺࢇ ࢈ሻ ൌ ࢇ ࢈
ࢇ ࢇ ࢈ ࢇ࢈ ࢈ ൌ ࢇ ࢈
ࢇ ࢈ ࢇ࢈ ൌ
ࢇ࢈ሺࢇ ࢈ሻ ൌ ,
which leads to ࢇ ൌ , ࢈ ൌ , and ࢇ ൌ െ࢈.
Anytime ࢇ ൌ and/or ࢈ ൌ or ࢇ ൌ െ࢈, then ሺࢇ ࢈ሻ ൌ ࢇ ࢈, and the equation is true.
d. Is this true for all positive and negative values of ࢇ and ࢈? Explain and prove by choosing positive and
negative values for the variables.
Yes, since ൌ െ࢈ , if ࢇ is positive, the equation would be true if ࢈ was negative. Likewise, if ࢇ is negative, the
equation would be true if ࢈ was positive. Answers will vary. If ࢇ ൌ and ࢈ ൌ െ,
൫ ሺെሻ൯ ൌ ሺሻ ሺെሻ, meaning ൌ ૡ ሺെૡሻ or ൌ . If ࢇ ൌ െ and ࢈ ൌ ,
൫ሺെሻ ൯ ൌ ሺെሻ ሺሻ, meaning ൌ ሺെૡሻ ૡ, or ൌ . Therefore, this statement is true for all
positive and negative values of ࢇ and ࢈.
e. Does ࢌሺ࢞ሻ ൌ ࢞ display ideal linear properties? Explain.
No, because ሺࢇ ࢈ሻ ് ࢇ ࢈ for all real values of the variables.
Extension Discussion (14 minutes, optional)
As a class, watch this video (7 minutes) that shows another way to justify Exercise 1
(http://www.jamestanton.com/?p=677). Discuss what the groups discovered in the exercises and what was shown in
the video. If time allows, let the groups present their findings and discuss similarities and differences.
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Closing (3 minutes)
Ask students to perform a 30-second Quick Write explaining what was learned today using these questions as a guide.
When does ሺܽ ܾሻଶ ൌ ܽଶ ܾ ଶ? How do you know?
à When ܽ ൌ Ͳ and/or ܾ ൌ Ͳ
When does ξܽ ܾ ൌ ξܽ ξܾ? How do you know?
à When ܽ ൌ Ͳ and/or ܾ ൌ Ͳ
Are ܽ ൌ Ͳ and/or ܾ ൌ Ͳ always the values when functions display ideal linear properties?
à No. It depends on the function. Sometimes these values work, and other times they do not. Sometimes
there are additional values that work such as with the function ݂ሺݔሻ ൌ ݔଷ , when ܽ ൌ െܾ also works.
When does a function display ideal linear properties?
à When ݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ and ݂ሺ݇ݔሻ ൌ ݂݇ሺݔሻ
Exit Ticket (5 minutes)
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Name Date
Lesson 1: Wishful Thinking—Does Linearity Hold?
Exit Ticket
1. Xavier says that ሺܽ ܾሻଶ ് ܽଶ ܾ ଶ but that ሺܽ ܾሻଷ ൌ ܽଷ ܾ ଷ . He says that he can prove it by using the values
ܽ ൌ ʹ and ܾ ൌ െʹ. Shaundra says that both ሺܽ ܾሻଶ ൌ ܽଶ ܾ ଶ and ሺܽ ܾሻଷ ൌ ܽଷ ܾ ଷ are true and that she can
prove it by using the values of ܽ ൌ and ܾ ൌ Ͳ and also ܽ ൌ Ͳ and ܾ ൌ ͵. Who is correct? Explain.
2. Does ݂ሺݔሻ ൌ ͵ ݔ ͳ display ideal linear properties? Explain.
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Exit Ticket Sample Solutions
1. Xavier says that ሺࢇ ࢈ሻ ് ࢇ ࢈ but that ሺࢇ ࢈ሻ ൌ ࢇ ࢈ . He says that he can prove it by using the values
ࢇ ൌ and ࢈ ൌ െ. Shaundra says that both ሺࢇ ࢈ሻ ൌ ࢇ ࢈ and ሺࢇ ࢈ሻ ൌ ࢇ ࢈ are true and that she can
prove it by using the values of ࢇ ൌ ૠ and ࢈ ൌ and also ࢇ ൌ and ࢈ ൌ . Explain.
Neither is correct. Both have chosen values that just happen to work in one or both of the equations. In the first
equation, anytime ࢇ ൌ and/or ࢈ ൌ , the statement is true. In the second equation, anytime ࢇ ൌ and/or ࢈ ൌ
and also when ࢇ ൌ െ࢈, the statement is true. If they tried other values such as ࢇ ൌ and ࢈ ൌ , neither statement
would be true.
2. Does ࢌሺ࢞ሻ ൌ ࢞ display ideal linear properties? Explain.
No. ࢌሺࢇ࢞ሻ ൌ ࢇ࢞ , but ࢇࢌሺ࢞ሻ ൌ ࢇ࢞ ࢇ These are not equivalent.
Also, ࢌሺ࢞ ࢟ሻ ൌ ሺ࢞ ࢟ሻ ൌ ࢞ ࢟ , but ࢌሺ࢞ሻ ࢌሺ࢟ሻ ൌ ࢞ ࢟ ൌ ࢞ ࢟ .
They are not equivalent, so the function does not display ideal linear properties.
Problem Set Sample Solutions
Assign students some or all of the functions to investigate. All students should attempt Problem 4 to set up the next
lesson. It is hoped that students may give some examples that are studied in Lesson 2.
Study the statements given in Problems 1–3. Prove that each statement is false, and then find all values of ࢇ and ࢈ for
which the statement is true. Explain your work and the results.
1. If ࢌሺ࢞ሻ ൌ ࢞ , does ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ?
Answers that prove the statement false will vary but could include ࢇ ൌ and ࢈ ൌ െ.
This statement is true when ࢇ ൌ and/or ࢈ ൌ .
2. If ࢌሺ࢞ሻ ൌ ࢞ , does ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ?
Answers that prove the statement false will vary but could include ࢇ ൌ and ࢈ ൌ .
This statement is true when ࢇ ൌ and/or ࢈ ൌ and when ࢇ ൌ െ࢈.
3. If ࢌሺ࢞ሻ ൌ ξ࢞, does ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ?
Answers that prove the statement false will vary but could include ࢇ ൌ െ and ࢈ ൌ .
This statement is true when ࢇ ൌ and/or ࢈ ൌ .
4. Think back to some mistakes that you have made in the past simplifying or expanding functions. Write the
statement that you assumed was correct that was not, and find numbers that prove your assumption was false.
Answers will vary but could include ܖܑܛሺ࢞ ࢟ሻ ൌ ܖܑܛሺ࢞ሻ ܖܑܛሺ࢟ሻ, which is false when ࢞ and ࢟ equal ι,
ܗܔሺࢇሻ ൌ ԝܗܔሺࢇሻ, which is false for ࢇ ൌ .
ࢇା࢈ ൌ ࢇ ࢈ , which is false for ࢇ,࢈ ൌ , ൌ , which is false for ࢇ, ࢈ ൌ .
ࢇା࢈ ࢇ ࢈
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A STORY OF FUNCTIONS Lesson 2 M1
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Lesson 2: Wishful Thinking—Does Linearity Hold?
Student Outcomes
Students learn when ideal linearity properties do and do not hold for classes of functions studied in previous
years.
Students develop familiarity with linearity conditions.
Lesson Notes
This is the second day of a two-day lesson that looks at common mistakes that students make, all based on assuming
linearity holds for all functions. In Lesson 1, students were introduced to a new definition of linearity. ݂ሺݔሻ is a linear
transformation if ݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ and ݂ሺ݇ݔሻ ൌ ݂݇ሺݔሻ. Students continue to explore linearity by looking at
common student mistakes. In Lesson 1, students explored polynomials and radical equations. Lesson 2 extends this
exploration to trigonometric, rational, and logarithmic functions. The last exercise in this lesson, Exercise 4, has no real
solutions, leading to the discovery of complex solutions, and this launches the study of complex numbers. This study
includes operations on complex numbers as well as using the conjugates to find moduli and quotients.
Classwork
In this Exploratory Challenge, students work individually while discussing the steps as a class. The exercises are
completed in pairs with the class coming together at the end to present their findings and to watch a video.
Opening Exercise (8 minutes)
In the last problem of the Problem Set from Lesson 1, students were asked to use what they had learned in Lesson 1 and
then to think back to some mistakes that they had made in the past simplifying or expanding functions and to show that
the mistakes were based on false assumptions. In this Opening Exercise, students give examples of some of their
misconceptions. Have students put the examples on the board. Some of these are studied directly in Lesson 2, and
others can be assigned as part of classwork, homework, or as extensions.
In the last problem of the Problem Set from Lesson 1, you were asked to think back to some mistakes that you
had made in the past simplifying or expanding functions. Show me some examples that you wrote down, and I
will ask some of you to put your work on the board.
à Answers will vary but could include mistakes such as ሺ ݔ ݕሻ ൌ ሺݔሻ ሺݕሻ,
ଵ ଵ ଵ
ሺʹܽሻ ൌ ʹԝሺܽሻ, ͳͲା ൌ ͳͲ ͳͲ , ൌ , and many others.
ା
Note: Emphasize that these examples are errors, not true mathematical statements.
Pick a couple of the simpler examples of mistakes that are not covered in class, and talk about those, going through the
steps of Lesson 1. For each example of a mistake, have students verify with numbers that the equation is not true for all
real numbers and then find a solution that is true for all real numbers. Indicate to students the statements that are to be
reviewed in class. List other statements that may be reviewed later.
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Exploratory Challenge (14 minutes)
In Lesson 1, we discovered that not all functions are linear transformations.
Scaffolding:
Today, we will study some different functions.
Students may need a reminder
Let’s start by looking at a trigonometric function. Is ݂ሺݔሻ ൌ ሺݔሻ a linear of how to convert between
transformation? Explain why or why not. radians and degrees and critical
à No, ሺ ݔ ݕሻ ് ሺݔሻ ሺݕሻ and ሺܽݔሻ ് ܽ ሺݔሻ. trigonometric function values.
Does ሺ ݔ ݕሻ ൌ ሺݔሻ ሺݕሻ for all real values of ݔand ?ݕ Create a chart for students to
à Answers may vary. complete that lists degrees,
radian measure, ሺݔሻ, and
Substitute some values of ݔand ݕinto this equation.
ሺݔሻ. A copy of a table
Some students may use degrees and others radians. Allow students to choose. follows this lesson in the
Alternatively, assign half of students to use degrees and the other half to use radians. student materials.
Compare answers.
Did anyone find values of ݔand ݕthat produced a true statement?
à Answers will vary but could include ݔൌ Ͳιǡ ݕൌ Ͳι or ݔൌ ͳͺͲιǡ ݕൌ ͳͺͲι or ݔൌ Ͳιǡ ݕൌ ͻͲι or the
ߨ
equivalent in radians ݔൌ Ͳǡ ݕൌ Ͳ or ݔൌ ߨǡ ݕൌ ߨ or ݔൌ Ͳǡ ݕൌ .
ʹ
If you used degrees, compare your answers to the answers of a neighbor who used radians. What do you
notice?
à The answers will be the same but a different measure. For example, ݔൌ ͳͺͲιǡ ݕൌ ͳͺͲι is the same as
ݔൌ ߨǡ ݕൌ ߨ because ͳͺͲι ൌ ߨ. For example, ሺߨ ߨሻ ൌ ሺʹߨሻ ൌ Ͳ, and
ሺߨሻ ሺߨሻ ൌ Ͳ Ͳ ൌ Ͳ, so the statement is true.
Did anyone find values of ݔand ݕthat produced a false statement? Explain.
à Answers will vary but could include ݔൌ Ͷͷιǡ ݕൌ Ͷͷι or ݔൌ ͵Ͳιǡ ݕൌ ͵Ͳιor ݔൌ ͵Ͳιǡ ݕൌ Ͳι or the
ߨ ߨ ߨ ߨ ߨ ߨ
equivalent in radians ݔൌ ǡ ݕൌ or ݔൌ ǡ ݕൌ or ݔൌ ǡ ݕൌ . For example,
Ͷ Ͷ ͵
ߨ ߨ ߨ ߨ ߨ ඥʹ ඥʹ
ቀ ቁ ൌ ቀ ቁ ൌ ͳ, but ቀ ቁ ቀ ቁ ൌ ൌ ξʹ, so the statement is false.
Ͷ Ͷ ʹ Ͷ Ͷ ʹ ʹ
Is this function a linear transformation? Explain this to your neighbor.
à This function is not a linear transformation because ሺ ݔ ݕሻ ് ሺݔሻ ሺݕሻ for all real numbers.
Exercises (15 minutes)
In the exercises below, allow students to work through the problems in pairs. Circulate and give students help as
needed. Call the class back together, and have groups present their results. All groups can be assigned Exercises 1–4.
For advanced groups, ask students to find the imaginary solutions to Exercise 4, and/or assign some of the more
complicated examples that students brought to class from the Lesson 1 Problem Set and presented in the Opening
Exercise.
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Exercises
1. Let ࢌሺ࢞ሻ ൌ ܖܑܛሺ࢞ሻ. Does ࢌሺ࢞ሻ ൌ ࢌሺ࢞ሻ for all values of ࢞? Is it true for any values of ࢞?
Show work to justify your answer.
Scaffolding:
࣊ ࣊ ࣊ For advanced learners,
No. If ࢞ ൌ , ܖܑܛ൬ ቀ ቁ൰ ൌ ܖܑܛሺ࣊ሻ ൌ , but ܖܑܛቀ ቁ ൌ ሺሻ ൌ , so the statement does
have students determine
not hold for every value of ࢞. It is true anytime ܖܑܛሺ࢞ሻ ൌ , so for ࢞ ൌ ,
the general solution that
࢞ ൌ േ࣊ǡ ࢞ ൌ േ࣊.
works for all real numbers.
2. Let ࢌሺ࢞ሻ ൌ ܗܔሺ࢞ሻ. Find a value for ࢇ such that ࢌሺࢇሻ ൌ ࢌሺࢇሻ. Is there one? Show work
Monitor group work, and
to justify your answer. target some groups with
ܗܔሺࢇሻ ൌ ԝܗܔሺࢇሻ more specific questions to
ܗܔሺࢇሻ ൌ ܗܔሺࢇ ሻ
help them with the
algebra needed. Students
ࢇ ൌ ࢇ
may need a reminder of
ࢇ െ ࢇ ൌ
the properties of
ࢇሺࢇ െ ሻ ൌ
logarithms such as
Thus, ࢇ ൌ or ࢇ ൌ . Because is not in the domain of the logarithmic function, the only ܽԝሺݔሻ ൌ ሺ ݔ ሻ.
solution is ࢇ ൌ .
Some groups may need to
complete the
3. Let ࢌሺ࢞ሻ ൌ ࢞ . Show that ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ is true for ࢇ ൌ ࢈ ൌ ܗܔሺሻ and that it
is not true for
trigonometry value table
ࢇ ൌ ࢈ ൌ . before starting the
For ࢇ ൌ ࢈ ൌ ܗܔሺሻ exercises.
ࢌሺࢇ ࢈ሻ ൌ ൫ܗܔሺሻାܗܔሺሻ൯ ൌ ԝܗܔሺሻ ൌ ܗܔ൫ ൯ ൌ ൌ
ࢌሺࢇሻ ࢌሺ࢈ሻ ൌ ܗܔ ܗܔ ൌ ൌ
Therefore, ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ.
For ࢇ ൌ ࢈ ൌ
ࢌሺࢇ ࢈ሻ ൌ ା ൌ ൌ
ࢌሺࢇሻ ࢌሺ࢈ሻ ൌ ൌ ൌ
Therefore, ࢌሺࢇ ࢈ሻ ് ࢌሺࢇሻ ࢌሺ࢈ሻ.
4. Let ࢌሺ࢞ሻ ൌ . Are there any real numbers ࢇ and ࢈ so that ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ? Explain.
࢞
Neither ࢇ nor ࢈ can equal zero since they are in the denominator of the rational expressions.
ൌ
ࢇ࢈ ࢇ ࢈
ࢇ࢈ሺࢇ ࢈ሻ ൌ ࢇ࢈ሺࢇ ࢈ሻ ࢇ࢈ሺࢇ ࢈ሻ
ࢇ࢈ ࢇ ࢈
ࢇ࢈ ൌ ࢇሺࢇ ࢈ሻ ࢈ሺࢇ ࢈ሻ
ࢇ࢈ ൌ ࢇ ࢇ࢈ ࢇ࢈ ࢈
ࢇ࢈ ൌ ࢇ ࢇ࢈ ࢈
ࢇ࢈ ൌ ሺࢇ ࢈ሻ
This means that ࢇ࢈ must be a positive number. Simplifying further, we get ൌ ࢇ ࢇ࢈ ࢈ .
The sum of three positive numbers will never equal zero, so there are no real solutions for ࢇ and ࢈.
5. What do your findings from these exercises illustrate about the linearity of these functions? Explain.
Answers will vary but should address that in each case, the function is not a linear transformation because it does
not hold to the conditions ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ and ࢌሺࢉ࢞ሻ ൌ ࢉ൫ࢌሺ࢞ሻ൯ for all real-numbered inputs.
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A STORY OF FUNCTIONS Lesson 2 M1
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Closing (3 minutes)
As a class, have a discussion using the following questions.
What did you notice about the solutions of trigonometric functions? Why?
à There are more solutions that work for trigonometric functions because they are cyclical.
Which functions were hardest to find solutions that worked? Why?
à Answers will vary, but many students may say logarithmic or exponential functions.
Are ܽ ൌ Ͳ and/or ܾ ൌ Ͳ always solutions? Explain.
à No. It depends on the function.
à For example, ሺͲ Ͳሻ ് ሺͲሻ ሺͲሻ and ͳͲሺାሻ ് ͳͲ ͳͲ .
Are trigonometric, exponential, and logarithmic functions linear transformations? Explain.
à No. They do not meet the conditions required for linearity:
݂ሺܽ ܾሻ ൌ ݂ሺܽሻ ݂ሺܾሻ and ݂ሺܿݔሻ ൌ ܿ൫݂ሺݔሻ൯ for all real-numbered inputs
Exit Ticket (5 minutes)
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A STORY OF FUNCTIONS Lesson 2 M1
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Name Date
Lesson 2: Wishful Thinking—Does Linearity Hold?
Exit Ticket
1. Koshi says that he knows that ሺ ݔ ݕሻ ൌ ሺݔሻ ሺݕሻ because he has substituted in multiple values for ݔand
ݕ, and they all work. He has tried ݔൌ Ͳι and ݕൌ Ͳι, but he says that usually works, so he also tried ݔൌ Ͷͷι and
ݕൌ ͳͺͲι, ݔൌ ͻͲι and ݕൌ ʹͲι, and several others. Is Koshi correct? Explain your answer.
2. Is ݂ሺݔሻ ൌ ሺݔሻ a linear transformation? Why or why not?
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Exit Ticket Sample Solutions
1. Koshi says that he knows that ܖܑܛሺ࢞ ࢟ሻ ൌ ܖܑܛሺ࢞ሻ ܖܑܛሺ࢟ሻ because he has substituted in multiple values for ࢞
and ࢟, and they all work. He has tried ࢞ ൌ ι and ࢟ ൌ ι, but he says that usually works, so he also tried ࢞ ൌ ι
and ࢟ ൌ ૡι, ࢞ ൌ ૢι and ࢟ ൌ ૠι, and several others. Is Koshi correct? Explain your answer.
Koshi is not correct. He happened to pick values that worked, most giving at least one value of ܖܑܛሺ࢞ሻ ൌ . If he
had chosen other values such as ࢞ ൌ ι and ࢟ ൌ ι, ܖܑܛሺι ιሻ ൌ ܖܑܛሺૢιሻ ൌ , but
ඥ
ܖܑܛሺιሻ ܖܑܛሺιሻ ൌ , so the statement that ܖܑܛሺι ιሻ ൌ ܖܑܛሺιሻ ܖܑܛሺιሻ is false.
2. Is ࢌሺ࢞ሻ ൌ ܖܑܛሺ࢞ሻ a linear transformation? Why or why not?
No. ܖܑܛሺ࢞ ࢟ሻ ് ܖܑܛሺ࢞ሻ ܖܑܛሺ࢟ሻ and ܖܑܛሺࢇ࢞ሻ ് ࢇ ܖܑܛሺ࢞ሻ
Problem Set Sample Solutions
Assign students some or all of the functions to investigate. Problems 1–4 are all trigonometric functions, Problem 5 is a
rational function, and Problems 6 and 7 are logarithmic functions. These can be divided up. Problem 8 sets up Lesson 3
but is quite challenging.
Examine the equations given in Problems 1–4, and show that the functions ࢌሺ࢞ሻ ൌ ܛܗ܋ሺ࢞ሻ and ࢍሺ࢞ሻ ൌ ܖ܉ܜሺ࢞ሻ are not
linear transformations by demonstrating that they do not satisfy the conditions indicated for all real numbers. Then, find
values of ࢞ and/or ࢟ for which the statement holds true.
1. ܛܗ܋ሺ࢞ ࢟ሻ ൌ ܛܗ܋ሺ࢞ሻ ܛܗ܋ሺ࢟ሻ
Answers that prove the statement false will vary but could include ࢞ ൌ and ࢟ ൌ .
This statement is true when ࢞ ൌ Ǥ ૢ, or Ǥ ૠι, and ࢟ ൌ Ǥ ૢ, or Ǥ ૠι. This will be difficult for
students to find without technology.
2. ܛܗ܋ሺ࢞ሻ ൌ ԝܛܗ܋ሺ࢞ሻ
࣊
Answers that prove the statement false will vary but could include ࢞ ൌ or ࢞ ൌ .
This statement is true when ࢞ ൌ Ǥ ૢ, or Ǥ ૠι. This will be difficult for students to find without technology.
3. ܖ܉ܜሺ࢞ ࢟ሻ ൌ ܖ܉ܜሺ࢞ሻ ܖ܉ܜሺ࢟ሻ
࣊ ࣊
Answers that prove the statement false will vary but could include ࢞ ൌ and ࢟ ൌ .
This statement is true when ࢞ ൌ and ࢟ ൌ .
4. ܖ܉ܜሺ࢞ሻ ൌ ԝܖ܉ܜሺ࢞ሻ
࣊ ࣊
Answers that prove the statement false will vary but could include ࢞ ൌ and ࢟ ൌ .
This statement is true when ࢞ ൌ and ࢟ ൌ .
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5. Let ࢌሺ࢞ሻ ൌ . Are there any real numbers ࢇ and ࢈ so that ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ? Explain.
࢞
Neither ࢇ nor ࢈ can equal zero since they are in the denominator of the fractions.
If ࢌሺࢇ ࢈ሻ ൌ ࢌሺࢇሻ ࢌሺ࢈ሻ, then ൌ .
ሺࢇା࢈ሻ ࢇ ࢈
Multiplying each term by ࢇ ࢈ ሺࢇ ࢈ሻ, we get
ࢇ ࢈ ሺࢇ ࢈ሻ ൌ ࢇ ࢈ ሺࢇ ࢈ሻ ࢇ ࢈ ሺࢇ ࢈ሻ
ሺࢇ ࢈ሻ ࢇ ࢈
ࢇ ࢈ ൌ ࢈ ሺࢇ ࢈ሻ ࢇ ሺࢇ ࢈ሻ
ࢇ ࢈ ൌ ሺࢇ ࢈ ሻሺࢇ ࢈ሻ
ࢇ ࢈ ൌ ࢇ ࢇ ࢈ ࢇ ࢈ ࢇ࢈ ࢈
ࢇ ࢈ ൌ ࢇ ࢇ࢈ሺࢇ ࢈ ሻ ࢇ ࢈ ࢈
ൌ ࢇ ࢇ࢈ሺࢇ ࢈ ሻ ࢇ ࢈ ࢈
The terms ࢇ , ࢇ ࢈, and ࢈ are positive because they are even-numbered powers of nonzero numbers. We
established in the lesson that ࢇ࢈ ൌ ሺࢇ ࢈ሻ and, therefore, is also positive.
The product ࢇ࢈ሺࢇ ࢈ ሻ must then also be positive.
The sum of four positive numbers will never equal zero, so there are no real solutions for ࢇ and ࢈.
6. Let ࢌሺ࢞ሻ ൌ ܗܔሺ࢞ሻ. Find values of ࢇ such that ࢌሺࢇሻ ൌ ࢌሺࢇሻ.
ܗܔሺࢇሻ ൌ ܗܔሺࢇሻ
ܗܔሺࢇሻ ൌ ܗܔሺࢇሻ
ࢇ ൌ ࢇ
ൌ ࢇ
ࢇ ൌ ξ
This is true for the value of ࢇ when ࢇ ൌ ࢇ that is in the domain, which is ࢇ ൌ ξ.
7. Let ࢌሺ࢞ሻ ൌ ܗܔሺ࢞ሻ. Find values of ࢇ such that ࢌሺࢇሻ ൌ ࢌሺࢇሻ.
This is true for the values of ࢇ when ࢇ ൌ ࢇ that are in the domain of the function.
8. Based on your results from the previous two problems, form a conjecture about whether ࢌሺ࢞ሻ ൌ ܗܔሺ࢞ሻ represents
a linear transformation.
The function is not an example of a linear transformation. The condition ࢌሺࢇሻ ൌ ࢌሺࢇሻ does not hold for all values
of ࢇ, for example, nonzero values of ࢉ and ࢇ ൌ .
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9. Let ࢌሺ࢞ሻ ൌ ࢇ࢞ ࢈࢞ ࢉ.
a. Describe the set of all values for ࢇ, ࢈, and ࢉ that make ࢌሺ࢞ ࢟ሻ ൌ ࢌሺ࢞ሻ ࢌሺ࢟ሻ valid for all real numbers ࢞
and ࢟.
This is challenging for students, but the goal is for them to realize that ࢇ ൌ , ࢉ ൌ , and any real number ࢈.
They may understand that ࢇ ൌ , but ࢉ ൌ could be more challenging. The point is that it is unusual for
functions to satisfy this condition for all real values of ࢞ and ࢟. This is discussed in detail in Lesson 3.
ࢌሺ࢞ ࢟ሻ ൌ ࢇሺ࢞ ࢟ሻ ࢈ሺ࢞ ࢟ሻ ࢉ ൌ ࢇ࢞ ࢇ࢞࢟ ࢇ࢟ ࢈࢞ ࢈࢟ ࢉ
ࢌሺ࢞ሻ ࢌሺ࢟ሻ ൌ ࢇ࢞ ࢈࢞ ࢉ ࢇ࢟ ࢈࢟ ࢉ
ࢌሺ࢞ ࢟ሻ ൌ ࢌሺ࢞ሻ ࢌሺ࢟ሻ
ࢇ࢞ ࢇ࢞࢟ ࢇ࢟ ࢈࢞ ࢈࢟ ࢉ ൌ ࢇ࢞ ࢈࢞ ࢉ ࢇ࢟ ࢈࢟ ࢉ
ࢇ࢞࢟ ࢉ ൌ ࢉ ࢉ
ࢇ࢞࢟ ൌ ࢉ
Therefore, the set of values that satisfies this equation for all real numbers ࢞ and ࢟ is ࢇ ൌ , any real number
࢈, and ࢉ ൌ .
b. What does your result indicate about the linearity of quadratic functions?
Answers will vary but should address that quadratic functions are not linear transformations, since they only
meet the condition ࢌሺ࢞ ࢟ሻ ൌ ࢌሺ࢞ሻ ࢌሺ࢟ሻ when ࢇ ൌ .
Trigonometry Table
Angle Measure Angle Measure
ܖܑܛሺ࢞ሻ ܛܗ܋ሺ࢞ሻ
(࢞ Degrees) (࢞ Radians)
࣊ ξ
࣊ ξ ξ
࣊ ξ
࣊
ૢ
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A STORY OF FUNCTIONS Lesson 3 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 3: Which Real Number Functions Define a Linear
Transformation?
Student Outcomes
Students develop facility with the properties that characterize linear transformations.
Students learn that a mapping ܮǣ Թ ՜ Թ is a linear transformation if and only if ܮሺݔሻ ൌ ܽ ݔfor some real
number ܽ.
Lesson Notes
This lesson begins with two examples of functions that were explored in Lessons 1–2, neither of which is a linear
transformation. Next, students explore the function ݂ሺݔሻ ൌ ͷݔ, followed by the more general ݂ሺݔሻ ൌ ܽݔ, proving that
these functions satisfy the requirements for linear transformations. The rest of the lesson is devoted to proving that
functions of the form ݂ሺݔሻ ൌ ܽ ݔare, in fact, the only linear transformations from Թ to Թ.
Classwork
Opening Exercise (4 minutes)
Opening Exercise
Recall from the previous two lessons that a linear transformation is a function ࢌ that satisfies two conditions:
(1) ࢌሺ࢞ ࢟ሻ ൌ ࢌሺ࢞ሻ ࢌሺ࢟ሻ and (2) ࢌሺ࢞ሻ ൌ ࢌሺ࢞ሻ. Here, refers to any real number, and ࢞ and ࢟ represent arbitrary
elements in the domain of ࢌ.
a. Let ࢌሺ࢞ሻ ൌ ࢞ . Is ࢌ a linear transformation? Explain why or why not.
Let ࢞ be and ࢟ be . ࢌሺ ሻ ൌ ࢌሺሻ ൌ ൌ , but ࢌሺሻ ࢌሺሻ ൌ ൌ ૢ ൌ . Since these
two values are different, we can conclude that ࢌ is not a linear transformation.
b. Let ࢍሺ࢞ሻ ൌ ξ࢞. Is ࢍ a linear transformation? Explain why or why not.
Let ࢞ be and ࢟ be . ࢍሺ ሻ ൌ ࢍሺሻ ൌ ξ, but ࢍሺሻ ࢍሺሻ ൌ ξ ξ, which is not equal to ξ. This
means that ࢍ is not a linear transformation.
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Discussion (9 minutes): A Linear Transformation
The exercises you just did show that neither ݂ሺݔሻ ൌ ݔଶ nor ݃ሺݔሻ ൌ ξ ݔis a linear transformation. Let’s look at
a third function together.
Let ݄ሺݔሻ ൌ ͷݔ. Does ݄ satisfy the requirements for a linear transformation? Take a minute to explore this
question on your own, and then explain your thinking with a partner.
à First, we need to check the addition requirement: ݄ሺ ݔ ݕሻ ൌ ͷሺ ݔ ݕሻ ൌ ͷ ݔ ͷݕ.
݄ሺݔሻ ݄ሺݕሻ ൌ ͷ ݔ ͷݕ. Thus, we do indeed have ݄ሺ ݔ ݕሻ ൌ ݄ሺݔሻ ݄ሺݕሻ. So far, so good.
Now, we need to check the multiplication requirement: ݄ሺ݇ݔሻ ൌ ͷሺ݇ݔሻ ൌ ͷ݇ݔ.
݄݇ሺݔሻ ൌ ݇ ή ͷ ݔൌ ͷ݇ݔ. Thus, we also have ݄ሺ݇ݔሻ ൌ ݄݇ሺݔሻ.
Therefore, ݄ satisfies both of the requirements for a linear transformation.
So, now we know that ݄ሺݔሻ ൌ ͷ ݔis a linear transformation. Can you generate your own example of a linear
transformation? Write down a conjecture, and share it with another student.
à Answers will vary.
Do you think that every function of the form ݄ሺݔሻ ൌ ܽ ݔis a linear transformation? Let’s check to make sure
that the requirements are satisfied.
à ݄ሺ ݔ ݕሻ ൌ ܽሺ ݔ ݕሻ ൌ ܽ ݔ ܽݕ
݄ሺݔሻ ݄ሺݕሻ ൌ ܽ ݔ ܽݕ
Thus, ݄ሺ ݔ ݕሻ ൌ ݄ሺݔሻ ݄ሺݕሻ, as required.
݄ሺ݇ݔሻ ൌ ܽሺ݇ݔሻ ൌ ܽ݇ݔ
݄݇ሺݔሻ ൌ ݇ ή ܽ ݔൌ ܽ݇ݔ
Thus, ݄ሺ݇ݔሻ ൌ ݄݇ሺݔሻ, as required.
This proves that ݄ሺݔሻ ൌ ܽݔ, with ܽ any real number, is indeed a linear transformation.
What about ݂ሺݔሻ ൌ ͷ ݔ ͵? Since the graph of this equation is a straight line, we know that it represents a
linear function. Does that mean that it automatically meets the technical requirements for a linear
transformation? Write down a conjecture, and then take a minute to see if you are correct.
à If ݂ is a linear transformation, then it must have the addition property.
݂ሺ ݔ ݕሻ ൌ ͷሺ ݔ ݕሻ ͵ ൌ ͷ ݔ ͷ ݕ ͵
݂ሺݔሻ ݂ሺݕሻ ൌ ሺͷ ݔ ͵ሻ ሺͷ ݕ ͵ሻ ൌ ͷ ݔ ͷ ݕ
Clearly, these two expressions are not the same for all values of ݔand ݕ, so ݂ fails the requirements for
a linear transformation.
A bit surprising, isn’t it? The graph is a straight line, and it is ͳͲͲΨ correct to say that ݂ is a linear function.
But at the same time, it does not meet the technical requirements for a linear transformation. It looks as
though some linear functions are considered linear transformations, but not all of them are. Let’s try to
understand what is going on here.
Does anything strike you about the graph of ݂ሺݔሻ ൌ ͷ ݔas compared to the graph of ݂ሺݔሻ ൌ ͷ ݔ ͵?
à The first graph passes through the origin; the second one does not.
Do you think it is necessary for a graph to pass through the origin in order to be considered a linear
transformation? Let’s explore this question together. We have shown that every function of the form
݂ሺݔሻ ൌ ܽ ݔis a linear transformation. Are there other functions that map real numbers to real numbers that
are linear in this sense, or are these the only kind that do? Let’s see what we can learn about these questions.
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Discussion (6 minutes): The Addition Property
Suppose we have a linear transformation ܮthat takes a real number as an input and produces a real number as
an output. We can write ܮǣ Թ ՜ Թ to denote this.
Now, suppose that ܮtakes ʹ to ͺ and ͵ to ͳʹ; that is, ܮሺʹሻ ൌ ͺ and ܮሺ͵ሻ ൌ ͳʹ.
Where does ܮtake ͷ? Can you calculate the value of ܮሺͷሻ?
à Since ͷ ൌ ʹ ͵, we know that ܮሺͷሻ ൌ ܮሺʹ ͵ሻ. Since ܮis a linear transformation, this must be the
same as ܮሺʹሻ ܮሺ͵ሻ, which is ͺ ͳʹ ൌ ʹͲ. So, ܮሺͷሻ must be ʹͲ.
For practice, find out where ܮtakes and ͺ. That is, find ܮሺሻ and ܮሺͺሻ.
à ܮሺሻ ൌ ܮሺͷ ʹሻ ൌ ܮሺͷሻ ܮሺʹሻ ൌ ʹͲ ͺ ൌ ʹͺ
à ܮሺͺሻ ൌ ܮሺͷ ͵ሻ ൌ ܮሺͷሻ ܮሺ͵ሻ ൌ ʹͲ ͳʹ ൌ ͵ʹ
What can we learn about linear transformations through these examples? Let’s dig a little deeper.
We used the facts that ܮሺʹሻ ൌ ͺ and ܮሺ͵ሻ ൌ ͳʹ to figure out that ܮሺͷሻ ൌ ʹͲ. What is the relationship
between the three inputs here? What is the relationship among the three outputs?
à ͷ ൌ ʹ ͵, so the third input is the sum of the first two inputs.
à ʹͲ ൌ ͺ ͳʹ, so the third output is the sum of the first two outputs.
Do these examples give you a better understanding of the property ܮሺ ݔ ݕሻ ൌ ܮሺݔሻ ܮሺݕሻ? This statement
is saying that if you know what a linear transformation does to any two inputs ݔand ݕ, then you know for sure
what it does to their sum ݔ ݕ. In particular, to get the output for ݔ ݕ, you just have to add the outputs
ܮሺݔሻ and ܮሺݕሻ, just as we did in the example above, where we figured out that ܮሺͷሻ must be ʹͲ.
What do you suppose all of this means in terms of the graph of ?ܮLet’s plot each of the input-output pairs we
have generated so far and then see what we can learn.
L(5+3)
L(5+2)
L(2+3)
L(3)
L(2)
2 3 2+3 5+2 5+3 x
What do you notice about this graph?
à It looks as though the points lie on a line through the origin.
Can we be absolutely sure of this? Let’s keep exploring to find out if this is really true.
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Discussion (4 minutes): The Multiplication Property
Let’s again suppose that ܮሺʹሻ ൌ ͺ.
Can you figure out where ܮtakes ?
à Since ൌ ͵ ή ʹ, we know that ܮሺሻ ൌ ܮሺ͵ ή ʹሻ. Since ܮis a linear transformation, this must be the
same as ͵ ή ܮሺʹሻ, which is ͵ ή ͺ ൌ ʹͶ. So, ܮሺሻ must be ʹͶ.
For practice, find out where ܮtakes Ͷ and ͺ. That is, find ܮሺͶሻ and ܮሺͺሻ.
à ܮሺͶሻ ൌ ܮሺʹ ή ʹሻ ൌ ʹ ή ܮሺʹሻ ൌ ʹ ή ͺ ൌ ͳ
à ܮሺͺሻ ൌ ܮሺͶ ή ʹሻ ൌ Ͷ ή ܮሺʹሻ ൌ Ͷ ή ͺ ൌ ͵ʹ
We computed ܮሺͺሻ earlier using the addition property, and now we have computed it again using the
multiplication property. Are the results the same?
à Yes. In both cases, we have ܮሺͺሻ ൌ ͵ʹ.
Does this work give you a feel for what the multiplication property is all about? Let’s summarize our work in
the last few examples. Suppose you know that, for a certain input ݔ, ܮproduces output ݕ, so that ܮሺݔሻ ൌ ݕ.
The multiplication property is saying that if you triple the input from ݔto ͵ݔ, you will also triple the output
from ݕto ͵ݕ. This is the meaning of the statement ܮሺ͵ݔሻ ൌ ͵ ή ܮሺݔሻ, or more generally, ܮሺ݇ݔሻ ൌ ݇ܮሺݔሻ.
Once again, let’s see what all of this means in terms of the graph of ܮ. We will plot the input-output pairs we
generated.
4*L(2)
3*L(2)
2*L(2)
L(2)
2 2*2 3*2 4*2 x
Does this graph look like you expected it to?
à Yes. It is a straight line through the origin, just like before.
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Discussion (4 minutes): Opposites
So, we used the fact that ܮሺʹሻ ൌ ͺ to figure out that ܮሺͶሻ ൌ ͳ, ܮሺሻ ൌ ʹͶ, and Scaffolding:
ܮሺͺሻ ൌ ͵ʹ.
If students are struggling to
What about the negative multiples of ʹ? Can you figure out ܮሺെʹሻ? compute ܮሺെʹሻ, point out that
à ܮሺെʹሻ ൌ ܮሺെͳ ή ʹሻ ൌ െͳ ή ܮሺʹሻ ൌ െͳ ή ͺ ൌ െͺ െʹ ൌ െͳ ή ʹ, and then ask
For practice, find ܮሺെͶሻ, ܮሺെሻ, and ܮሺെͺሻ. them to apply the
à ܮሺെͶሻ ൌ ܮሺെͳ ή Ͷሻ ൌ െͳ ή ܮሺͶሻ ൌ െͳ ή ͳ ൌ െͳ multiplication property for
linear transformations.
à ܮሺെሻ ൌ ܮሺെͳ ή ሻ ൌ െͳ ή ܮሺሻ ൌ െͳ ή ʹͶ ൌ െʹͶ
à ܮሺെͺሻ ൌ ܮሺെͳ ή ͺሻ ൌ െͳ ή ܮሺͺሻ ൌ െͳ ή ͵ʹ ൌ െ͵ʹ
Look carefully at what ܮdoes to a number and its opposite. For instance, compare the outputs for ʹ and െʹ,
for Ͷ and െͶ, etc. What do you notice?
à We see that ܮሺʹሻ ൌ ͺ and ܮሺെʹሻ ൌ െͺ. We also see that ܮሺͶሻ ൌ ͳ and ܮሺെͶሻ ൌ െͳ.
Can you take your observation and formulate a general conjecture?
à It looks as though ܮሺെݔሻ ൌ െܮሺݔሻ.
This says that if you know what ܮdoes to a particular input ݔ, then you know for sure that ܮtakes the opposite
input, െݔ, to the opposite output, െܮሺݔሻ.
Now, prove that your conjecture is true in all cases.
à ܮሺെݔሻ ൌ ܮሺെͳ ή ݔሻ ൌ െͳ ή ܮሺݔሻ ൌ െܮሺݔሻ
Once again, let’s collect all of this information in graphical form.
All signs point to a straight-line graph that passes through the origin. But we have not yet shown that the
graph actually contains the origin. Let’s turn our attention to that question now.
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Discussion (5 minutes): Zero
If the graph of ܮcontains the origin, then ܮmust take Ͳ to Ͳ. Does this really have to be the case?
How can we use the addition property to our advantage here? Can you form the number Ͳ from the inputs we
already have information about?
à We know that ʹ െʹ ൌ Ͳ, so maybe that can help. Since ܮሺʹሻ ൌ ͺ and ܮሺെʹሻ ൌ െͺ, we can now
figure out ܮሺͲሻ. ܮሺͲሻ ൌ ܮሺʹ െʹሻ ൌ ܮሺʹሻ ܮሺെʹሻ ൌ ͺ െͺ ൌ Ͳ.
So, it really is true that ܮሺͲሻ ൌ Ͳ. What does this tell us about the graph of ?ܮ
à The graph contains the point ሺͲǡͲሻ, which is the origin.
In summary, if you give the number Ͳ as an input to a linear transformation ܮሺݔሻ, then the output is sure to be
Ͳ.
Quickly: Is ݂ሺݔሻ ൌ ݔ ͳ a linear transformation? Why or why not?
à No. It cannot be a linear transformation because ݂ሺͲሻ ൌ Ͳ ͳ ൌ ͳ, and a linear transformation
cannot transform Ͳ into ͳ.
For practice, use the fact that ܮሺሻ ൌ ʹͶ to show that ܮሺͲሻ ൌ Ͳ.
à We already showed that ܮሺെሻ ൌ െʹͶ, so ܮሺͲሻ ൌ ܮሺ െሻ ൌ ܮሺሻ ܮሺെሻ ൌ ʹͶ െ ʹͶ ൌ Ͳ.
We have used the addition property to show that ܮሺͲሻ ൌ Ͳ. Do you think it is possible to use the
multiplication property to reach the same conclusion?
à Yes. Ͳ is a multiple of ʹ, so we can write ܮሺͲሻ ൌ ܮሺͲ ή ʹሻ ൌ Ͳ ή ܮሺʹሻ ൌ Ͳ ή ͺ ൌ Ͳ.
So now, we have two pieces of evidence that corroborate our hypothesis that the graph of ܮpasses through
the origin. We can now officially add ሺͲǡͲሻ to our graph.
We originally said that the graph looks like a line through the origin. What is the equation of that line?
à The equation of the line that contains all of these points is ݕൌ Ͷݔ.
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Discussion (3 minutes): The Complete Graph of ࡸ
How can we be sure that the graph of ܮሺݔሻ is identical to the graph of ݕൌ Ͷ ?ݔIn theory, Scaffolding:
there could be points on ܮሺݔሻ that are not on ݕൌ Ͷݔ, or vice versa. Are these graphs in
fact identical? Perhaps we can show once and for all that ܮሺݔሻ ൌ Ͷݔ. Advanced students could be
challenged to pursue this
We know that ܮሺʹሻ ൌ ͺ. What is ܮሺͳሻ? question without specific
ͳ ͳ ͳ guidance. In other words, ask
à ܮሺͳሻ ൌ ܮቀ ή ʹቁ ൌ ή ܮሺʹሻ ൌ ή ͺ ൌ Ͷ
ʹ ʹ ʹ students if the graph of ݕൌ Ͷݔ
How might we use the multiplication property to compute ܮሺݔሻ for an arbitrary is identical to the graph of
input ?ݔ ܮሺݔሻ, and then let them
à Since ݔൌ ݔή ͳ, we have ܮሺݔሻ ൌ ܮሺ ݔή ͳሻ ൌ ݔή ܮሺͳሻ ൌ ݔή Ͷ. investigate on their own and
We have shown that, for any input ݔ, ܮሺݔሻ ൌ Ͷ ݔis a formula that gives the justify their responses.
output under the linear transformation ܮ. So, the complete graph of ܮlooks like
this:
Discussion (2 minutes): General Linear Transformations Թ ՜ Թ
All of the work we did to reach the conclusion that ܮሺݔሻ ൌ Ͷ ݔwas based on just one assumption: We took
ܮሺʹሻ ൌ ͺ as a given, and the rest of our conclusions were worked out from the properties of linear
transformations. Now, let’s show that every linear transformation ܮǣ Թ ՜ Թ has the form ܮሺݔሻ ൌ ܽݔ.
Show that ܮሺݔሻ ൌ ݔή ܮሺͳሻ.
à Since ݔൌ ݔή ͳ, we have ܮሺݔሻ ൌ ܮሺ ݔή ͳሻ ൌ ݔή ܮሺͳሻ.
Since ܮproduces real numbers as outputs, there is some number ܽ corresponding to ܮሺͳሻ. So, let’s define
ܽ ൌ ܮሺͳሻ. Now, we have that ܮሺݔሻ ൌ ݔή ܮሺͳሻ ൌ ݔή ܽ, which means that every linear transformation looks like
ܮሺݔሻ ൌ ܽݔ. There are no other functions Թ ՜ Թ that can possibly satisfy the requirements for a linear
transformation.
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Closing (2 minutes)
Write down a summary of what you learned in the lesson today, and then share your summary with a partner.
à Every function of the form ܮሺݔሻ ൌ ܽ ݔis a linear transformation.
à Every linear transformation ܮǣ Թ ՜ Թ corresponds to a formula ܮሺݔሻ ൌ ܽݔ.
à Linear transformations take the origin to the origin; that is, ܮሺͲሻ ൌ Ͳ.
à Linear transformations are odd functions; that is, ܮሺെݔሻ ൌ െܮሺݔሻ.
à The graph of a linear transformation ܮǣ Թ ՜ Թ is a straight line.
Exit Ticket (6 minutes)
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A STORY OF FUNCTIONS Lesson 3 M1
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Name Date
Lesson 3: Which Real Number Functions Define a Linear
Transformation?
Exit Ticket
Suppose you have a linear transformation ݂ǣ Թ ՜ Թ, where ݂ሺ͵ሻ ൌ ͻ and ݂ሺͷሻ ൌ ͳͷ.
1. Use the addition property to compute ݂ሺͺሻ and ݂ሺͳ͵ሻ.
2. Find ݂ሺͳʹሻ and ݂ሺͳͲሻ. Show your work.
3. Find ݂ሺെ͵ሻ and ݂ሺെͷሻ. Show your work.
4. Find ݂ሺͲሻ. Show your work.
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5. Find a formula for ݂ሺݔሻ.
6. Draw the graph of the function ݕൌ ݂ሺݔሻ.
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Exit Ticket Sample Solutions
Suppose you have a linear transformation ࢌǣ Թ ՜ Թ, where ࢌሺሻ ൌ ૢ and ࢌሺሻ ൌ .
1. Use the addition property to compute ࢌሺૡሻ and ࢌሺሻ.
ࢌሺૡሻ ൌ ࢌሺ ሻ ൌ ࢌሺሻ ࢌሺሻ ൌ ૢ ൌ
ࢌሺሻ ൌ ࢌሺૡ ሻ ൌ ࢌሺૡሻ ࢌሺሻ ൌ ൌ ૢ
2. Find ࢌሺሻ and ࢌሺሻ. Show your work.
ࢌሺሻ ൌ ࢌሺ ή ሻ ൌ ή ࢌሺሻ ൌ ή ૢ ൌ
ࢌሺሻ ൌ ࢌሺ ή ሻ ൌ ή ࢌሺሻ ൌ ή ൌ
3. Find ࢌሺെሻ and ࢌሺെሻ. Show your work.
ࢌሺെሻ ൌ െࢌሺሻ ൌ െૢ
ࢌሺെሻ ൌ െࢌሺሻ ൌ െ
4. Find ࢌሺሻ. Show your work.
ࢌሺሻ ൌ ࢌሺ െሻ ൌ ࢌሺሻ ࢌሺെሻ ൌ ૢ െૢ ൌ
5. Find a formula for ࢌሺ࢞ሻ.
We know that there is some number ࢇ such that ࢌሺ࢞ሻ ൌ ࢇ࢞, and since ࢌሺሻ ൌ ૢ, the value of ࢇ ൌ . In other words,
ࢌሺ࢞ሻ ൌ ࢞. We can also check to see if ࢌሺሻ ൌ is consistent with ࢇ ൌ , which it is.
6. Draw the graph of the function ࢟ ൌ ࢌሺ࢞ሻ.
y
x
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Problem Set Sample Solutions
The first problem provides students with practice in the core skills for this lesson. The second problem is a series of
exercises in which students explore concepts of linearity in the context of integer-valued functions as opposed to real-
valued functions. The third problem plays with the relation ݂ሺ ݔ ݕሻ ൌ ݂ሺݔሻ ݂ሺݕሻ, exchanging addition for
multiplication in one or both expressions.
1. Suppose you have a linear transformation ࢌǣ Թ ՜ Թ, where ࢌሺሻ ൌ and ࢌሺሻ ൌ .
a. Use the addition property to compute ࢌሺሻ, ࢌሺૡሻ, ࢌሺሻ, and ࢌሺሻ.
ࢌሺሻ ൌ ࢌሺ ሻ ൌ ࢌሺሻ ࢌሺሻ ൌ ൌ
ࢌሺૡሻ ൌ ࢌሺ ሻ ൌ ࢌሺሻ ࢌሺሻ ൌ ൌ
ࢌሺሻ ൌ ࢌሺ ሻ ൌ ࢌሺሻ ࢌሺሻ ൌ ൌ
ࢌሺሻ ൌ ࢌሺ ሻ ൌ ࢌሺሻ ࢌሺሻ ൌ ൌ
b. Find ࢌሺሻ, ࢌሺሻ, and ࢌሺሻ. Show your work.
ࢌሺሻ ൌ ࢌሺ ή ሻ ൌ ή ࢌሺሻ ൌ ή ൌ
ࢌሺሻ ൌ ࢌሺ ή ሻ ൌ ή ࢌሺሻ ൌ ή ൌ
ࢌሺሻ ൌ ࢌሺ ή ሻ ൌ ή ࢌሺሻ ൌ ή ൌ
c. Find ࢌሺെሻ, ࢌሺെሻ, and ࢌሺെૡሻ. Show your work.
ࢌሺെሻ ൌ ࢌሺെ ή ሻ ൌ െ ή ࢌሺሻ ൌ െ ή ൌ െ
ࢌሺെሻ ൌ ࢌሺെ ή ሻ ൌ െ ή ࢌሺሻ ൌ െ ή ൌ െ
ࢌሺെૡሻ ൌ ࢌሺെ ή ሻ ൌ െ ή ࢌሺሻ ൌ െ ή ൌ െ
d. Find a formula for ࢌሺ࢞ሻ.
We know there is some number ࢇ such that ࢌሺ࢞ሻ ൌ ࢇ࢞, and since ࢌሺሻ ൌ , then the value of ࢇ ൌ .
࢞
In other words, ࢌሺ࢞ሻ ൌ . We can also check to see if ࢌሺሻ ൌ is consistent with ࢇ ൌ , which it is.
e. Draw the graph of the function ࢌሺ࢞ሻ.
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2. The symbol Ժ represents the set of integers, and so ࢍǣ Ժ ՜ Ժ represents a function that takes integers as inputs and
produces integers as outputs. Suppose that a function ࢍǣ Ժ ՜ Ժ satisfies ࢍሺࢇ ࢈ሻ ൌ ࢍሺࢇሻ ࢍሺ࢈ሻ for all integers ࢇ
and ࢈. Is there necessarily an integer such that ࢍሺሻ ൌ for all integer inputs ?
a. Let ൌ ࢍሺሻ. Compute ࢍሺሻ and ࢍሺሻ.
ࢍሺሻ ൌ ࢍሺ ሻ ൌ ࢍሺሻ ࢍሺሻ ൌ ൌ
ࢍሺሻ ൌ ࢍሺ ሻ ൌ ࢍሺሻ ࢍሺሻ ࢍሺሻ ൌ ൌ
b. Let be any positive integer. Compute ࢍሺሻ.
ࢍሺሻ ൌ ࢍሺ ڮ ሻ ൌ ࢍሺሻ ڮ ࢍሺሻ ൌ ڮ ൌ
c. Now, consider ࢍሺሻ. Since ࢍሺሻ ൌ ࢍሺ ሻ, what can you conclude about ࢍሺሻ?
ࢍሺሻ ൌ ࢍሺ ሻ ൌ ࢍሺሻ ࢍሺሻ. By subtracting ࢍሺሻ from both sides of the equation, we get ࢍሺሻ ൌ .
d. Lastly, use the fact that ࢍሺ െሻ ൌ ࢍሺሻ to learn something about ࢍሺെሻ, where is any positive integer.
ࢍሺሻ ൌ ࢍሺ െሻ ൌ ࢍሺሻ ࢍሺെሻ. Since we know that ࢍሺሻ ൌ , we have ࢍሺሻ ࢍሺെሻ ൌ . This tells
us that ࢍሺെሻ ൌ െࢍሺሻ.
e. Use your work above to prove that ࢍሺሻ ൌ for every integer . Be sure to consider the fact that could
be positive, negative, or .
We showed that if is a positive integer, then ࢍሺሻ ൌ , where ൌ ࢍሺሻ. Also, since ή ൌ and we
showed that ࢍሺሻ ൌ , we have ࢍሺሻ ൌ ή . Finally, if is a negative integer, then െ is positive, which
means ࢍሺെሻ ൌ ሺെሻ ൌ െ. But, since ࢍሺെሻ ൌ െࢍሺሻ, we have ࢍሺሻ ൌ െࢍሺെሻ ൌ െሺെሻ ൌ .
Thus, in all cases, ࢍሺሻ ൌ .
3. In the following problems, be sure to consider all kinds of functions: polynomial, rational, trigonometric,
exponential, logarithmic, etc.
a. Give an example of a function ࢌǣ Թ ՜ Թ that satisfies ࢌሺ࢞ ή ࢟ሻ ൌ ࢌሺ࢞ሻ ࢌሺ࢟ሻ.
Any logarithmic function works; for instance, ࢌሺ࢞ሻ ൌ ܗܔሺ࢞ሻ.
b. Give an example of a function ࢍǣ Թ ՜ Թ that satisfies ࢍሺ࢞ ࢟ሻ ൌ ࢍሺ࢞ሻ ή ࢍሺ࢟ሻ.
Any exponential function works; for instance, ࢍሺ࢞ሻ ൌ ࢞ .
c. Give an example of a function ࢎǣ Թ ՜ Թ that satisfies ࢎሺ࢞ ή ࢟ሻ ൌ ࢎሺ࢞ሻ ή ࢎሺ࢟ሻ.
Any power of ࢞ works; for instance, ࢎሺ࢞ሻ ൌ ࢞.
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A STORY OF FUNCTIONS Lesson 4 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 4: An Appearance of Complex Numbers
Student Outcomes
Students solve quadratic equations with complex solutions.
Students understand the geometric origins of the imaginary unit ݅ in terms of ͻͲ-degree rotations. Students
use this understanding to see why ݅ ଶ ൌ െͳ.
Lesson Notes
This lesson begins with an exploration of an equation that arose in Lesson 1 in the context of studying linear
transformations. To check the solutions to this equation, students need a variety of skills involving the arithmetic of
complex numbers. The purpose of this phase of the lesson is to point to the need to review and extend students’
knowledge of complex number arithmetic. This phase of the lesson continues with a second example of a quadratic
equation with complex solutions, which is solved by completing the square.
The second phase of the lesson involves a review of the theory surrounding complex numbers. In particular, ݅ is
introduced as a multiplier that induces a ͻͲ-degree rotation of the coordinate plane and which satisfies the equation
݅ ଶ ൌ െͳ.
Classwork
Scaffolding:
Opening Exercise (2 minutes)
If students need help
answering the question in the
Opening Exercise
Opening Exercise, ask them,
Is ࡾሺ࢞ሻ ൌ a linear transformation? Explain how you know. “What are the properties of a
࢞
linear transformation?” If
ࡾሺ ሻ ൌ ࡾሺሻ ൌ , but ࡾሺሻ ࡾሺሻ ൌ ൌ , which is not the same as . This means necessary, cue them to check
that the reciprocal function does not preserve addition, and so it is not a linear transformation. whether or not
ܴሺܽ ܾሻ ൌ ܴሺܽሻ ܴሺܾሻ.
Example 1 (8 minutes)
Apparently, it is not true in general that ܴሺʹ ݔሻ ൌ ܴሺʹሻ ܴሺݔሻ, since this statement is false when ݔൌ ͵.
But this does not mean that there are no values of ݔthat make the equation true. Let’s see if we can produce
at least one solution.
ଵ ଵ ଵ
Solve the equation ൌ .
ଶା௫ ଶ ௫
What is the first step in solving this equation?
à We can multiply both sides by ʹݔሺʹ ݔሻ:
ʹ ݔൌ ݔሺʹ ݔሻ ʹሺʹ ݔሻ
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How might we continue?
à We can apply the distributive property:
ʹ ݔൌ ʹ ݔ ݔଶ Ͷ ʹݔ
Evidently, we are dealing with a quadratic equation. What are some techniques you know for solving
quadratic equations?
à We could try factoring, we could complete the square, or we could use the quadratic formula.
Let’s solve this equation by completing the square. Work on this problem until you have a perfect square
equal to a number, and then stop.
Circulate throughout the class, monitoring students’ work and providing assistance as needed.
െͶ ൌ ݔଶ ʹ ݔ
ͳ െ Ͷ ൌ ݔଶ ʹ ݔ ͳ Scaffolding:
െ͵ ൌ ሺ ݔ ͳሻଶ An area diagram can be used to
help students understand why
ݔ ͳ it was necessary to add ͳ to
both sides of the equation to
create a perfect square.
ݔ ݔଶ ݔ
ͳ ݔ ǫ
Do you notice anything unusual about the equation at this point?
à We have a quantity whose square is equal to a negative number.
What does that tell you about the solutions to the equation?
à No real number can satisfy the equation, so the solutions must be complex numbers.
Go ahead and find the solutions.
ඥሺ ݔ ͳሻଶ ൌ ξെ͵
ݔ ͳ ൌ േ݅ξ͵
ݔൌ െͳ േ ݅ξ͵
Do these solutions satisfy the original equation ܴሺʹ ݔሻ ൌ ܴሺʹሻ ܴሺݔሻ? How can we tell?
ଵ ଵ ଵ
à We need to check to see whether or not ൌ
ଶାሺିଵାξଷሻ ଶ ିଵାξଷ
In order to ascertain whether or not these two expressions are equal, we need to review and extend the things
we learned about complex numbers in Algebra II. But first, let’s do some additional work with quadratics that
have complex solutions.
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Example 2 (13 minutes)
Solve the equation ͵ ݔଶ ͷ ݔ ൌ ͳ.
Recall that we can use an area diagram to help us visualize the process of completing the square. A first
attempt might look something like this:
ǫ ǫ ξ͵ݔ ǫ
ͷ ͷ
ǫ ͵ ݔଶ ݔ ξ͵ݔ ͵ ݔଶ ݔ
ʹ ʹ
ͷ ͷ
ǫ ݔ ǫ ǫ ݔ ǫ
ʹ ʹ
Do you notice anything awkward about this initial diagram?
à The square roots and the fractions are a bit awkward to work with.
How can we get around these awkward points? What would make this problem easier to handle?
à If the coefficient of ݔଶ were a perfect square, then we would not have a radical to contend with. If the
coefficient of ݔwere even, then we would not have a fraction to contend with.
We can multiply both sides of the equation by any number we choose. Let’s be strategic about this. What
multiplier could we choose that would create a perfect square for the ݔଶ -term?
à If we multiply both sides of the equation by ͵, the leading coefficient becomes ͻ, a perfect square.
Go ahead and multiply by ͵, and see what you get.
͵ ݔଶ ͷ ݔ ൌ ͳ
͵ሺ͵ ݔଶ ͷ ݔ ሻ ൌ ͵ሺͳሻ
ͻ ݔଶ ͳͷ ݔ ʹͳ ൌ ͵
Now, let’s deal with the ݔ-term. What multiplier could we choose that would make the ݔ-term even, without
disturbing the requirement about having a perfect square in the leading term?
à We could multiply both sides of the equation by Ͷ, which is both even and a perfect square.
Go ahead and multiply by Ͷ, and see what you get.
ͻ ݔଶ ͳͷ ݔ ʹͳ ൌ ͵
Ͷሺͻ ݔଶ ͳͷ ݔ ʹͳሻ ൌ Ͷሺ͵ሻ
͵ଶ Ͳ ݔ ͺͶ ൌ ͳʹ
͵ଶ Ͳ ݔ ͺͶ ൌ ͳʹ
Because we took the simple steps of multiplying by ͵ and then by Ͷ, the algebra will now be much easier to
handle.
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Go ahead and complete the square now. Use an area diagram to help you do this. Then, solve the equation
completely.
ǫ ǫ ݔ ͷ
ǫ ͵ ݔଶ ͵Ͳݔ ݔ ͵ ݔଶ ͵Ͳݔ
ǫ ͵Ͳݔ ǫ ͷ ͵Ͳݔ ǫ
͵ ݔଶ Ͳ ݔ ͺͶ ൌ ͳʹ
͵ ݔଶ Ͳ ݔൌ ͳʹ െ ͺͶ ൌ െʹ
͵ ݔଶ Ͳ ݔ ʹͷ ൌ െʹ ʹͷ
ሺ ݔ ͷሻଶ ൌ െͶ
ݔ ͷ ൌ േ݅ξͶ
െͷ േ ݅ξͶ
ݔൌ
ͷ ξͶ
ݔൌെ േ݅
Quickly answer the following question in your notebook:
When you want to complete the square of a quadratic expression ܽ ݔଶ ܾ ݔ ܿ, what conditions on ܽ and ܾ
make the process go smoothly?
à It is desirable to convert ܽ into a perfect square and to convert ܾ into an even number.
Let’s generalize the work we did with the example above.
Take the expression ܽ ݔଶ ܾ ݔ ܿ, and multiply each term by Ͷܽ. What do you get?
à Ͷܽሺܽ ݔଶ ܾ ݔ ܿሻ ൌ Ͷܽଶ ݔଶ Ͷܾܽ ݔ Ͷܽܿ
How does this connect to the summary point you wrote in your notebook?
à Ͷܽଶ ൌ ሺʹܽሻଶ , so it is a perfect square. Also, Ͷܾܽ ൌ ʹሺʹܾܽሻ, so it is even.
You may also recognize that the expression Ͷܽܿ is a component from the general quadratic formula. Using Ͷܽ
as a multiplier is useful indeed.
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Exercise 1 (4 minutes)
Exercises
1. Solve ࢞ െ ࢞ ૠ ൌ ૢ.
࢞ െ
࢞ െ ࢞ ૠ ൌ ૢ
࢞ െ ࢞ ൌ ૢ െ ૠ ൌ െૡ
ሺ࢞ െ ࢞ሻ ൌ ሺെૡሻ ࢞ ࢞ െ࢞
࢞ െ ࢞ ൌ െ
࢞ െ ࢞ ૢ ൌ െ ૢ
ሺ࢞ െ ሻ ൌ െ െ െ࢞ ǫ
േ ξ ξ
࢞ൌ ൌ േ
Take about 30 seconds to write down what you have learned so far today, and then share what you wrote with
another student.
Now that we have practiced solving a few equations with complex solutions, we are going to conduct a general
review of things we know about complex numbers, starting with the definition of ݅.
Discussion (5 minutes): The Geometry of Multiplication by
Recall that multiplying by െͳ rotates the number line about the origin through ͳͺͲ degrees.
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You may remember that the number ݅ is the multiplier that rotates the number line through ͻͲ degrees.
If we take a point on the vertical axis and multiply it by ݅, what would you expect to see geometrically?
à This should produce another ͻͲ-degree rotation.
Now we have performed two ͻͲ-degree rotations, which is the same as a ͳͺͲ-degree rotation. This means
that multiplying a number by ݅ twice is the same as multiplying the number by െͳ.
Knowing that ݅ ή ݅ ݔൌ ݅ ଶ ή ݔ, what do the above observations suggest must be true about the number ݅?
à If ݔis any real number, we have ݅ ଶ ή ݔൌ െͳ ή ݔ, which means that ݅ ଶ ൌ െͳ.
Example 3 (2 minutes)
We know that multiplying by ݅ rotates a point through ͻͲ degrees, and multiplying by ݅ ଶ rotates a point
through ͳͺͲ degrees. What do you suppose multiplying by ݅ ଷ does? What about ݅ ସ ?
à It would seem as though this should produce three ͻͲ-degree rotations, which is ʹͲ degrees. If
multiplying by ݅ ସ is the same as doing four ͻͲ-degree rotations, then that would make ͵Ͳ degrees.
So, ݅ ସ takes a point back to where it started. In light of the fact that ݅ ଶ ൌ െͳ, does this make sense?
à Yes, because ݅ ସ ൌ ݅ ଶ ή ݅ ଶ , which is ሺെͳሻሺെͳሻ, which is just ͳ. Multiplying by ͳ takes a point to itself, so
the ͵Ͳ-degree rotation does indeed make sense.
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Exercises 2–4 (3 minutes)
Ask students to work independently on these problems, and then discuss them as a whole class.
2. Use the fact that ൌ െ to show that ൌ െ. Interpret this statement geometrically.
We have ൌ ή ൌ ሺെሻ ή ൌ െ. Multiplying by rotates a point through ૢ degrees, and multiplying by െ
rotates it ૡ degrees farther. This makes sense with our earlier conjecture that multiplying by would induce a
ૠ-degree rotation.
3. Calculate
ൌ ή ή ൌ ሺെሻሺെሻሺെሻ ൌ ሺሻሺെሻ ൌ െ
4. Calculate
ൌ ή ή ൌ ሺെሻሺെሻሺሻ ൌ ሺሻሺሻ ൌ
Closing (3 minutes)
Ask students to write responses to the following questions, and then have them share their responses in pairs. Then,
briefly discuss the responses as a whole class.
What is important to know about ݅ from a geometric point of view?
à Multiplication by ݅ rotates a point in the plane counterclockwise about the origin through ͻͲ degrees.
What is important to know about ݅ from an algebraic point of view?
à The number ݅ satisfies the equation ݅ ଶ ൌ െͳ.
Exit Ticket (5 minutes)
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Name Date
Lesson 4: An Appearance of Complex Numbers
Exit Ticket
1. Solve the equation below.
ʹ ݔଶ െ ͵ ݔ ͻ ൌ Ͷ
2. What is the geometric effect of multiplying a number by ݅ ସ ? Explain your answer using words or pictures, and then
confirm your answer algebraically.
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Exit Ticket Sample Solutions
1. Solve the equation below.
࢞ െ ࢞ ૢ ൌ
࢞ െ ࢞ ൌ െ
࢞ െ ࢞ ൌ െ
࢞ െ ࢞ ૢ ൌ െ ૢ
ሺ࢞ െ ሻ ൌ െ
࢞ െ ൌ േξ
േ ξ ξ
࢞ൌ ൌ േ
2. What is the geometric effect of multiplying a number by ? Explain your answer using words or pictures, and then
confirm your answer algebraically.
If you multiply a number by four times, you would expect to see four ૢ-degree rotations. This amounts to a
-degree rotation. In other words, each point is mapped back to itself. This makes sense algebraically as well
since the work below shows that ൌ .
ൌ ή ൌ െ ή െ ൌ
Problem Set Sample Solutions
1. Solve the equation below.
࢞ െ ૠ࢞ ૡ ൌ
࢞ െ ૠ࢞ ൌ െ
࢞ െ ࢞ ൌ െ
࢞ െ ࢞ ૢ ൌ െ ૢ
ሺ࢞ െ ૠሻ ൌ െૠ
ૠ േ ξૠ ૠ ξૠ
࢞ൌ ൌ േ
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2. Consider the equation ࢞ ൌ ૡ.
a. What is the first solution that comes to mind?
It is easy to see that ൌ ૡ, so is a solution.
b. It may not be easy to tell at first, but this equation actually has three solutions. To find all three solutions, it
is helpful to consider ࢞ െ ૡ ൌ , which can be rewritten as ሺ࢞ െ ሻሺ࢞ ࢞ ሻ ൌ (check this for
yourself). Find all of the solutions to this equation.
࢞ ࢞ ൌ
࢞ ࢞ ൌ െ
࢞ ࢞ ൌ െ
ሺ࢞ ሻ ൌ െ
࢞ ൌ െ േ ξ
The solutions to ࢞ െ ૡ ൌ are , െ ξ, and െ െ ξ.
3. Make a drawing that shows the first powers of (i.e., ǡ ǡ ǥ ǡ ), and then confirm your results algebraically.
ൌ
ൌ െ
ൌ ή ൌ െ ή ൌ െ
ൌ ή ൌ െ ή െ ൌ
ൌ ή ൌ ή ൌ
4. What is the value of ૢૢ? Explain your answer using words or drawings.
Multiplying by four times is equivalent to rotating through ή ૢ ൌ degrees, which is a complete rotation.
Since ૢૢ ൌ ή , multiplying by for ૢૢ times is equivalent to performing complete rotations, followed by
three ૢ-degree rotations. Thus, ૢૢ ൌ െ.
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5. What is the geometric effect of multiplying a number by െ? Does your answer make sense to you? Give an
explanation using words or drawings.
If we multiply a number by and then by െ, we get a quarter turn followed by a half turn. This is equivalent to a
three-quarters turn in the counterclockwise direction, which is the same as a quarter turn in the clockwise direction.
This makes sense because we would expect multiplication by െ to have the opposite effect as multiplication by ,
and so it feels right to say that multiplying by െ rotates a point in the opposite direction by the same amount.
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Lesson 5: An Appearance of Complex Numbers
Student Outcomes
Students describe complex numbers and represent them as points in the complex plane.
Students perform arithmetic with complex numbers, including addition, subtraction, scalar multiplication, and
complex multiplication.
Lesson Notes
In this lesson, complex numbers are formally described, and students review how to represent complex numbers as
points in the complex plane. Students look for and make use of structure as they see similarities between plotting
ordered pairs of real numbers in the coordinate plane and plotting complex numbers in the complex plane.
Next, students review the mechanics involved in adding complex numbers, subtracting complex numbers, multiplying a
complex number by a scalar, and multiplying a complex number by a second complex number. Students look for and
make use of structure as they see similarities between the process of multiplying two binomials and the process of
multiplying two complex numbers.
Classwork
Opening Exercise (2 minutes)
Opening Exercise
Write down two fundamental facts about that you learned in the previous lesson.
Multiplication by induces a ૢ-degree counterclockwise rotation about the origin. Also, satisfies the equation
ൌ െ.
Discussion (5 minutes): Describing Complex Numbers
What do you recall about the meanings of the following terms? Briefly discuss what you remember with a
partner, providing examples of each kind of number as you are able. After you have had a minute to share
with one another, we will review each term as a whole class.
à Real number
à Imaginary number
à Complex number
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After students talk in pairs, bring the class together, and ask a few individual students to share an example of each kind
of number.
ଵ
à Examples of real numbers: ͷ, ԝ, ͲǤͺͷ, െͶ, Ͳ
ଶ
à Examples of imaginary numbers: ͵݅, ͷ݅, െʹ݅
à Examples of complex numbers: ͵ Ͷ݅, ͷ െ ݅
In the previous lesson, we reviewed the definition of the imaginary unit ݅. We can also form multiples of ݅, such as
ʹ݅ǡ ͵݅ǡ Ͷ݅ǡ െͳͲ݅. The multiples of ݅ are called imaginary numbers. As you know, the term real number refers to numbers
ଷ
like ͵, െͳʹ, Ͳ, ԝ, ξʹ, and so forth, none of which have an imaginary component. If we combine a real number and an
ହ
imaginary number, we get expressions like these: ͷ ʹ݅, Ͷ െ ͵݅, െ ͳͲ݅. These numbers are called complex
numbers. In general, a complex number has the form ܽ ܾ݅, where ܽ and ܾ are both real numbers. The number ܽ is
called the real component, and the number ܾ is called the imaginary component.
Definition in your own words Facts/characteristics
Complex
Examples Number Non-examples
Discussion (5 minutes): Visualizing Complex Numbers
Visualization is an extremely important tool in mathematics. How do we visually represent real numbers?
à Real numbers can be represented as points on a number line.
How do you suppose we could visually represent a complex number? Do you think we could use a number line
just like the one we use for real numbers?
à Since it takes two real numbers ܽ and ܾ to describe a complex number ܽ ܾ݅, we cannot just use a
single number line.
In fact, we need two number lines to represent a complex number. The standard way to represent a complex number is
to create what mathematicians call the complex plane. In the complex plane, the ݔ-axis is used to represent the real
component of a complex number, and the ݕ-axis is used to represent its imaginary component. For instance, the
complex number ͷ ʹ݅ has a real component of ͷ, so we take the point that is ͷ units along the ݔ-axis. The imaginary
component is ʹ, so we take the point that is ʹ units along the ݕ-axis. In this way, we can associate the complex number
ͷ ʹ݅ with the point ሺͷǡʹሻ as shown on the next page. This seemingly simple maneuver, associating complex numbers
with points in the plane, turns out to have profound implications for our studies in this module.
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Discussion: Visualizing Complex Numbers
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Exercises 1–3 (2 minutes)
Allow students time to respond to the following questions and to discuss their responses with a partner. Then, bring the
class together, and allow a few individual students to share their responses with the class.
Exercises 1–7
1. Give an example of a real number, an imaginary number, and a complex number. Use examples that have not
already been discussed in the lesson.
Answers will vary.
2. In the complex plane, what is the horizontal axis used for? What is the vertical axis used for?
The horizontal axis is used to represent the real component of a complex number. The vertical axis is used to
represent the imaginary component.
3. How would you represent െ in the complex plane?
The complex number െ corresponds to the point ሺെǡ ሻ in the coordinate plane.
Example 1 (6 minutes): Scalar Multiplication
Let’s consider what happens when we multiply a real number by a complex number.
ͻሺെͺ ͳͲ݅ሻ
Does this remind you of a situation that is handled by a property of real numbers?
à This expression resembles the form ܽሺܾ ܿሻ, which can be handled using the distributive property.
The distributive property tells us that ܽሺܾ ܿሻ ൌ ܾܽ ܽܿ, but the ordinary version of this property only
applies when ܽ, ܾ, and ܿ are real numbers. In fact, we can extend the use of the distributive property to
include cases that involve complex numbers.
ͻሺെͺ ͳͲ݅ሻ ൌ ͻሺെͺሻ ͻሺͳͲ݅ሻ ൌ െʹ ͻͲ݅
Let’s explore this operation from a geometric point of view.
If ݓൌ ͵ ݅, what do you suppose ʹ ݓlooks like in the complex plane? Compute ʹݓ, and then plot both ݓ
and ʹ ݓin the complex plane.
à ʹ ݓൌ ʹሺ͵ ݅ሻ ൌ ʹ݅
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How would you describe the relationship between ݓand ʹ?ݓ
à The points representing ݓand ʹ ݓare on the same line through the origin. The distance from Ͳ to ʹݓ
is twice as long as the distance from Ͳ to ݓ.
Notice that the real component of ݓwas transformed from ͵ to ʹሺ͵ሻ ൌ , and the imaginary component of ݓ
was transformed from ͳ to ʹ. We could say that each component got scaled up by a factor of ʹ. For this
reason, multiplying by a real number is referred to as scalar multiplication, and since real numbers have this
kind of scaling effect, they are sometimes called scalars.
Example 2 (7 minutes): Multiplying Complex Numbers
Let’s look now at an example that involves multiplying a complex number by Scaffolding:
another complex number. Advanced learners may be
ሺͺ ݅ሻሺͳͲ െ ͷ݅ሻ challenged to perform the
multiplication without any cues
What situation involving real numbers does this remind you of?
from the teacher.
à It resembles the situation where you are multiplying two binomials:
ሺܽ ܾሻሺܿ ݀ሻ.
Although we are working with complex numbers, the distributive property still applies. Multiply the terms
using the distributive property.
ሺͺ ݅ሻሺͳͲ െ ͷ݅ሻ ൌ ͺሺͳͲ െ ͷ݅ሻ ݅ሺͳͲ െ ͷ݅ሻ ൌ ͺͲ െ ͶͲ݅ Ͳ݅ െ ͵ͷ݅ ଶ
What could we do next?
Scaffolding:
à We can combine the two ݅-terms into a single term:
The diagram below can be used
ͺͲ െ ͶͲ݅ Ͳ݅ െ ͵ͷ݅ ଶ ൌ ͺͲ ͵Ͳ݅ െ ͵ͷ݅ ଶ to multiply complex numbers
Does anything else occur to you to try here? in much the same way that it
can be used to multiply two
à We know that ݅ ଶ ൌ െͳ, so we can write
binomials. Some students may
ͺͲ ͵Ͳ݅ െ ͵ͷ݅ ଶ ൌ ͺͲ ͵Ͳ݅ െ ͵ͷሺെͳሻ find it helpful to organize their
ͺͲ ͵Ͳ݅ െ ͵ͷሺെͳሻ ൌ ͺͲ ͵Ͳ݅ ͵ͷ ൌ ͳͳͷ ͵Ͳ݅ work in this way.
Summing up: We started with two complex numbers ͺ ݅ and ͳͲ െ ͷ݅, we
multiplied them together, and we produced a new complex number ͳͳͷ ͵Ͳ݅.
ͺ ݅
ͳͲ 80 Ͳ݅
െͷ݅ െͶͲ݅ െ͵ͷ݅ ଶ
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Example 3 (5 minutes): Addition and Subtraction
Do you recall the procedures for adding and subtracting complex numbers? Go ahead and give these two
problems a try.
ሺെͳͲ െ ͵݅ሻ ሺെ ݅ሻ
ሺͻ െ ݅ሻ െ ሺെ͵ െ ͳͲ݅ሻ
Give students an opportunity to try these problems. Walk around the room, and monitor students’ work, and then call
on students to share what they have done.
ሺെͳͲ െ ͵݅ሻ ሺെ ݅ሻ ൌ ሺെͳͲ െሻ ሺെ͵݅ ݅ሻ ൌ െͳ ͵݅
ሺͻ െ ݅ሻ െ ሺെ͵ െ ͳͲ݅ሻ ൌ ሾͻ െ ሺെ͵ሻሿ ሾെ݅ െ ሺെͳͲ݅ሻሿ ൌ ͳʹ Ͷ݅
The key points to understand here are these:
– To add two complex numbers, add the real components and the imaginary components separately.
– To subtract two complex numbers, subtract the real components and the imaginary components
separately.
In the lesson today, we saw that when we multiply a complex number by a scalar, we get a new complex
number that is simply a scaled version of the original number. Now, suppose we were to take two complex
numbers ݖand ݓ. How do you suppose ݖ ݓ, ݖെ ݓ, and ݖή ݓare related geometrically? These questions
will be explored in the upcoming lessons.
Exercises 4–7 (4 minutes)
Tell students to perform the following exercises for practice and then to compare their answers with a partner. Call on
students at random to share their answers.
For Exercises 4–7, let ࢇ ൌ and ࢈ ൌ െ .
4. Find ࢇ ࢈. Then, plot ࢇ, ࢈, and ࢇ ࢈ in the complex plane.
ࢇ ࢈ ൌ
5. Find ࢇ െ ࢈. Then, plot ࢇ, ࢈, and ࢇ െ ࢈ in the complex plane.
ࢇ െ ࢈ ൌ െ
6. Find ࢇ. Then, plot ࢇ and ࢇ in the complex plane.
ࢇ ൌ
7. Find ࢇ ή ࢈. Then, plot ࢇ, ࢈, and ࢇ ή ࢈ in the complex plane.
ࢇ ή ࢈ ൌ ሺ ሻሺ െ ሻ
ൌ െ െ
ൌ
ൌ
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Closing (3 minutes)
Ask students to respond to the following questions in their notebooks, and then give them a minute to share their
responses with a partner.
What is the complex plane used for?
à The complex plane is used to represent complex numbers visually.
What operations did you learn to perform on complex numbers?
à We learned how to add, subtract, and multiply two complex numbers, as well as how to perform scalar
multiplication on complex numbers.
Which of the four fundamental operations was not discussed in this lesson? This topic will be treated in an
upcoming lesson.
à We did not discuss how to divide two complex numbers.
Exit Ticket (6 minutes)
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Name Date
Lesson 5: An Appearance of Complex Numbers
Exit Ticket
In Problems 1–4, perform the indicated operations. Write each answer as a complex number ܽ ܾ݅.
1. Let ݖଵ ൌ െʹ ݅, ݖଶ ൌ ͵ െ ʹ݅, and ݓൌ ݖଵ ݖଶ . Find ݓ, and graph ݖଵ , ݖଶ , and ݓin the complex plane.
2. Let ݖଵ ൌ െͳ െ ݅, ݖଶ ൌ ʹ ʹ݅, and ݓൌ ݖଵ െ ݖଶ . Find ݓ, and graph ݖଵ , ݖଶ , and ݓin the complex plane.
3. Let ݖൌ െʹ ݅ and ݓൌ െʹݖ. Find ݓ, and graph ݖand ݓin the complex plane.
4. Let ݖଵ ൌ ͳ ʹ݅, ݖଶ ൌ ʹ െ ݅, and ݓൌ ݖଵ ݖ ڄଶ . Find ݓ, and graph ݖଵ , ݖଶ , and ݓin the complex plane.
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Exit Ticket Sample Solutions
In Problems 1–4, perform the indicated operations. Write each answer as a complex number ࢇ ࢈.
1. Let ࢠ ൌ െ , ࢠ ൌ െ , and ࢝ ൌ ࢠ ࢠ . Find ࢝, and graph ࢠ , ࢠ , and ࢝ in the complex plane.
࢝ൌെ
2. Let ࢠ ൌ െ െ , ࢠ ൌ , and ࢝ ൌ ࢠ െ ࢠ . Find ࢝, and graph ࢠ , ࢠ , and ࢝ in the complex plane.
࢝ ൌ െ െ
3. Let ࢠ ൌ െ and ࢝ ൌ െࢠ. Find ࢝, and graph ࢠ and ࢝ in the complex plane.
࢝ ൌ െ
4. Let ࢠ ൌ , ࢠ ൌ െ ܑ, and ࢝ ൌ ࢠ ࢠ ڄ. Find ࢝, and graph ࢠ , ࢠ , and ࢝ in the complex plane.
࢝ ൌ ሺ ሻሺ െ ሻ
ൌ െ
ൌ
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Problem Set Sample Solutions
Problems 1–4 involve the relationships among the set of real numbers, the set of imaginary numbers, and the set of
complex numbers. Problems 5–9 involve practice with the core set of arithmetic skills from this lesson. Problems 10–12
involve the complex plane. A reproducible complex plane is provided at the end of the lesson should the teacher choose
to hand out copies for the Problem Set.
1. The number is a real number. Is it also a complex number? Try to find values of ࢇ and ࢈ so that ൌ ࢇ ࢈.
Because ൌ , is a complex number.
2. The number is an imaginary number and a multiple of . Is it also a complex number? Try to find values of ࢇ and
࢈ so that ൌ ࢇ ࢈.
Because ൌ , is a complex number.
3. Daria says that every real number is a complex number. Do you agree with her? Why or why not?
For any real number ࢇ, ࢇ ൌ ࢇ , so Daria is correct.
4. Colby says that every imaginary number is a complex number. Do you agree with him? Why or why not?
An imaginary number ࢈ ൌ ࢈, so Colby is correct.
In Problems 5–9, perform the indicated operations. Report each answer as a complex number ࢝ ൌ ࢇ ࢈, and graph it
in a complex plane.
5. Given ࢠ ൌ െૢ , ࢠ ൌ െ െ , find ࢝ ൌ ࢠ ࢠ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ െૢ
6. Given ࢠ ൌ െ , ࢠ ൌ െૠ െ , find ࢝ ൌ ࢠ െ ࢠ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ ሺെ ሻ െ ሺെૠ െ ሻ
ൌ
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7. Given ࢠ ൌ ξ , ࢠ ൌ ξ െ , find ࢝ ൌ ࢠ െ ࢠ , and
graph ࢠ , ࢠ , and ࢝.
࢝ ൌ ൫ξ ൯ െ ൫ξ െ ൯
ൌ ξ
8. Given ࢠ ൌ , ࢠ ൌ െ ૡ, find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ െ െ
9. Given ࢠ ൌ , ࢠ ൌ െ , find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ൌെ
10. Given ࢠ ൌ െ, ࢠ ൌ , find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ െ െ
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11. Given ࢠ ൌ , ࢠ ൌ െ െ , find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ ሺ ሻሺെ െ ሻ
ൌ െ െ െ െ
ൌ െ െ െ ሺെሻ
ൌ െ െ
12. Given ࢠ ൌ , ࢠ ൌ , find ࢝ ൌ ࢠ ࢠ ڄ, and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ ሺ ሻሺ ሻ
ൌ
ൌ െ
ൌ
13. Given ࢠ ൌ , ࢠ ൌ , find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ൌڄ
ൌ
14. Given ࢠ ൌ , ࢠ ൌ , find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ ሺ ሻ
ൌ െ
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15. Given ࢠ ൌ ξ ξ, ࢠ ൌ െξ ξ, find ࢝ ൌ ࢠ ࢠ ڄ , and graph ࢠ , ࢠ , and ࢝.
࢝ ൌ ൫ξ ξ൯൫െξ ξ൯
ൌ െ െ െ
ൌ െૡ
16. Represent ࢝ ൌ െ as a point in the complex plane.
17. Represent ࢝ as a point in the complex plane. ࢝ ൌ ሺെ ሻ ൌ െૡ
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18. Compare the positions of ࢝ and ࢝ from Problems 10 and 11. Describe what you see. (Hint: Draw a segment from
the origin to each point.)
The points , ࢝, and ࢝ all lie on the same line. The distance from to ࢝ is twice as great as the distance from
to ࢝. The segment to ࢝ is a scaled version of the segment to ࢝, with scale factor .
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Complex Plane Reproducible
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Lesson 6: Complex Numbers as Vectors
Student Outcomes
Students represent complex numbers as vectors.
Students represent complex number addition and subtraction geometrically using vectors.
Lesson Notes
Students studied vectors as directed line segments in Grade 8, and in this lesson, vectors are used to represent complex
numbers in the coordinate plane. This representation presents a geometric interpretation of addition and subtraction of
complex numbers and is needed to make the case in Lesson 15 that when multiplying two complex numbers ݖand ݓ,
the argument of the product is the sum of the arguments: ሺݓݖሻ ൌ ሺݖሻ ሺݓሻ.
The following vocabulary terms from Grade 8 are needed in this lesson:
VECTOR: A vector associated to the directed line segment ሬሬሬሬሬԦ
ܤܣis any directed segment that is congruent to the directed
segment ሬሬሬሬሬԦ
ܤܣusing only translations of the plane.
DIRECTED SEGMENT: A directed segment ሬሬሬሬሬԦ
ܤܣis the line segment ܤܣtogether with a direction given by connecting an initial
point ܣto a terminal point ܤ.
The study of vectors forms a vital part of this course; notation for vectors varies across different contexts and curricula.
Ͷ Ͷ
These materials refer to a vector as ( ܞlowercase, bold, non-italicized) or ۃͶǡ ͷ ۄor in column format, ቀ ቁ or ቂ ቃ.
ͷ ͷ
“Let ܞൌ ۃͶǡͷ ”ۄis used to establish a name for the vector ۃͶǡ ͷۄ.
This curriculum avoids stating ܞൌ ۃͶǡͷ ۄwithout the word let preceding the equation when naming a vector unless it is
absolutely clear from the context that a vector is being named. However, the “ൌ” continues to be used to describe
vector equations, like ܞ ܟൌ ܟ ܞ, as has been done with equations throughout all other grades.
The vector from ܣto ܤis referred to as “vector ሬሬሬሬሬԦ
—” ܤܣnotice, this is a ray with a full arrow. This notation is consistent
with the way vectors are introduced in Grade 8 and is also widely used in postsecondary textbooks to describe both a ray
and a vector depending on the context. To avoid confusion in this curriculum, the context is provided or strongly
implied, so it is clear whether the full arrow indicates a vector or a ray. For example, when referring to a ray from ܣ
passing through ܤ, “ray ሬሬሬሬሬԦ
” ܤܣis used, and when referring to a vector from ܣto ܤ, “vector ሬሬሬሬሬԦ
” ܤܣis used. Students should
be encouraged to think about the context of the problem and not just rely on a hasty inference based on the symbol.
The magnitude of a vector is signified as ԡܞԡ (lowercase, bold, non-italicized).
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Classwork
Opening Exercise (4 minutes)
The Opening Exercise reviews complex number arithmetic. This example is revisited later in the lesson when the
geometric interpretation of complex addition and subtraction using a vector representation of complex numbers is
studied. Students should work on these exercises either individually or in pairs.
Opening Exercise
Perform the indicated arithmetic operations for complex numbers ࢠ ൌ െ and ࢝ ൌ െ െ .
a. ࢠ࢝
ࢠ ࢝ ൌ െ
b. ࢠെ࢝
ࢠ െ ࢝ ൌ െ ૠ
c. ࢠ ࢝
ࢠ ࢝ ൌ െ
d. ࢠ െ ࢠ
ࢠ െ ࢠ ൌ
e. Explain how you add and subtract complex numbers.
Add or subtract the real components and the imaginary components separately.
Discussion (6 minutes)
In Lesson 5, we represented a complex number ܽ ܾ݅ as the
point ሺܽǡ ܾሻ in the coordinate plane. Another way we can
represent a complex number in the coordinate plane is as a
vector. Recall the definition of a vector from Grade 8, which is
that a vector ሬሬሬሬሬԦ
ܤܣis a directed segment from point ܣin the
plane to point ܤ, which we draw as an arrow from point ܣto
point ܤ. Since we can represent a complex number
ݖൌ ܽ ܾ݅ as the point ሺܽǡ ܾሻ in the plane, and we can use a
vector to represent the directed segment from the origin to the
point ሺܽǡ ܾሻ, we can represent a complex number as a vector in
the plane. The vector representing the complex number
ݖൌ െ͵ Ͷ݅ is shown.
The length of a vector ሬሬሬሬሬԦ
ܤܣis the distance from the tail ܣof the
vector to the tip ܤ. For our purposes, the tail is the origin, and
the tip is the point ݖൌ ሺܽ ܾ݅ሻ in the coordinate plane.
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A vector consists of a length and a direction. To get from point ܣto point ܤ, you move a distance ܤܣin the
direction of the vector ሬሬሬሬሬԦ
ܤܣ. So, to move from the origin to point ݖൌ ܽ ܾ݅, we move the length of ܽ ܾ݅ in
the direction of the line from the origin to ሺܽǡ ܾሻ. (This idea is important later in the lesson when we use
vectors to add and subtract complex numbers.)
What is the length of the vector that represents the complex number ݖଵ ൌ െ͵ Ͷ݅?
à Since ඥሺെ͵ െ Ͳሻଶ ሺͶ െ Ͳሻଶ ൌ ξͻ ͳ ൌ ͷ, the length of the vector that represents ݖଵ is ͷ.
What is the length of the vector that represents the complex number ݖଶ ൌ ʹ െ ݅?
à Since ඥሺʹ െ Ͳሻଶ ሺെ െ Ͳሻଶ ൌ ξͶ Ͷͻ ൌ ξͷ͵, the length of the vector that represents ݖଶ is ξͷ͵.
What is the length of the vector that represents the complex number ݖଷ ൌ ܽ ܾ݅?
à Since ඥሺܽ െ Ͳሻଶ ሺܾ െ Ͳሻଶ ൌ ξܽଶ ܾ ଶ , the length of the vector that represents ݖଷ is ξܽଶ ܾ ଶ.
Exercise 1 (8 minutes)
Have students work Exercise 1 in pairs or small groups. Circulate to be sure that students are correctly plotting the
complex numbers in the plane and correctly computing the lengths of the resulting vectors.
Exercises
1. The length of the vector that represents ࢠ ൌ െ ૡ is because ඥ ሺെૡሻ ൌ ξ ൌ .
a. Find at least seven other complex numbers that can be represented as vectors that
have length . Scaffolding:
There are an infinite number of complex numbers that meet this criteria; the most Students struggling to find
obvious are , ૡ, ૡ , , െ ૡ, െૡ , െ, െૡ െ , െ െ ૡ, these values may want to work
െ, ૡ െ , and െૡ . The associated vectors for these numbers are shown in
the sample response for part (b). with the more general formula
ܽଶ ܾ ଶ ൌ ͳͲͲ. Choose either
ܽ or ܾ to create an equation
b. Draw the vectors on the coordinate axes provided below.
they can solve. This also helps
students see the relation to a
circle in part (c).
c. What do you observe about all of these vectors?
Students should observe that the tips of the vectors lie on the circle of radius centered at the origin.
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Discussion (10 minutes)
Proceed slowly through this Discussion, drawing plenty of figures to clarify the process of adding two vectors.
Using vectors, how can we add two complex numbers? We know from the Opening Exercise that if
ݖൌ െͶ ͷ݅ and ݓൌ െͳ െ ʹ݅, ݖ ݓൌ െͷ ͵݅. How could we show this is true using vectors?
à We know that if we think of vectors as a length and a direction, the sum ݖ ݓis the distance we need
to move from the origin in the direction of the vector ݖ ݓto get to the point ݖ ݓ. We can get from
the origin to point ݖ ݓby first moving from the origin to point ݖand then moving from point ݖto
point ݓ.
Using coordinates, this means that to find ݖ ݓ, we do the following:
1. Starting at the origin, move Ͷ units left and ͷ units up to the tip of the vector representing ݖ.
2. From point ݖ, move ͳ unit left and ʹ units down.
3. The resulting point is ݖ ݓ.
Using vectors, we can locate point ݖ ݓby the tip-to-tail method: Translate the vector that represents ݓso
that the tip of ݖis at the same point as the tail of the new vector that represents ݓ. The tip of this new
translated vector is the sum ݖ ݓ. See the sequence of graphs below.
How would we find ݖെ ݓusing vectors?
Scaffolding:
à We could think of ݖെ ݓas ݖ ሺെݓሻ since we already know how to add
For advanced students, discuss
vectors.
how to find ݖെ ݓusing the
Thinking of a vector as a length and a direction, how does െ ݓrelate to ?ݓ original parallelogram; that is,
à The vector െ ݓwould have the same length as ݓbut the opposite ݖെ ݓis the vector from the tip
direction. of ݓto the tip of ݖand then
translated to the origin.
Discuss how this is the same as
subtraction in one dimension.
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Yes. So, we need to add െ ݓto ݖ. To find െݓ, we reverse its direction. See the sequence of graphs below.
In the Opening Exercise, we found that ݖെ ݓൌ െ͵ ݅. Does that agree with our calculation using vectors?
à Yes
Exercises 2–6 (8 minutes)
Have students work on these exercises in pairs or small groups. Scaffolding:
Model an example such as
2. In the Opening Exercise, we computed ࢠ ࢝. Calculate this sum using vectors. ʹ ݖ ݓfor struggling students
before asking them to work on
these exercises.
3. In the Opening Exercise, we also computed ࢠ െ ࢠ. Calculate this sum using vectors.
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4. For the vectors ܝand ܞpictured below, draw the specified sum or difference on the coordinate axes provided.
a. ܝܞ
b. ܞെܝ
c. ܝെ ܞ
d. െ ܝെ ܞ
5. Find the sum of and െ geometrically.
6. Show that ሺૠ ሻ െ ሺ െ ሻ ൌ by representing the complex numbers as vectors.
Closing (4 minutes)
Ask students to write in their journals or notebooks to explain the process of representing a complex number by a vector
and the processes for adding and subtracting two vectors.
Exit Ticket (5 minutes)
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Name Date
Lesson 6: Complex Numbers as Vectors
Exit Ticket
Let ݖൌ െͳ ʹ݅ and ݓൌ ʹ ݅. Find the following, and verify each geometrically by graphing ݖ, ݓ, and each result.
a. ݖݓ
b. ݖെݓ
c. ʹ ݖെ ݓ
d. ݓെݖ
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Exit Ticket Sample Solutions
Let ࢠ ൌ െ and ࢝ ൌ . Find the following, and verify each geometrically by graphing ࢠ, ࢝, and each result.
a. ࢠ࢝
b. ࢠെ࢝
െ
c. ࢠ െ ࢝
െ
d. ࢝െࢠ
െ
Problem Set Sample Solutions
1. Let ࢠ ൌ and ࢝ ൌ െ . Find the following. Express your answers in ࢇ ࢈ form.
a. ࢠ࢝
െ ൌ െ
b. ࢠെ࢝
െ ሺ െ ሻ ൌ െ
ൌ
c. ࢝
ሺ െ ሻ ൌ െ
d. ࢠ ࢝
ሺ ሻ െ ൌ െ
ൌ
e. െ࢝ െ ࢠ
െሺ െ ሻ െ ሺ ሻ ൌ െ െ െ
ൌ െ
f. What is the length of the vector representing ࢠ?
The length of the vector representing ࢠ is ξ ൌ ξ.
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g. What is the length of the vector representing ࢝?
The length of the vector representing ࢝ is ඥ ሺെሻ ൌ ξ.
2. Let ࢛ ൌ , ࢜ ൌ , and ࢝ ൌ െ െ . Find the following. Express your answer in ࢇ ࢈ form, and represent
the result in the plane.
a. ࢛ െ ࢜
െ ሺ ሻ ൌ െ െ
ൌ
b. ࢛ െ ࢝
െ ሺെ െ ሻ ൌ
ൌ ૠ
c. ࢛࢜࢝
െ െ ൌ
d. ࢛െ࢜࢝
െ ሺ ሻ െ െ ൌ െ െ െ െ
ൌ
e. What is the length of the vector representing ࢛?
The length of the vector representing ࢛ is ξ ൌ ξ.
f. What is the length of the vector representing ࢛ െ ࢜ ࢝?
The length of the vector representing ࢛ െ ࢜ ࢝ ൌ ξ ൌ ξ ൌ .
3. Find the sum of െ െ and geometrically.
െ
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4. Show that ሺെ െ ሻ െ ሺെૡ െ ሻ ൌ െ by representing the complex numbers as vectors.
5. Let ࢠ ൌ ࢇ ࢈ , ࢠ ൌ ࢇ ࢈ , and ࢠ ൌ ࢇ ࢈ . Prove the following using algebra or by showing with vectors.
a. ࢠ ࢠ ൌ ࢠ ࢠ
ࢠ ࢠ ൌ ሺࢇ ࢈ ሻ ሺࢇ ࢈ ሻ
ൌ ሺࢇ ࢈ ሻ ሺࢇ ࢈ ሻ
ൌ ࢠ ࢠ
b. ࢠ ሺࢠ ࢠ ሻ ൌ ሺࢠ ࢠ ሻ ࢠ
ࢠ ሺࢠ ࢠ ሻ ൌ ሺࢇ ࢈ ሻ ൫ሺࢇ ࢈ ሻ ሺࢇ ࢈ ሻ൯
ൌ ൫ሺࢇ ࢈ ሻ ሺࢇ ࢈ ሻ൯ ሺࢇ ࢈ ሻ
ൌ ሺࢠ ࢠ ሻ ࢠ
6. Let ࢠ ൌ െ െ and ࢝ ൌ െ .
a. Draw vectors representing ࢠ and ࢝ on the same set of axes.
b. What are the lengths of the vectors representing ࢠ and ࢝?
The length of the vector representing ࢠ is ඥሺെሻ ሺെሻ ൌ ξ ൌ .
The length of the vector representing ࢝ is ඥሺെሻ ൌ ξ ൌ .
c. Find a new vector, ࢛ࢠ , such that ࢛ࢠ is equal to ࢠ divided by the length of the vector representing ࢠ.
െ െ െ
࢛ࢠ ൌ ൌ െ
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A STORY OF FUNCTIONS Lesson 6 M1
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d. Find ࢛࢝ , such that ࢛࢝ is equal to ࢝ divided by the length of the vector representing ࢝.
െ െ
࢛࢝ ൌ ൌ
e. Draw vectors representing ࢛ࢠ and ࢛࢝ on the same set of axes as part (a).
࢝
࢛࢝
ࢠ ࢛ࢠ
f. What are the lengths of the vectors representing ࢛ࢠ and ࢛࢝ ?
ૢ
The length of the vector representing ࢛ࢠ is ටቀെ ቁ ቀെ ቁ ൌ ට ൌ ට ൌ ξ ൌ .
ૢ
The length of the vector representing ࢛࢝ is ටቀെ ቁ ቀ ቁ ൌ ට ൌ ට ൌ ξ ൌ .
g. Compare the vectors representing ࢛ࢠ to ࢠ and ࢛࢝ to ࢝. What do you notice?
The vectors representing ࢛ࢠ and ࢛࢝ are in the same direction as ࢠ and ࢝, respectively, but their lengths are
only .
h. What is the value of ࢛ࢠ times ࢛࢝ ?
ૢ
൬ െ ൰ ൬ ൰ ൌ ൬ ൰ െ ൬ ൰ ൌ ൌ
i. What does your answer to part (h) tell you about the relationship between ࢛ࢠ and ࢛࢝ ?
Since their product is , we know that ࢛ࢠ and ࢛࢝ are reciprocals of each other.
7. Let ࢠ ൌ ࢇ ࢈.
a. Let ࢛ࢠ be represented by the vector in the direction of ࢠ with length . How can you find ࢛ࢠ ? What is the
value of ࢛ࢠ ?
Find the length of ࢠ, and then divide ࢠ by its length.
ࢇ ࢈
࢛ࢠ ൌ
ξࢇ ࢈
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b. Let ࢛࢝ be the complex number that when multiplied by ࢛ࢠ , the product is . What is the value of ࢛࢝ ?
ࢇ᩺െ᩺࢈
From Problem 4, we expect ࢛࢝ ൌ . Multiplying, we get
ටࢇ ᩺᩺࢈
ࢇ ࢈ ࢇ െ ࢈ ࢇ െ ሺ࢈ሻ
ڄ ൌ
ξࢇ ࢈ ξࢇ ࢈ ࢇ ࢈
ࢇ ࢈
ൌ
ࢇ ࢈
ൌ
c. What number could we multiply ࢠ by to get a product of ?
Since we know that ࢛ࢠ is equal to ࢠ divided by the length of ࢠ and that ࢛ࢠ ࢛࢝ ڄൌ , we get
ࢇ െ ࢈ ࢇ െ ࢈
ࢠڄ ڄ ൌࢠ ڄ ൌ
ξࢇ ࢈ ξࢇ ࢈ ࢇ ࢈
ࢇ᩺ି᩺࢈
So, multiplying ࢠ by will result in a product of .
ࢇ ᩺ା᩺࢈
8. Let ࢠ ൌ െ .
a. Draw a picture representing ࢠ ࢝ ൌ ૡ .
ࢠ
ࢠ࢝
b. What is the value of ࢝?
࢝ ൌ െ
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A STORY OF FUNCTIONS Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 7: Complex Number Division
Student Outcomes
Students determine the multiplicative inverse of a complex number.
Students determine the conjugate of a complex number.
Lesson Notes
This is the first day of a two-day lesson on complex number division and applying this knowledge to further questions
about linearity. In this lesson, students find the multiplicative inverse of a complex number. Students see the
connection between the conjugate of a complex number and its multiplicative inverse. This sets the stage for our study
of complex number division in Lesson 8.
Classwork
Opening Exercise (5 minutes)
To get ready for our work in this lesson, we review complex number operations that students have previously studied in
Algebra II, as well as ܽ ܾ݅ form. For our work in Lessons 7 and 8, students need to understand the real and imaginary
components of complex numbers.
Opening Exercise
Perform the indicated operations. Write your answer in ࢇ ࢈ form. Identify the real part of your answer and the
imaginary part of your answer.
a. ሺ ሻ ሺെૠ െ ሻ
െ െ , െ is real, and െ is imaginary.
b. ሺെሻ
, there is no real component, and is imaginary.
c. െ ሺെ ሻ
െ , is real, and െ is imaginary.
d. ሺ െ ሻሺെૠ ሻ
െ , െ is real, and is imaginary.
e. ሺെ െ ሻሺെ ሻ
, is real, and there is no imaginary component.
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Discussion (5 minutes)
In real number arithmetic, what is the multiplicative inverse of ͷ? Scaffolding:
ଵ If students do not see the
à pattern, have them do a
ହ
few additional examples.
How do you know? In other words, what is a multiplicative inverse?
Find the multiplicative
ͳ inverses of െͳ ʹ݅,
à ͷ ቀ ቁ ൌ ͳ; a number times its multiplicative inverse is always equal to ͳ.
ͷ
െʹ െ ݅, ͵ ͳͲ݅, and
The role of the multiplicative inverse is to get back to the identity.
Ͷ െ ݅.
Is there a multiplicative inverse of ݅?
To help students see the
Allow students to really think about this and discuss this among themselves. Then, follow pattern of the
with the questions below. multiplicative inverse,
have them compare the
Is there a complex number ݖsuch that ݅ ڄ ݖൌ ͳ?
inverses of ʹ ͵݅, ʹ െ ͵݅,
ଵ
à െʹ ͵݅, and െʹ െ ͵݅.
For advanced students,
ଵ
Can you find another way to say ? Explain your answer. have them work
independently in pairs
à െ݅ because ݅ ڄെ݅ ൌ െሺ݅ ଶ ሻ ൌ െሺെͳሻ ൌ ͳ. Students could also mention through the examples and
݅ ଷ as a possibility. exercises without leading
questions. Be sure to
In today’s lesson, we look further at the multiplicative inverse of complex
check to make sure their
numbers.
general formula is correct
before they begin the
exercises.
Exercise 1 (2 minutes)
Exercises
1. What is the multiplicative inverse of ?
ൌ ڄൌ ڄሺെሻ ൌ െ
Example 1 (8 minutes)
Students were able to reason what the multiplicative inverse of ݅ was in the Discussion, but the multiplicative inverse of
a complex number in the form of ݅ݍis more difficult to find. In this example, students find the multiplicative inverse
of a complex number by multiplying by a complex number in general form and solving the resulting system of equations.
Does ͵ Ͷ݅ have a multiplicative inverse?
ଵ
à Yes,
ଷାସ
Is there a complex number ݅ݍsuch that ሺ͵ Ͷ݅ሻሺ ݅ݍሻ ൌ ͳ?
Students will have to think about this answer. Give them a couple of minutes, and then proceed with the example.
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Let’s begin by expanding this binomial. What equation do you get?
à ͵ ͵݅ ݍ Ͷ݅ Ͷ݅ ଶ ݍൌ ͳ, so ͵ ͵݅ ݍ Ͷ݅ െ Ͷ ݍൌ ͳ.
Group the real terms and the complex terms, and rewrite the equation.
à ሺ͵ െ Ͷݍሻ ሺ͵ ݍ Ͷሻ݅ ൌ ͳ
Look at the right side of the equation. What do you notice?
à The number ͳ is real, and there is no imaginary component.
What would the real terms have to be equal to? The imaginary terms?
à The real terms must equal ͳ, and the imaginary terms must equal Ͳ.
Set up that system of equations.
à ͵ െ Ͷ ݍൌ ͳ and Ͷ ͵ ݍൌ Ͳ
Solve this system of equations for and ݍ.
͵ Ͷ
à ൌ , ݍൌെ
ʹͷ ʹͷ
ଵ
This suggests that ൌ…?
ଷାସ
ଷ ିସ ଷିସ
à ݅ൌ
ଶହ ଶହ ଶହ
ଷିସ
Does the product of ͵ Ͷ݅ and equal ͳ? Confirm that they are multiplicative inverses by performing
ଶହ
this calculation. Check your work with a neighbor.
Students should confirm that their result was correct.
Note: Students can use their prior knowledge of conjugates and radicals from Algebra II for a simpler method of finding
the inverse for Examples 1 and 2. If students see this connection, allow this, but be sure that students see the
connection and understand the math behind this concept.
Exercise 2 (3 minutes)
2. Find the multiplicative inverse of .
െ
Example 2 (8 minutes)
In Example 2, students look at patterns between the complex numbers and their multiplicative inverses from Example 1
and Exercise 2 and then find the general formula for the multiplicative inverse of any number.
Without doing any work, can you tell me what the multiplicative inverse of ͵ െ Ͷ݅ and ͷ െ ͵݅ would be?
ଷାସ ହାଷ
à and
ଶହ ଷସ
Explain to your neighbor in words how to find the multiplicative inverse of a complex number.
à Change the sign between the real and imaginary terms, and then divide by the sum of the squares of
the coefficients of each term.
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A STORY OF FUNCTIONS Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS
If ݖൌ ܽ ܾ݅, do you remember the name of ݖҧ ൌ ܽ െ ܾ݅ from Algebra II?
à The conjugate
Let’s develop a general formula for the multiplicative inverse of any number of the form ݖൌ ܽ ܾ݅. Using
what we did earlier in this example, what might we do?
à Multiply by another complex number ሺ ݅ݍሻ, and set the product equal to ͳ.
Solve ሺܽ ܾ݅ሻሺ ݅ݍሻ ൌ ͳ. Show each step, and explain your work to your neighbor.
à ܽ ܽ ݅ݍ ܾ ݅ ܾ ݅ݍଶ ൌ ͳ Expand the binomial.
à ܽ ܽ ݅ݍ ܾ ݅െ ܾ ݍൌ ͳ Simplify the equation.
à ܽ െ ܾ ݍൌ ͳ and ܽ ݍ ܾ ൌ Ͳ Set the real terms equal to ͳ and the imaginary terms equal to Ͳ.
ܽ ܾ
à ൌ ʹ and ݍൌ െ ʹ Solve the system of equations for and ݍ.
ܽʹܾ ܽʹܾ
What is the general formula of the multiplicative inverse of ݖൌ ܽ ܾ݅?
ି ି
à ݅ or
మ ାమ మ ାమ మ ାమ
Does this agree with what you discovered earlier in the example?
à Yes
Explain how to find the multiplicative inverse of a complex number using the term conjugate.
à To find the multiplicative inverse of a complex number, ܽ ܾ݅, take the conjugate of the number, and
divide by ܽଶ ܾ ଶ .
Exercises 3–7 (6 minutes)
In these exercises, students practice using the general formula for finding the multiplicative inverse of a number. The
goal is to show students that this formula works for all numbers, real or complex.
State the conjugate of each number, and then using the general formula for the multiplicative inverse of ࢠ ൌ ࢇ ࢈, find
the multiplicative inverse.
3.
െ െ
െ ; ൌ
4. ૠ െ
ૠ െ ሺെሻ ૠ
ૠ ; ൌ
ૠ ሺെሻ
5.
െ െ
െ; ൌ ൌ െ
ሺሻ
6.
െ
; ൌ ൌ
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7. Show that ࢇ ൌ െ ξ and ࢈ ൌ satisfy ൌ .
ࢇା࢈ ࢇ ࢈
Finding a common denominator of the right
side, and then simplifying:
ൌ ൌ
ࢇ ࢈ െ ξ ࢇ ࢈ െ ξ
ሺሻ ൫െ ξ൯ ή
ൌ ൌ
ሺሻ ൫െ ξ൯ ή ξ
ξ
ൌ
ሺെ ξሻ
ξ െ ξ
ൌ ή
൫െ ξ൯ െ ξ
െ
ൌ
൫െ ξ ξ െ ൯
ൌ
൫െ ξ ൯
ൌ
൫ ξ൯
ൌ
൫ ξ൯
ൌ
ξ
The two expressions are equal for the given values of ࢇ and ࢈.
Closing (3 minutes)
Allow students to think about the questions below in pairs, and then pull the class together to wrap up the discussion.
Was it necessary to use the formula for Exercise 6? Explain.
à No. The number ʹ is a real number, so the multiplicative inverse was its reciprocal.
Look at Exercises 3–6. What patterns did you discover in the formats of the real and complex numbers and
their multiplicative inverses?
ି
à For any complex number ݖൌ ܽ ܾ݅, the multiplicative inverse has the format when simplified.
మ ାమ
à This formula works for real and complex numbers, but for real numbers it is easier just to find the
reciprocal.
Exit Ticket (5 minutes)
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A STORY OF FUNCTIONS Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS
Name Date
Lesson 7: Complex Number Division
Exit Ticket
1. Find the multiplicative inverse of ͵ െ ʹ݅. Verify that your solution is correct by confirming that the product of
͵ െ ʹ݅ and its multiplicative inverse is ͳ.
2. What is the conjugate of ͵ െ ʹ݅?
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A STORY OF FUNCTIONS Lesson 7 M1
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Exit Ticket Sample Solutions
1. Find the multiplicative inverse of െ . Verify that your solution is correct by confirming that the product of
െ and its multiplicative inverse is .
If ࢇ ࢈ is the multiplicative inverse of െ , then
ሺ െ ሻሺࢇ ࢈ሻ ൌ ሺ െ ሻሺࢇ ࢈ሻ
ࢇ ࢈ െ ࢇ െ ࢈ ൌ
ࢇ ࢈ െ ࢇ ࢈ ൌ .
ࢇ ࢈ ൌ and ሺ࢈ െ ࢇሻ ൌ , so ࢈ െ ࢇ ൌ .
ࢇൌ , ࢈ ൌ , so the multiplicative inverse ࢇ ࢈ ൌ .
ૢ ૢ
Verification: ሺ െ ሻ ቀ ቁൌ െ െ ൌ ൌ
2. What is the conjugate of െ ?
Problem Set Sample Solutions
Problems 1 and 2 are easy entry problems that allow students to practice operations on complex numbers and the
algebra involved in such operations, including solving systems of equations. These problems also reinforce that complex
numbers have a real component and an imaginary component. Problem 3 is more difficult. Most students should
attempt part (a), but part (b) is optional and sets the stage for the next lesson. All skills practiced in this Problem Set are
essential for success in Lesson 8.
1. State the conjugate of each complex number. Then, find the multiplicative inverse of each number, and verify by
multiplying by ࢇ ࢈ and solving a system of equations.
a. െ
Conjugate:
െሺࢇ ࢈ሻ ൌ
െࢇ െ ࢈ ൌ
െࢇ ࢈ ൌ
െࢇ ൌ ǡ ࢈ ൌ
ࢇ ൌ ǡ ࢈ ൌ
Multiplicative inverse: ൌ
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A STORY OF FUNCTIONS Lesson 7 M1
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b. െ ξ
Conjugate: ൫ ξ൯
൫ െ ξ൯ሺࢇ ࢈ሻ ൌ
ࢇ ࢈ െ ξࢇ െ ξ࢈ ൌ
ࢇ ࢈ െ ξࢇ ξ࢈ ൌ
ࢇ ξ࢈ ൌ ǡ ࢈ െ ξࢇ ൌ
ξ
ࢇൌ ǡ࢈ ൌ
ૡ ૡ
ାξ
Multiplicative Inverse:
ૡ
2. Find the multiplicative inverse of each number, and verify using the general formula to find multiplicative inverses
of numbers of the form ࢠ ൌ ࢇ ࢈.
a.
ൌ െ ൌ െ
ିሺିሻ
Multiplicative inverse: ൌ ൌ
ାሺିሻ
b.
ൌ
ି ૢ
Multiplicative inverse: ൌ ൌ ή ൌ
ቀ ቁ ା ૢ
ඥെ
c.
ξ െ ξ െ
ൌ
ξ ష ξ ξశ ξశ
ିቀ ቁ ା ξା
Multiplicative inverse: ൌ ൌ ൌ ൌ ή ൌ ξ
ξ ష ା
൬ ൰ ାቀ ቁ
d.
ି ି
Multiplicative inverse: ൌ ൌ െ
ሺሻ ାሺିሻ
e. െ
ା ା
Multiplicative inverse: ൌ ൌ
ሺሻ ାሺିሻ
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f.
ି ି
Multiplicative inverse: ൌ ൌ െ
ሺሻ ାሺିሻ
g. െ െ
ିା ିା
Multiplicative inverse: ൌ ൌെ
ሺିሻ ାሺିሻ
h. െ
ିି ିି
Multiplicative inverse: ൌ ൌെ െ
ሺିሻ ାሺሻ
i. ξ
ξି ξି ξ
Multiplicative inverse: ൌ ൌ െ
൫ξ൯ ାሺሻ
j. െ ξ ή
ାξ ାξ ξ
Multiplicative inverse: ൌ ൌ
ሺሻ ା൫ξ൯
k. ξ ξ ή
ξିξ ξିξ ξ ξ
Multiplicative inverse: ൌ ൌ െ
൫ξ൯ ା൫ିξ൯ ૡ ૡ ૡ
3. Given ࢠ ൌ and ࢠ ൌ .
a. Let ࢝ ൌ ࢠ ࢠ ڄ . Find ࢝ and the multiplicative inverse of ࢝.
࢝ ൌ ሺ ሻሺ ሻ ൌ െ
ିି
Multiplicative inverse: ൌെ െ
ା
b. Show that the multiplicative inverse of ࢝ is the same as the product of the multiplicative inverses of ࢠ
and ࢠ .
ି ି
ࢠ ൌ ; Multiplicative inverse: ൌ
ା
ି ି
ࢠ ൌ ; Multiplicative inverse: ൌ
ାૢ
െ െ
ࢠ ࢠ ڄ ൌ ൬ ൰൬ ൰
െ െ െ
ൌ
െ െ
ൌ
ൌെ െ
Lesson 7: Complex Number Division 93
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A STORY OF FUNCTIONS Lesson 8 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Complex Number Division
Student Outcomes
Students determine the modulus and conjugate of a complex number.
Students use the concept of conjugate to divide complex numbers.
Lesson Notes
This is the second day of a two-day lesson on complex number division and applying this knowledge to further questions
about linearity. In Lesson 7, students studied the multiplicative inverse. In this lesson, students study the numerator
and denominator of the multiplicative inverse and their relationship to the conjugate and modulus. The lesson
culminates with complex number division.
Classwork
Opening Exercise (3 minutes)
Students practice using the formula for the multiplicative inverse derived in Lesson 7 as a lead into this lesson.
Opening Exercise
Use the general formula to find the multiplicative inverse of each complex number.
a.
െ
b. െૠ െ
െૠ
c. െ
െ െ
Discussion (2 minutes)
Look at the complex numbers given in the Opening Exercise and the numerators Scaffolding:
of the multiplicative inverses. Do you notice a pattern? Explain. Use a Frayer diagram to define
à The real term is the same in both the original complex number and its conjugate. See Lesson 5 for an
multiplicative inverse, but the imaginary term in the multiplicative example.
inverse is the opposite of the imaginary term in the original complex
number.
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A STORY OF FUNCTIONS Lesson 8 M1
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If the complex number ݖൌ ܽ ܾ݅, what is the numerator of its multiplicative inverse?
à ܽ െ ܾ݅
Features of the multiplicative inverse formula often reappear in complex number arithmetic, so
mathematicians have given these features names. The conjugate of a complex number ܽ ܾ݅ is ܽ െ ܾ݅.
Repeat that with me.
à The conjugate of ܽ ܾ݅ is ܽ െ ܾ݅.
Exercises 1–4 (2 minutes)
Have students quickly complete the exercises individually, and then follow up with the questions below. This would be a
good exercise to do as a rapid white board exchange.
Exercises 1–11
Find the conjugate, and plot the complex number and its conjugate in the complex plane. Label the conjugate with a
prime symbol.
1. :
ᇱ : െ
2. : െ െ
ᇱ : െ
3. : ૠ
ᇱ : ૠ
4. ࡰ:
ࡰᇱ : െ
Discussion (8 minutes)
Does have a complex conjugate? If so, what is it? Explain your answer.
à Yes. ൌ Ͳ݅, so the complex conjugate would be െ Ͳ݅ ൌ .
What is the complex conjugate of Ͷ݅? Explain.
à Ͷ݅ ൌ Ͳ Ͷ݅; the complex conjugate is Ͳ െ Ͷ݅ ൌ െͶ݅. Scaffolding:
If ݖൌ ܽ ܾ݅, then the conjugate of ݖis denoted ݖҧ. That means ݖҧ ൌ ܽ െ ܾ݅. Describe how ͵ Ͷ݅ is
related to ͵ െ Ͷ݅.
What is the geometric effect of taking the conjugate of a complex number?
Describe how െʹ െ ݅ is
à The complex conjugate reflects the complex number across the real axis.
related to െʹ ݅.
What can you say about the conjugate of the conjugate of a complex number?
In the Discussion, instead
à The conjugate of the conjugate is the original number. of using variables, use
numbers.
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Is ݖതതതതതതതത
ݓൌ ݖҧ ݓ
ഥ always true? Explain.
à Yes. Answers will vary. Students could plug in different complex numbers for ݖand ݓand show that
they work or use a general formula argument. If ݖൌ ܽ ܾ݅ and ݓൌ ܿ ݀݅,
ݖ ݓൌ ሺܽ ܿሻ ሺܾ ݀ሻ݅, and ݖതതതതതതതത
ݓൌ ሺܽ ܿሻ െ ሺܾ ݀ሻ݅. ݖҧ ൌ ܽ െ ܾ݅, and ݓ ഥ ൌ ܿ െ ݀݅, so
ഥ ൌ ሺܽ ܿሻ െ ሺܾ ݀ሻ݅. Therefore, തതതതതതതത
ݖҧ ݓ ݖ ݓൌ ݖҧ ݓഥ is always true.
Is ݖതതതതതത
ή ݓൌ ݖҧ ή ݓ
ഥ always true? Explain.
à Yes. Answers will vary. Using the general formula argument: If ݖൌ ܽ ܾ݅ and ݓൌ ܿ ݀݅, then
ݖή ݓൌ ܽܿ ܽ݀݅ ܾܿ݅ െ ܾ݀ ൌ ሺܽܿ െ ܾ݀ሻ ሺܽ݀ ܾܿሻ݅. തതതതതത ݖή ݓൌ ሺܽܿ െ ܾ݀ሻ െ ሺܽ݀ ܾܿሻ݅.
ݖҧ ൌ ܽ െ ܾ݅, and ݓ ഥ ൌ ܽܿ െ ܽ݀݅ െ ܾܿ݅ െ ܾ݀ ൌ ሺܽܿ െ ܾ݀ሻ െ ሺܽ݀ ܾܿሻ݅. Therefore,
ഥ ൌ ܿ െ ݀݅, so ݖҧ ή ݓ
ݖή ݓൌ ݖҧ ή ݓ
തതതതതത ഥ is always true.
Now, let’s look at the denominator of the multiplicative inverse. Remind me how we find the denominator.
à ܽଶ ܾ ଶ , the sum of the squares of the real term and the coefficient of the imaginary term
Does this remind of you something that we have studied?
à The Pythagorean theorem, and if we take the square root,
the distance formula
Mathematicians have given this feature a name, too. The
modulus of a complex number ܽ ܾ݅ is the real number
ξܽଶ ܾ ଶ. Repeat that with me.
à The modulus of a complex number ܽ ܾ݅ is the real
number ξܽଶ ܾ ଶ .
Exercises 5–8 (4 minutes)
Have students quickly complete the exercises individually, and then follow up with the questions below. This would also
be a good exercise to do as a rapid white board exchange.
Find the modulus.
5.
ඥ ൌ ξ ൌ
6. െ െ
ඥሺെሻ ሺെሻ ൌ ξ
7. ૠ
ඥૠ ൌ ξૢ ൌ ૠ
8.
ඥ ሺሻ ൌ ξ ൌ
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Discussion (3 minutes)
If ݖൌ ܽ ܾ݅, then the modulus of ݖis denoted ȁݖȁ. This means ȁݖȁ ൌ ξܽଶ ܾ ଶ.
If ݖൌ ܽ ܾ݅ is a point in the complex plane, what is the geometric interpretation of ȁݖȁ?
à The modulus is the distance of the point from the origin in the complex plane.
The notation for the modulus of a complex number matches the notation for the absolute value of a real
number. Do you think this is a coincidence? If a complex number is real, what can you say about its modulus?
à The modulus is the number.
Explain to your neighbor what you have learned about the conjugate and the modulus of a complex number.
à The conjugate of a complex number ܽ ܾ݅ is ܽ െ ܾ݅; taking the conjugate of a complex number reflects
the number over the real axis.
à The modulus of complex number ܽ ܾ݅ is ξܽଶ ܾ ଶ; the modulus represents the distance from the
origin to the point ܽ ܾ݅ in the complex plane.
Exercises 9–11 (6 minutes)
Students should complete Exercises 9–11 in pairs. For advanced learners, assign all problems. Assign only one problem
to other groups. Bring the class back together to debrief.
Given ࢠ ൌ ࢇ ࢈.
9. Show that for all complex numbers ࢠ, ȁࢠȁ ൌ ȁࢠȁ.
ȁࢠȁ ൌ ȁሺࢇ ࢈ሻȁ ൌ ȁࢇ െ ࢈ȁ ൌ ȁെ࢈ ࢇȁ ൌ ඥሺെ࢈ሻ ࢇ ൌ ඥࢇ ࢈ ൌ ȁࢠȁ
10. Show that for all complex numbers ࢠ, ࢠ ή ࢠത ൌ ȁࢠȁ .
ࢠ ή ࢠത ൌ ሺࢇ ࢈ሻሺࢇ െ ࢈ሻ ൌ ࢇ ࢈
ȁࢠȁ ൌ ቀඥࢇ ࢈ ቁ ൌ ࢇ ࢈
ࢠ ή ࢠത ൌ ȁࢠȁ
ࢠത
11. Explain the following: Every nonzero complex number ࢠ has a multiplicative inverse. It is given by ൌ .
ࢠ ȁࢠȁ
ࢇെ࢈ ࢠഥ
The multiplicative inverse of ࢇ ࢈ ൌ ൌ ȁࢠȁ.
ࢇ ࢈
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Example (5 minutes)
In this example, students divide complex numbers by multiplying the numerator and Scaffolding:
denominator by the conjugate. Do this as a whole-class discussion. For advanced learners,
assign this example
Example without leading questions.
െ Target some groups for
individual instruction.
െ െ
ૢ
In this example, we are going to divide these two complex numbers. Complex number division is different
from real number division, and the quotient also looks different.
To divide complex numbers, we want to make the denominator a real number. We need to multiply the
denominator by a complex number that makes it a real number. Multiply the denominator by its conjugate.
What type of product do you get?
à ሺʹ ͷ݅ሻሺʹ െ ͷ݅ሻ ൌ Ͷ െ ͳͲ݅ ͳͲ݅ െ ʹͷ݅ ଶ ൌ Ͷ ʹͷ ൌ ʹͻ
You get a real number.
The result of multiplying a complex number by its conjugate is always a real number.
ଶି
The goal is to rewrite this expression as an equivalent expression with a denominator that is a real
ଶାହ
number. We now know that we must multiply the denominator by its conjugate. What about the numerator?
What must we multiply the numerator by in order to obtain an equivalent expression?
à We must multiply the numerator by the same expression, ʹ െ ͷ݅.
Perform that operation, and check your answer with a neighbor.
ଶି ଶିହ ସିଵିଵଶାଷ మ ସିଶଶିଷ ିଶିଶଶ
à ή ൌ ൌ ൌ
ଶାହ ଶିହ ସିଶହ మ ସାଶହ ଶଽ
Tell your neighbor how to divide complex numbers.
à Multiply the numerator and denominator by the conjugate of the denominator.
Exercises 12–13 (5 minutes)
Have students complete the exercises and then check answers and explain their work to a neighbor.
Exercises 12–13
Divide.
ା
12.
ିିૠ
െ ૠ െ െ െ െ ૠ
ή ൌ ൌ
െ െ ૠ െ ૠ ૢ
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13.
ି
ૢ ૢ
ή ൌ ൌ
െ ૢ
Closing (2 minutes)
Allow students to think about the questions below in pairs, and then pull the class together to wrap up the discussion.
What is the conjugate of ܽ ܾ݅? What is the geometric effect of this conjugate in the complex plane?
à ܽ െ ܾ݅; the conjugate is a reflection of the complex number across the real axis.
What is the modulus of ܽ ܾ݅? What is the geometric effect of the modulus in the complex plane?
à ξܽଶ ܾ ଶ; the modulus is the distance of the point from the origin in the complex plane.
How is the conjugate used in complex number division?
à Multiply by a ratio in which both the numerator and denominator are the conjugate.
Exit Ticket (5 minutes)
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Name Date
Lesson 8: Complex Number Division
Exit Ticket
1. Given ݖൌ Ͷ െ ͵݅
a. What does ݖҧ mean?
b. What does ݖҧ do to ݖgeometrically?
c. What does ȁݖȁ mean both algebraically and geometrically?
2. Describe how to use the conjugate to divide ʹ െ ݅ by ͵ ʹ݅, and then find the quotient.
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Exit Ticket Sample Solutions
1. Given ࢠ ൌ െ .
a. What does ࢠത mean?
ࢠത means the conjugate of ࢠ, which is .
b. What does ࢠത do to ࢠ geometrically?
ࢠത is the reflection of ࢠ across the real axis.
c. What does ȁࢠȁ mean both algebraically and geometrically?
ȁࢠȁ is a modulus of ࢠ, which is a real number.
ȁࢠȁ is the distance from the point ࢠ ൌ െ to the origin in the complex plane.
ȁࢠȁ ൌ ඥࢇ ࢈
ൌ ඥሺሻ ሺെሻ
ൌ ξ ૢ
ൌ ξ
ൌ
2. Describe how to use the conjugate to divide െ by , and then find the quotient.
When is multiplied by its conjugate of െ , the denominator is a real number, which is necessary.
ି
Multiply by .
ି
െ ሺ െ ሻሺ െ ሻ െ െ െ െ ૠ ૠ
ൌ ൌ ൌ ൌ െ
ሺ ሻሺ െ ሻ ૢ
Problem Set Sample Solutions
Problems 1–3 are easy problems and allow students to practice finding the conjugate and modulus and dividing complex
numbers. Problems 4–6 are more difficult. Students can use examples or a geometrical approach to explain their
reasoning. Problem 5 is a preview of the effect of adding or subtracting complex numbers in terms of geometrical
interpretations. Students need to find and compare the modulus, ݎ , and ߮ in order to come to their assumptions.
1. Let ࢠ ൌ െ and ࢝ ൌ െ . Show that
a. ȁࢠȁ ൌ ȁࢠതȁ
ȁࢠȁ ൌ ඥሺሻ ሺെሻ ൌ ξ ૢ ൌ ξ ൌ
ࢠത ൌ ǡ ȁࢠതȁ ൌ ඥሺሻ ሺሻ ൌ ξ ૢ ൌ ξ ൌ
Therefore, ȁࢠȁ ൌ ȁࢠതȁ.
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b. ቚ ቚ ൌ ȁഥȁ
ࢠ ࢠ
ሺାሻ ା
ൌ ൌ ൌ ൌ ; therefore,
ࢠ ି ሺିሻሺାሻ
ૢ
ቚ ቚ ൌ ටቀ ቁ ቀ ቁ ൌඨ ൌඨ ൌ .
ࢠ ൫൯ ൫൯ ൫൯
Since ȁࢠതȁ ൌ ; therefore, ൌ , which equals ቚ ቚ ൌ .
ȁࢠതȁ ࢠ
c. If ȁࢠȁ ൌ , must it be that ࢠ ൌ ?
Yes. Let ࢠ ൌ ࢇ ࢈, and then ȁࢠȁ ൌ ඥሺࢇሻ ሺ࢈ሻ. If ȁࢠȁ ൌ , it indicates that ඥሺࢇሻ ሺ࢈ሻ ൌ . Since
ሺࢇሻ ሺ࢈ሻ both are positive real numbers, the only values of ࢇ and ࢈ that will make the equation true is
that ࢇ and ࢈ have to be , which means ࢠ ൌ ൌ .
d. Give a specific example to show that ȁࢠ ࢝ȁ usually does not equal ȁࢠȁ ȁ࢝ȁ.
Answers vary, but ࢠ ൌ and ࢝ ൌ െ will work.
ࢠ ࢝ ൌ
ȁࢠ ࢝ȁ ൌ ඥሺሻ ሺሻ ൌ
ȁࢠȁ ȁ࢝ȁ ൌ ඥሺሻ ሺሻ ඥሺሻ ሺെሻ ൌ ξ, which is not equal to .
2. Divide.
ି
a.
ሺିሻሺሻ ା
ൌ or െ െ
ሺሻ ି
ି
b.
ା
ሺିሻሺିሻ ିି ି
ൌ ൌ or െ
ሺାሻሺିሻ ା ૢ ૢ ૢ
ξି
c.
ିିξ
ሺξ െ ሻ൫െ ξ൯ െξ ξ ૠ
ൌ ൌ ൌ
൫െ െ ξ൯൫െ ξ൯ ૠ
3. Prove that ȁࢠ࢝ȁ ൌ ȁࢠȁ ή ȁ࢝ȁ for complex numbers ࢠ and ࢝.
Since ȁࢠȁ ൌ ࢠ ࢠ ڄത; therefore, ȁࢠ࢝ȁ ൌ ሺࢠ࢝ሻሺࢠ࢝
തതതതሻ ൌ ሺࢠ࢝ሻሺࢠത࢝ ഥ ൌ ȁࢠȁ ή ȁ࢝ȁ .
ഥ ሻ ൌ ࢠࢠത࢝࢝
Now we have ȁࢠ࢝ȁ ൌ ȁࢠȁ ή ȁ࢝ȁ ; therefore, ȁࢠ࢝ȁ ൌ ȁࢠȁ ή ȁ࢝ȁ.
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4. Given ࢠ ൌ , ࢝ ൌ .
a. Find ࢠ ࢝, and graph ࢠ, ࢝, and ࢠ ࢝ on the same complex plane. Explain what you discover if you draw
line segments from the origin to those points ࢠ, ࢝, and ࢠ ࢝. Then, draw line segments to connect ࢝ to
ࢠ and ࢠ ࢝ to ࢠ.
ࢠ ࢝ ൌ
Students should discover that the lines form a parallelogram. They then can graphically see that the lengths
of the two sides are greater than the diagonal, ȁࢠ ࢝ȁ ȁࢠȁ ȁ࢝ȁ.
b. Find െ࢝, and graph ࢠ, ࢝, and ࢠ െ ࢝ on the same complex plane. Explain what you discover if you draw line
segments from the origin to those points ࢠ, ࢝, and ࢠ െ ࢝. Then, draw line segments to connect ࢝ to ࢠ െ ࢝
and ࢠ െ ࢝ to ࢠ.
ࢠ െ ࢝ ൌ െ
Students should discover that the lines form a parallelogram. They then can graphically see that the lengths
of the two sides are greater than the diagonal, ȁࢠ െ ࢝ȁ ȁࢠȁ ȁ࢝ȁ.
5. Explain why ȁࢠ ࢝ȁ ȁࢠȁ ȁ࢝ȁ and ȁࢠ െ ࢝ȁ ȁࢠȁ ȁ࢝ȁ geometrically. (Hint: Triangle inequality theorem)
By using Example 5, we can apply the triangle inequality theorem into these two formulas.
Lesson 8: Complex Number Division 103
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^dKZzK&&hEd/KE^
Mathematics Curriculum
PRECALCULUS AND ADVANCED TOPICS ͻ MODULE 1
Topic B
Complex Number Operations and
Transformations
Focus Standards: (+) Find the conjugate of a complex number; use conjugates to find moduli and
quotients of complex numbers.
(+) Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
(+) Represent addition, subtraction, multiplication, and conjugation of complex
numbers geometrically on the complex plane; use properties of this representation for
ଷ
computation. For example, ൫െͳ ξ͵݅൯ ൌ ͺ because ൫െͳ ξ͵݅൯ has a modulus of ʹ
and an argument of ͳʹͲι.
(+) Calculate the distance between numbers in the complex plane as the modulus of the
difference and the midpoint of a segment as the average of the number at its
endpoints.
Instructional Days: 9
Lessons 9–10: The Geometric Effect of Some Complex Arithmetic (P, P) 1
Lessons 11–12: Distance and Complex Numbers (P, E)
Lesson 13: Trigonometry and Complex Numbers (P)
Lesson 14: Discovering the Geometric Effect of Complex Multiplication (E)
Lesson 15: Justifying the Geometric Effect of Complex Multiplication (S)
Lesson 16: Representing Reflections with Transformations (P)
Lesson 17: The Geometric Effect of Multiplying by a Reciprocal (E)
In Topic B, students develop an understanding of the geometric effect of operations on complex numbers. In
Lesson 9, students explore what happens to a point in the complex plane when complex numbers are added
and subtracted, leading to Lesson 10’s study of the effect of multiplication. Students revisit the
1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
104 Topic : Complex Number Operations and Transformations
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idea of linearity in Lesson 10, determining whether given complex functions are linear transformations.
Students discover that some complex functions are linear transformations and others are not. Students
understand that when complex numbers are considered points in the Cartesian plane, complex number
multiplication has the geometric effect of a rotation followed by a dilation in the complex plane. In Lesson 11,
students use the distance and midpoint formulas they studied in Geometry to find the distance of a complex
point from the origin and the midpoint of two complex numbers. Lesson 12 extends this concept as students
play the Leap Frog game, repeatedly finding the midpoints of pairs of complex numbers. They discover that
when starting with three fixed midpoints, a series of moves jumping across midpoints leads back to the
starting point. Students then explore what happens when they start with only one or two fixed midpoints
and find that a series of jumps never returns to the starting point. Students verify these results using the
midpoint formula derived earlier in the lesson. Lesson 13 introduces the modulus and argument and polar
coordinates as students study the geometric effect of complex number multiplication, leading to writing the
complex number in polar form. If a complex number ݖhas argument ߠ and modulusݎ, then it can be written
in polar form, ݖൌݎሺ ሺߠሻ ݅ሺߠሻሻǤStudents explain why complex numbers can be written in either
rectangular or polar form and why the two forms are equivalent. Lessons 14–16 continue the study of
multiplication by complex numbers, leading students to the understanding that the geometric effect of
multiplying by a complex number, ݓ, is a rotation of the argument of ݓfollowed by a dilation with scale
factor the modulus of ݓ. Lesson 17 concludes this topic as students discover that the multiplicative inverse
of a complex number (i.e., its reciprocal) provides the inverse geometric operation, which leads to the
complex conjugate and division of complex numbers. In ƚŚŝƐ ƚŽƉŝĐ , students reason abstractly and
quantitatively about complex numbers and the geometric effects of operations involving complex numbers
previous results to predict rotations and and use dilations produced.
Topic : Complex Number Operations and Transformations 105
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Lesson 9: The Geometric Effect of Some Complex
Arithmetic
Student Outcomes
Students represent addition, subtraction, and conjugation of complex numbers geometrically on the complex
plane.
Lesson Notes
In the last few lessons, students have informally seen the geometric effects of complex conjugates and of multiplying by
݅. This is the first of a two-day lesson in which students further explore the geometric interpretations of complex
arithmetic. This lesson focuses on the geometric effects of adding and subtracting complex numbers.
Classwork
Opening (3 minutes)
Students have had previous exposure to some geometric effects of complex Scaffolding:
numbers. Ask them to answer the following questions and to share their
Use concrete examples such as
responses with a neighbor.
asking students to describe the
Describe the geometric effect of multiplying a complex number by ݅. geometric effect of multiplying
Describe the geometric effect of a complex conjugate. ʹ ݅ by ݅ and ͵ ʹ݅ by ͵ െ ʹ݅.
Help students see the effect of
multiplying by complex numbers by
Discussion (5 minutes) plotting (ʹ ݅ and െͳ ʹ݅) and
(ʹ ݅ and ʹ െ ݅) in the complex
The points in the complex plane are similar to points in the coordinate plane. The
plane and asking how they are
real part of the complex number is represented on the horizontal axis and the
geometrically related.
imaginary part on the vertical axis. This and the next lesson show that complex
arithmetic causes reflections, translations, dilations, and rotations to points in the
complex plane.
Begin the lesson by having students share their responses to the opening questions. As responses are shared, provide a
visual depiction of each effect on the board. Use this as an opportunity to review notation as well.
In coordinate geometry, what would happen to a point ሺݔǡ ݕሻ if we rotated it ͻͲι counterclockwise?
à The point ሺݔǡ ݕሻ would map to ሺെݕǡ ݔሻ.
Describe the geometric effect of multiplying a complex number by ݅.
à Multiplying a complex number by ݅ induces a ͻͲι rotation about the origin.
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What would happen if we continued to multiply by ݅?
à Each time we multiply by ݅ results in another counterclockwise rotation of ͻͲι about the origin; for
example, multiplying by ݅ twice results in a ͳͺͲι rotation about the origin, and multipliyng by ݅ four
times results in a full rotation about the origin.
Describe the geometric effect of taking the complex conjugate.
à A complex conjugate reflects the complex number across the real axis.
Remind students that the notation for the conjugate of ݖis ݖҧ.
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Exercise 1 (5 minutes)
Have students answer this exercise individually and share their responses with a neighbor. Scaffolding:
Then, continue with the following discussion.
Encourage advanced
learners to write the
Exercises
general rule for Exercise 1.
1. Taking the conjugate of a complex number corresponds to reflecting a complex number
If students struggle to
about the real axis. What operation on a complex number induces a reflection across the
imaginary axis? answer the question posed
in Exercise 1, encourage
For a complex number ࢇ ࢈, the reflection across the imaginary axis is െࢇ ࢈.
Alternatively, for a complex number ࢠ, the reflection across the imaginary axis is െࢠത. them to plot a complex
number like െ͵ Ͷ݅ and
to use it to find the
Students may have answered that the reflection of ܽ ܾ݅ across the imaginary axis is reflection.
െܽ ܾ݅. Discuss as a class how to write this in terms of the conjugate of the complex
number.
Is it possible to write െܽ ܾ݅ another way? (Recall that the complex number ݖcan be written as ܽ ܾ݅.)
Begin by factoring out െͳ: െͳሺܽ െ ܾ݅ሻ.
Replace ܽ െ ܾ݅ with ݖҧ: െݖҧ.
Exercises 2–3 (8 minutes)
In this exercise, students explore the geometric effects of addition and subtraction to the points in the complex plane.
Let students work in small groups. Before students begin, ask them to write a conjecture about the effect of adding a
real number (e.g., ʹ) to a complex number.
2. Given the complex numbers ࢝ ൌ െ and ࢠ ൌ െ ,
graph each of the following:
a. ࢝
b. ࢠ
c. ࢝
d. ࢠ
e. ࢝െ
f. ࢠെ
3. Describe in your own words the geometric effect adding or subtracting a real number has on a complex number.
Adding a real number to a complex number shifts the point to the right on the real (horizontal) axis, while
subtracting a real number shifts the point to the left.
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When students have finished the exercise, confirm as a class the answer to Exercise 3.
Did your conjecture match the answer to Exercise 3?
à Answers will vary.
Some students may no doubt have guessed that adding a positive real value (i.e., ݓ ʹ) to the complex number would
shift the point vertically instead of horizontally. They may be confusing the translation of a function, such as ݂ሺݔሻ ൌ ݔଶ ,
with that of a complex number. Make clear that even though comparisons are made between the complex and
coordinate planes, the geometric effects are different. Use the following discussion points to clarify.
What is the effect of adding a constant to a function like ݂ሺݔሻ ൌ ݔଶ ? (For example, ݂ሺݔሻ ൌ ݔଶ ʹ.)
à The graph of the parabola would shift upward ʹ units.
How does this differ from adding the real number ʹ to a complex number?
à The point representing the complex number would shift two units to the right, not vertically like the
function.
Exercises 4–5 (5 minutes)
Students continue to explore the geometric effects of addition and subtraction to the points in the complex plane. Let
students work in small groups.
4. Given the complex numbers ࢝ ൌ െ and
ࢠ ൌ െ , graph each of the following:
a. ࢝
b. ࢠ
c. ࢝
d. ࢠ
e. ࢝ െ
f. ࢠ െ
5. Describe in your own words the geometric effect adding or subtracting an imaginary number has on a complex
number.
Adding an imaginary number to a complex number shifts the point up the imaginary (vertical) axis, while subtracting
an imaginary number shifts the point down.
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Discussion (5 minutes)
Now that the class has explored the effect of adding and subtracting real and imaginary parts to a complex number,
bring both concepts together.
Given the complex numbers ݖൌ െ ݅ and ݓൌ ʹ ͷ݅, how would you describe the translation of the point ݖ
compared to ݖ ?ݓ
à The point ݖwould shift ʹ units to the right and ͷ units up.
Represent the translation on the complex plane, and point out that a right triangle is formed. Encourage students to
think about how to describe the translation other than simply stating that the point shifts left/right or up/down.
Note: At this time, do not explicitly state to students that the distance between the complex numbers is the modulus of
the difference, as that is covered in a later lesson.
In what other way could we describe or quantify the relationship between ݖand ݖ ?ݓ
à The distance between the two points. We could use the Pythagorean theorem to determine the missing
side of the right triangle.
Example 1 (6 minutes)
Work through this example as a whole-class discussion. Encourage advanced learners to attempt the whole problem on
their own.
Example
Given the complex number ࢠ, find a complex number ࢝ such that ࢠ ࢝ is shifted ξ units in a southwest direction.
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Begin by plotting the complex number. What does it mean
for the point to be shifted in a southwest direction?
à The point shifts to the left and down the same
number of units.
A right triangle is formed. What are the values of the legs
and the hypotenuse?
à The legs are both ݔǡ and the hypotenuse is ξʹ.
Give students an opportunity to solve for ݔon their own and use the ξʹ
information to determine the complex number ݓ.
ଶ
à ݔଶ ݔଶ ൌ ൫ξʹ൯
à ʹ ݔଶ ൌ ʹ, so ݔൌ ͳ.
à Since the point was shifted ͳ unit down and ͳ unit to the left, the complex number must be െͳ െ ݅.
Closing (3 minutes)
Have students summarize the key ideas of the lesson in writing or by talking to a neighbor. Take this opportunity to
informally assess student understanding. The Lesson Summary provides some of the key ideas from the lesson.
Lesson Summary
The conjugate, ࢠത, of a complex number ࢠ reflects the point across the real axis.
The negative conjugate, െࢠത, of a complex number ࢠ reflects the point across the imaginary axis.
Adding or subtracting a real number to a complex number shifts the point left or right on the real
(horizontal) axis.
Adding or subtracting an imaginary number to a complex number shifts the point up or down on the
imaginary (vertical) axis.
Exit Ticket (5 minutes)
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Name Date
Lesson 9: The Geometric Effect of Some Complex Arithmetic
Exit Ticket
1. Given ݖൌ ͵ ʹ݅ and ݓൌ െʹ െ ݅, plot the following
in the complex plane:
a. ݖ
b. ݓ
c. ݖെʹ
d. ݓ ͵݅
e. ݓݖ
2. Given ݖൌ ܽ ܾ݅, what complex number represents the reflection of ݖabout the imaginary axis? Give one example
to show why.
3. What is the geometric effect of ܶሺݖሻ ൌ ݖ ሺͶ െ ʹ݅ሻ?
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Exit Ticket Sample Solutions
1. Given ࢠ ൌ and ࢝ ൌ െ െ , plot the following in the complex plane:
a. ࢠ
b. ࢝
c. ࢠെ
d. ࢝
e. ࢝ࢠ
2. Given ࢠ ൌ ࢇ ࢈, what complex
number represents the reflection of ࢠ
about the imaginary axis? Give one
example to show why.
െࢠത, the negative conjugate of ࢠ. For
example, ࢠ ൌ ,
െࢠത ൌ െሺ െ ሻ ൌ െ , which is reflected about the imaginary axis.
3. What is the geometric effect of ࢀሺࢠሻ ൌ ࢠ ሺ െ ሻ?
ࢀሺࢠሻ shifts units to the right and the units downward.
Problem Set Sample Solutions
1. Given the complex numbers ࢝ ൌ െ and ࢠ ൌ െ , graph each of the following:
a. ࢝െ
࢝ െ ൌ െ െ ൌ െ
b. ࢠ
ࢠ ൌ െ ൌ െ
c. ࢝
࢝ ൌ െ ൌ െ
d. ࢠ െ
ࢠ െ ൌ െ െ ൌ െ െ
e. ࢝ࢠ
࢝ ࢠ ൌ െ ሺെ ሻ ൌ െ െ
f. ࢠെ࢝
ࢠ െ ࢝ ൌ െ െ ሺ െ ሻ ൌ െ
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2. Let ࢠ ൌ െ . Find ࢝ for each case.
a. ࢠ is a ૢι counterclockwise rotation about the origin of ࢝.
ࢠ െ
࢝ ή ൌ ࢠ; therefore, ࢝ ൌ ൌ ൌ ൌ െ െ .
െ
b. ࢠ is reflected about the imaginary axis from ࢝.
࢝ ൌ െࢠത; therefore, ࢝ ൌ െሺ ሻ ൌ െ െ .
c. ࢠ is reflected about the real axis from ࢝.
࢝ ൌ ࢠത; therefore, ࢝ ൌ .
3. Let ࢠ ൌ െ ,࢝ ൌ െ . Simplify the following expressions.
a. ࢠ࢝
ഥ
ࢠ࢝
ഥ ൌ െ ൌ
b. ȁ࢝ െ ࢠതȁ
ȁ࢝ െ ࢠതȁ ൌ ȁ െ െ ሺെ െ ሻȁ ൌ ȁ െ ȁ ൌ ȁ ȁ ൌ ඥሺሻ ሺሻ ൌ ξ
c. ࢠ െ ࢝
ࢠ െ ࢝ ൌ െ െ ሺ െ ሻ ൌ െ െ ൌ െ ૠ
ࢠ
d.
࢝
ࢠ െ ሺെ ሻሺ ሻ െ ૠ െ ૠ
ൌ ൌ ൌ ൌ
࢝ െ ሺ െ ሻሺ ሻ ૠ ૠ
4. Given the complex number ࢠ, find a complex number ࢝ where ࢠ ࢝ is shifted:
a. ξ units in a northeast direction.
࢞ ࢞ ൌ ൫ξ൯ , ࢞ ൌ ૡ, ࢞ ൌ േ. Therefore, ࢝ ൌ .
b. ξ units in a southeast direction.
࢞ ࢞ ൌ ൫ξ൯ , ࢞ ൌ , ࢞ ൌ േ. Therefore, ࢝ ൌ െ .
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Lesson 10: The Geometric Effect of Some Complex
Arithmetic
Student Outcomes
Students represent multiplication of complex numbers geometrically on the complex plane.
Lesson Notes
This is the second of a two-day lesson in which students continue to explore the geometric interpretations of complex
arithmetic. In Lesson 9, students studied the geometric effects of adding and subtracting complex numbers. Lesson 10
focuses on multiplication of complex numbers and the geometric effect. Students revisit the concept of linearity in this
lesson. Students need graph paper for each exercise and example.
Classwork
Opening Exercises (8 minutes)
Lesson 8 introduced the geometric effect of adding complex numbers. This opening
Scaffolding:
exercise helps students solidify that concept as well as the geometric effect of multiplying
by ݅ and taking the complex conjugate. Students need a firm grasp of all of these topics to Provide graphs with grid lines
understand the geometric effect of multiplying by a complex number. Students should or with original points or points
work in small groups or pairs. Each student needs a piece of graph paper. and images plotted for
students with eye-hand or
Opening Exercises
spatial difficulties.
1. Given ࢠ ൌ െ , plot and label the following, and describe the geometric effect of the operation.
a. ࢠ
units on the real axis to the right of the origin
and units on the imaginary axis below the
origin
b. ࢠെ
units on the real axis to the left of ࢠ
c. ࢠ
units on the imaginary axis above ࢠ
d. ࢠ ሺെ ሻ
units on the real axis to the left of ࢠ and units
on the imaginary axis above ࢠ
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2. Describe the geometric effect of the following:
a. Multiplying by
A ૢι rotation about the origin
b. Taking the complex conjugate
Reflects the imaginary number across the real axis
c. What operation reflects a complex number across the imaginary axis?
The opposite of the conjugate (െࢠത) reflects a complex number across the imaginary axis.
Example 1 (8 minutes)
In Example 1, students multiply complex numbers by a constant to discover the geometric Scaffolding:
effect. Allow students to work in pairs or in small groups. We confirm students’
When confirming dilations
conjectures that multiplying a complex number by a constant creates a dilation with the
in Example 1, assign
constant as the scale factor by calculating the length of the segment from the point to the
struggling students with
origin before and after transformation. Students need graph paper.
points on the axes or with
coefficients of ͳ and a
Example 1
constant that is a small
Plot the given points, and then plot the image ࡸሺࢠሻ ൌ ࢠ. whole number.
a. ࢠ ൌ Assign advanced groups
ࢠ ൌ points with larger and/or
negative coefficients and
b. ࢠ ൌ
fractional or decimal
constants.
ࢠ ൌ
c. ࢠ ൌ
ࢠ ൌ
d. ࢠ ൌ െ
ࢠ ൌ െૡ
e. ࢠ ൌ െ
ࢠ ൌ െ
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Algebraically, how did ܮሺݖሻ ൌ ʹ ݖaffect ?ݖ
à The coefficients doubled.
Geometrically, how did ܮሺݖሻ ൌ ʹ ݖaffect ?ݖ
à There was a dilation with scale factor ʹ.
Draw a segment from the origin to one point and its image. Confirm your conjecture by finding the length of
the segments.
Assign different groups different pairs of points so as a class, all pairs are checked. Students confirm that every segment
from the origin to the image is double the length of the segment from the origin to the original point.
Do you think the same would be true if we multiplied by a constant other than ʹ? Confirm your answer.
à This is true for any constant.
Assign different groups different points by which to multiply different constants, and then have groups report their
findings. This would be a good activity to use for differentiating instruction.
Exercises 1–7 (6 minutes)
Students already know that multiplying by ݅ produces a ͻͲι counterclockwise rotation about the origin. This lesson
solidifies that concept in preparation for Example 2 and future lessons. This exercise can be completed in groups or
pairs. Students need graph paper.
Exercises
Plot the given points, and then plot the image ࡸሺࢠሻ ൌ ࢠ.
1. ࢠ ൌ
ࢠ ൌ
2. ࢠ ൌ
ࢠ ൌ െ
3. ࢠ ൌ
ࢠ ൌ ሺ ሻ ൌ െ ൌ െ
4. ࢠ ൌ െ
ࢠ ൌ ሺെ ሻ ൌ െ െ ൌ െ െ
5. ࢠ ൌ Ȃ
ࢠ ൌ ሺ െ ሻ ൌ ൌ
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6. What is the geometric effect of the transformation? Confirm your conjecture using the slope of the segment joining
the origin to the point and then to its image.
Multiplying by rotates the point ૢι counterclockwise about the origin. This is confirmed because the slopes of the
segments joining the origin and the original points and the slopes of the segments joining the origin and the image
of those points are opposite reciprocals, which means the segments are perpendicular.
7. Is ࡸሺࢠሻ a linear transformation? Explain how you know.
Yes, ࡸሺࢠ ࢝ሻ ൌ ࡸሺࢠሻ ࡸሺ࢝ሻ and ࡸሺࢠሻ ൌ ࡸሺࢠሻ.
Example 2 (15 minutes)
Scaffolding:
Students have discovered the geometric effect of multiplying a complex number by a Allow advanced learners to do
constant and by ݅. In Example 2, we multiply by a complex number with a real and an Example 2 on their own and
imaginary part. Students may struggle with this, but in this lesson, we are just looking at then change ܮሺݖሻ ൌ ሺ͵ Ͷ݅ሻݖ,
the transformations graphically and letting students think about the geometric effect. We or give them this problem
want students to understand that there is a rotation and a dilation, but the exact effect is instead of the one listed in the
studied in Lessons 10–12. Students should work in small groups or pairs and need graph example.
paper.
Example 2
Describe the geometric effect of ࡸሺࢠሻ ൌ ሺ ሻࢠ given the following. Plot the images on graph paper, and describe the
geometric effect in words.
a. ࢠ ൌ
ࡸሺࢠ ሻ ൌ ሺ ሻሺሻ ൌ , no change
b. ࢠ ൌ
ࡸሺࢠ ሻ ൌ ሺ ሻ ൌ െ ൌ െ , a ૢι
counterclockwise rotation about the origin
c. ࢠ ൌ
ࡸሺࢠ ሻ ൌ ሺ ሻሺ ሻ ൌ െ ൌ , a ι
counterclockwise rotation about the origin and a
dilation with a scale factor of ξ
d. ࢠ ൌ
ࡸሺࢠ ሻ ൌ ሺ ሻሺ ሻ ൌ െ ൌ െ ,
a clockwise rotation about the origin of some angle
measure ࣂ and a dilation with a scale factor greater
than
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What was the geometric result of multiplying the complex number by ͳ?
à There was no change.
Are you surprised by this result?
à No, multiplying by ͳ always results in the number you started with. It is the multiplicative identity.
What was the geometric result of multiplying by ݅?
à The result was a ͻͲι counterclockwise rotation about the origin. The point was reflected across the
imaginary axis.
Did this result surprise you?
à No, multiplying by ݅ always results in a ͻͲι counterclockwise rotation about the origin, but it is not
always a reflection across the imaginary axis.
What would always create a reflection across the real axis?
à Taking the complex conjugate, ͳ െ ݅
What would create a reflection across the imaginary axis?
à The opposite of the complex conjugate, െͳ ݅
What was the geometric result of multiplying by ͳ ݅?
à There was a clockwise rotation and a dilation.
Can you guess how many degrees the point rotated? Explain.
It is ok if students do not yet fully grasp the degree of rotation; they study this in
great detail in future lessons.
à Ͷͷι. The angle of the segment joining the origin to the original
point is Ͷͷι because if you draw a right triangle, each leg is ͳ
unit, so the triangle is a special Ͷͷι–Ͷͷι–ͻͲι triangle. The
image is on the ݕ-axis, so that is a ͻͲι. If the original point
was at a Ͷͷι angle and the reflection is at an angle of ͻͲι, the
angle that the point rotated was Ͷͷι.
What was the scale factor of the dilation? Explain.
ξʹ
à The segment joining the origin and the original point was the
hypotenuse of a Ͷͷι–Ͷͷι–ͻͲι with legs of length ͳ unit, so it
was ξʹ units in length by the Pythagorean theorem. The
segment joining the origin and the image is ʹ units in length.
This means that the original length was multiplied by a scale
factor of ξʹ to get the length of the segment containing the
image.
What was the result of multiplying by Ͷ ݅?
à A clockwise rotation and a dilation greater than ͳ
Can you determine the angle of rotation and the dilation scale factor?
Students are probably not able to determine the answers for this result, but this is studied in detail later. It can be left
for them to think about.
à The scale factor is ξʹ, and ߠ ൌ ߨ െ ିଵ ሺͷሻ ൌ ͳǤͺ ൌ ͳͲͳǤ͵ͳι.
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Closing (3 minutes)
Have students explain the following questions to a neighbor, and then bring the class back together for a debrief.
Encourage students to draw diagrams to support their responses.
Explain the geometric effect of multiplying a complex number by the following:
1. ͳ
à There is no change. The number stays where it was.
2. ݅
à This produces a counterclockwise rotation of ͻͲι about the origin.
3. ܽ ܾ݅
à The result is a rotation about the origin and a dilation.
Exit Ticket (5 minutes)
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Name Date
Lesson 10: The Geometric Effect of Some Complex Arithmetic
Exit Ticket
1. Given ܶሺݖሻ ൌ ݖ, describe the geometric effect of the following:
a. ܶሺݖሻ ൌ ͷݖ
ݖ
b. ܶሺݖሻ ൌ
ʹ
c. ܶሺݖሻ ൌ ݅ ή ݖ
2. If ݖൌ െʹ ͵݅ is the result of a ͻͲι counterclockwise rotation
about the origin from ݓ, find ݓ. Plot ݖand ݓin the complex
plane.
3. Explain the geometric effect of ݖif you multiply ݖby ݓ, where
ݓൌ ͳ ݅.
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Exit Ticket Sample Solutions
1. Given ࢀሺࢠሻ ൌ ࢠ, describe the geometric effect of the following:
a. ࢀሺࢠሻ ൌ ࢠ
It has a dilation with a scale factor of .
ࢠ
b. ࢀሺࢠሻ ൌ
It has a dilation with a scale factor of .
c. ࢀሺࢠሻ ൌ ࢠ
It has a ૢι counterclockwise rotation about the origin.
2. If ࢠ ൌ െ is the result of a ૢι counterclockwise rotation about the origin from ࢝, find ࢝. Plot ࢠ and ࢝ in the
complex plane.
ࢠൌ ή࢝
െ ൌ ή ࢝
െ
࢝ ൌ
ሺെ ሻ ή
࢝ൌ
ή
െ െ
࢝ൌ
െ
࢝ ൌ
3. Explain the geometric effect of ࢠ if you multiply ࢠ by ࢝, where ࢝ ൌ .
It has a ι counterclockwise rotation about the origin and a dilation with a scale factor of ξ.
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Problem Set Sample Solutions
1. Let ࢠ ൌ െ . Simplify the following, and describe the geometric effect of the operation. Plot the result in the
complex plane.
a. ࢠ െ
െ െ ൌ െ െ
ࢠ is shifted units to the right and units
downward.
b. ࢠ െ െ
െ െ െ ൌ െ െ
ࢠ is shifted units to the left and units
downward.
c. ࢠ െ ሺ െ ሻ
െ െ ሺ െ ሻ ൌ െ െ ൌ െ
ࢠ is shifted units to the left and units upward.
d. ࢠ
ሺെ ሻ ൌ െૡ
ࢠ has a dilation with a scale factor of .
ࢠ
e.
െ
ൌ െ
ࢠ has a dilation with a scale factor of .
2. Let ࢠ ൌ . Simplify the following, and describe the geometric effect of the operation.
a. ࢠ
ࢠ ൌ െ
ࢠ is rotated ૢιcounterclockwise.
b. ࢠ
ࢠ ൌ ሺെሻሺ ሻ ൌ െ െ
OR
ࢠ ൌ ή ή ሺ ሻ ൌ ሺ െ ሻ ൌ െ െ
ࢠ is rotated ૡι counterclockwise.
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c. ࢠത
ࢠത ൌ െ
ࢠ is reflected about the real axis.
d. െࢠത
െࢠത ൌ െሺ െ ሻ ൌ െ
ࢠ is reflected about the imaginary axis.
e. ࢠത
ࢠത ൌ ሺ െ ሻ ൌ
ࢠ is reflected about the real axis first and then is rotated ૢι counterclockwise.
f. ࢠ
ࢠ ൌ ሺ ሻ ൌ െ
ࢠ is rotated ૢι counterclockwise and then has a dilation with a scale factor of .
g. ࢠ െ
ࢠ െ ൌ ሺ ሻ െ ൌ െ െ ൌ െ
ࢠ is rotated ૢι counterclockwise first and then shifted units to the right and units downward.
3. Simplify the following expressions.
a. ሺ െ ሻሺ െ ሻ
െ
b. ሺെ ሻሺെ െ ሻ
c. ሺ ሻ
d. ሺ ሻ (Hint: ࢈ ൌ ሺ࢈ ሻ )
ሺ ሻ ൌ ሺሺ ሻ ሻ ൌ ሺሻ ൌ
െ
e.
െ
ሺെ ሻሺ ሻ െ െ െ െ െ െሺ െ ሻ
ൌ ൌ ൌ െ or ൌ ൌ െ
ሺ െ ሻሺ ሻ െ െ
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࢞
f. , provided ࢞ ്
࢞െ
࢞ ሺ࢞ ሻሺ࢞ െ ሻ
ൌ ൌ ࢞
࢞ െ ࢞ െ
4. Given ࢠ ൌ , describe the geometric effect of the following. Plot the result.
a. ࢠሺ ሻ
ࢠ is rotated ι counterclockwise and has a scale factor ξ
multiplying to ȁࢠȁ.
ඥ
b. ࢠ൬ ൰
ࢠ is rotated ι counterclockwise and has a scale factor
multiplying to ȁࢠȁ.
5. We learned that multiplying by produces a ૢι counterclockwise rotation about the origin. What do we need to
multiply by to produce a ૢι clockwise rotation about the origin?
࢝
We need to multiply by . If ࢝ ൌ ࢠ (࢝ is a ૢι counterclockwise rotation about the origin from ࢠ), ࢠ ൌ , which
means that if we divide ࢝ by , we will get to ࢠ, which will create a ૢι clockwise rotation about the origin from ࢝.
Another response: ૢι clockwise is ૠι counterclockwise, so you could multiply by to map ሺࢇǡ ࢈ሻ to ሺ࢈ǡ െࢇሻ.
6. Given ࢠ is a complex number ࢇ ࢈, determine if ࡸሺࢠሻ is a linear transformation. Explain why or why not.
a. ࡸሺࢠሻ ൌ ࢠ
Yes. ࡸሺࢠሻ ൌ െࢠ, so ࡸሺࢠ ࢝ሻ ൌ െࢠ െ࢝ and ࡸሺࢠሻ ࡸሺ࢝ሻ ൌ െࢠ െ࢝.
ࡸሺࢠሻ ൌ ሺെࢠሻ and ࡸሺࢠሻ ൌ െሺࢠሻ.
Since ࡸሺࢠ ࢝ሻ ൌ ࡸሺࢠሻ ࡸሺ࢝ሻ and ࡸሺࢠሻ ൌ ࡸሺࢠሻ, the function is a linear transformation.
b. ࡸሺࢠሻ ൌ ࢠ
No. ࡸሺࢠ ࢝ሻ ൌ ࢠ ࢝ and ࡸሺࢠሻ ࡸሺ࢝ሻ ൌ ࢠ ࢝ . ࡸሺࢠሻ ൌ ሺࢠ ሻ and ࡸሺࢠሻ ൌ ࢠ .
Since ࡸሺࢠ ࢝ሻ ൌ ࡸሺࢠሻ ࡸሺ࢝ሻ and ࡸሺࢠሻ ് ࡸሺࢠሻ, the function is not a linear transformation.
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Lesson 11: Distance and Complex Numbers
Student Outcomes
Students calculate distances between complex numbers as the modulus of the difference.
Students calculate the midpoint of a segment as the average of the numbers at its endpoints.
Lesson Notes
In Topic A, students saw that complex numbers have geometric interpretations associated with them since points in the
complex plane seem analogous to points in the coordinate plane. In Lesson 6, students considered complex numbers as
vectors and learned to add them by the tip-to-tail method. In Lessons 8 and 9, students explored the idea that every
complex operation must have some geometric interpretation, eventually coming to the realization that complex addition
and subtraction have the geometric effect of performing a translation to points in the complex plane. The geometric
interpretation of complex multiplication was left unresolved as students realized it was not readily obvious. Later in the
module, students continue to explore the question, “What is the geometric action of multiplication by a complex
number ݓon all the points in the complex plane?” To understand this, students first explore the connection between
geometry and complex numbers. The coordinate geometry studied in Geometry was about points in the coordinate
plane, whereas now the thinking is about complex numbers in the complex plane.
Classwork
Opening Exercise (5 minutes)
Give students time to work independently on the Opening Exercise before discussing as a class.
Opening Exercise
a. Plot the complex number ࢠ ൌ on the complex plane. Plot the ordered pair ሺǡ ሻ on the coordinate
plane.
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b. In what way are complex numbers points?
When a complex number is plotted on a complex plane, it looks just like the corresponding ordered pair
plotted on a coordinate plane. For example, when is plotted on the complex plane, it looks exactly the
same as when the ordered pair ሺǡ ሻ is plotted on a coordinate plane. We can interchangeably think of a
complex number ࢞ ࢟ in the complex plane as a point ሺ࢞ǡ ࢟ሻ in the coordinate plane, and vice versa.
c. What point on the coordinate plane corresponds to the complex numberെ ૡ?
ሺെǡ ૡሻ
d. What complex number corresponds to the point located at coordinateሺǡ െૢሻ?
െ ૢ or െૢ
Discussion (7 minutes)
Draw the following on the board.
When we say that complex numbers are points in the complex plane, what do we really mean?
à When a complex number ݔ ݅ݕis plotted on the complex plane, it looks exactly the same as when the
ordered pair ሺݔǡ ݕሻ is plotted on a coordinate plane.
What does this mean in terms of connecting the ideas we learned in Geometry to complex numbers?
à Since we can interchangeably think of a complex number ݔ ݅ݕin the complex plane as a point ሺݔǡ ݕሻ
in the coordinate plane, and vice versa, all the work we did in Geometry can be translated into the
language of complex numbers, and vice versa. Therefore, any work we do with complex numbers
should translate back to results from Geometry.
In Geometry, it did not make sense to add two points together. If ܣሺݔଵ ǡ ݕଵ ሻ Scaffolding:
and ܤሺݔଶ ǡ ݕଶ ሻare points, what would the geometric meaning of ܣ ܤbe?
If needed, provide students with
à It does not seem to have any meaning. an example using specific
It does make sense to add two complex numbers together. If ܣൌ ݔଵ ݕଵ ݅ numbers rather than general
and ܤൌ ݔଶ ݕଶ ݅, then what is ܣ ?ܤ parameters.
à ሺݔଵ ݔଶ ሻ ሺݕଵ +ݕଶ ሻ݅ If ܣൌ ͵ Ͷ and
What is the geometric effect of transforming ܣwith the function ܤൌ െͳ ݅, then what is
ܣ ?ܤ
݂ሺݖሻ ൌ ݖ ܤfor constant complex number ?ܤ
à ሺ͵ ሺെͳሻሻ ሺͶ ሻ݅
à Applying the transformation to ܣhas the geometric effect of
performing a translation. So, adding ܤto ܣwill shift point ܣright ݔଶ or ʹ ͳͲ݅
units and up ݕଶ .
If we view points in the plane as complex numbers, then we can add points in geometry.
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Exercise 1 (3 minutes)
Have students work on Exercise 1 independently and then share results with a partner. If students do not recall how to
find the midpoint, have them draw the line segment and locate the midpoint from the graph rather than providing them
with the midpoint formula.
Exercise 1
The endpoints of തതതത തതതത?
are ሺǡ ૡሻ and ሺെǡ ሻ. What is the midpoint of
The midpoint of തതതത
is ቀെǡ ቁ.
How do you find the midpoint of തതതത
? ܤܣ
à You find the average of the ݔ-coordinates and the ݕ-coordinates to find the halfway point.
ݔͳݕ ʹݔʹݕ
In Geometry, we learned that for two points ܣሺݔଵ ǡ ݕଵ ሻ and ܤሺݔଶ ǡ ݕଶ ሻǡ the midpoint of തതതത
ܤܣis ቀ ͳ ǡ ቁ.
ʹ ʹ
Now, view these points as complex numbers: ܣൌ ݔଵ ݕଵ ݅ and ܤൌ ݔଶ ݕଶ ݅Ǥ
Exercise 2 (7 minutes)
Allow students time to work on part (a) independently and then share results Scaffolding:
with a partner. Have them work in partners on part (b) before discussing as a
Provide visual learners with a graph
class.
on the complex plane.
Exercise 2
a. What is the midpoint of ൌ ૡ and ൌ െ ?
െ ሺૡ ሻ
ࡹൌ ൌ െ
b. Using ൌ ࢞ ࢟ and ൌ ࢞ ࢟ , show that, in general, the
ା
midpoint of points and is , the arithmetic average of the two
numbers.
࢞ ࢞ ࢟ ࢟
ࡹൌ If students need additional
࢞ ࢟ ࢞ ࢟ practice, use this example before
ൌ
completing part (b) of Exercise 2:
ൌ What is the midpoint of
ܣൌ െ͵ ʹ݅ and ܤൌ ͳʹ െ ͳͲ݅?
ͻ
à ܯൌ െ Ͷ݅
ʹ
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Exercise 3 (5 minutes)
As with Exercise 1, have students work on Exercise 3 independently and then share results with a partner. If students do
not recall how to find the length of a line segment, have them draw the line segment, and instruct them to think of it as
the hypotenuse of a right triangle rather than providing them with the distance formula.
Exercise 3
The endpoints of തതതത തതതത?
are ሺǡ ૡሻ and ሺെǡ ሻ. What is the length of
തതതത is ξ.
The length of
How do you find the length of തതതത
? ܤܣ
à You use the Pythagorean theorem, which can be written as ܤܣൌ ඥሺݔଶ െ ݔଵ ሻଶ ሺݕଶ െ ݕଵ ሻଶ for two
points ܣሺݔଵ ǡ ݕଵ ሻ and ܤሺݔଶ ǡ ݕଶ ሻǤ
As we did previously, view these points as complex numbers: ܣൌ ݔଵ ݕଵ ݅ and ܤൌ ݔଶ ݕଶ ݅Ǥ
Exercise 4 (7 minutes)
As with Exercise 2, allow students time to work on part (a) independently and Scaffolding:
then share results with a partner. Have them work in partners on part (b) before
Provide visual learners with a graph
discussing as a class.
on the complex plane.
Exercise 4
a. What is the distance between ൌ ૡ and ൌ െ ?
ࢊ ൌ ට൫ െ ሺെሻ൯ ሺૡ െ ሻ ൌ ξ
b. Show that, in general, the distance between ൌ ࢞ ࢟ and
ൌ ࢞ ࢟ is the modulus of െ .
െ ൌ ሺ࢞ െ ࢞ ሻ ሺ࢟ െ ࢟ ሻ
ȁ െ ȁ ൌ ඥሺ࢞ െ ࢞ ሻ ሺ࢟ െ ࢟ ሻ ൌ distance between and If students need additional
practice, use this example before
completing part (b) of Exercise 4:
What is the distance between
ܣൌ െ͵ ʹ݅ and ܤൌ ͳʹ െ ͳͲ݅?
à ݀ ൌ ξ͵ͻ ൌ ͵ξͶͳ
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Exercise 5 (3 minutes)
Allow students time to work either independently or with a partner. Scaffolding:
Circulate the room to ensure that students understand the concepts. Have struggling students create a graphic
organizer comparing a coordinate plane and
Exercise 5 a complex plane.
Suppose ࢠ ൌ ૠ and ࢝ ൌ െ . Coordinate Complex
a. Find the midpoint of ࢠ and ࢝. plane plane
ሺെ͵ǡʹሻ and െ͵ ʹ݅ and
ൌ െ
ሺͳǡ͵ሻ ͳ ͵݅
b. Verify that ȁࢠ െ ȁ ൌ ȁ࢝ െ ȁ.
Graph
Midpoint
ȁࢠ െ ȁ ൌ ඨቆ െ ൬െ ൰ቇ ሺૠ െ ሻ
Distance
ൌඨ ૢ
ൌඨ
ȁ࢝ െ ȁ ൌ ඨቆെ െ ൬െ ൰ቇ ሺ െ ሻ
ൌඨ ૢ
ൌඨ
Closing (3 minutes)
Have students discuss each question with a partner. Then, elicit class responses.
In what way can complex numbers be thought of as points?
à When a complex number is plotted on a complex plane, it looks just like the corresponding ordered pair
plotted on a coordinate plane.
Why is it helpful to interchange between complex numbers and points on a plane?
à Unlike points on a plane, we can add and subtract complex numbers. Thus, we can use operations on
complex numbers to find geometric measurements such as midpoint and distance.
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Lesson Summary
Complex numbers can be thought of as points in a plane, and points in a plane can be thought of as
complex numbers.
ା
For two complex numbers ൌ ࢞ ࢟ and ൌ ࢞ ࢟ , the midpoint of points and is .
The distance between two complex numbers ൌ ࢞ ࢟ and ൌ ࢞ ࢟ is equal to ȁ െ ȁ.
Exit Ticket (5 minutes)
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Name Date
Lesson 11: Distance and Complex Numbers
Exit Ticket
1. Kishore said that he can add two points in the coordinate plane like adding complex numbers in the complex plane.
For example, for point ܣሺʹǡ ͵ሻ and point ܤሺͷǡ ͳሻ, he will get ܣ ܤൌ ሺǡ Ͷሻ. Is he correct? Explain your reasoning.
2. Consider two complex numbers ܣൌ െͶ ͷ݅ and ܤൌ Ͷ െ ͳͲ݅.
a. Find the midpoint of ܣand ܤ.
b. Find the distance between ܣand ܤ.
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Exit Ticket Sample Solutions
1. Kishore said that he can add two points in the coordinate plane like adding complex numbers in the complex plane.
For example, for point ሺǡ ሻ and point ሺǡ ሻ, he will get ൌ ሺૠǡ ሻ. Is he correct? Explain your reasoning.
No. Kishore is not correct because we cannot add two points in the rectangular plane. However, we can add two
complex numbers in the complex plane, which has the geometric effect of performing a translation to points in
complex numbers.
2. Consider two complex numbers ൌ െ and ൌ െ .
a. Find the midpoint of and .
ࡹൌ
െ െ
ൌ
െ
ൌ
ൌ െ or െ
b. Find the distance between and .
ࢊ ൌ ට൫ െ ሺെሻ൯ ሺെ െ ሻ ൌ ૠ
Problem Set Sample Solutions
1. Find the midpoint between the two given points in the rectangular coordinate plane.
a. and ૡ
ૡ
ࡹൌ ൌ
b. െ ૠ and െ
െ ૠ െ
ࡹൌ ൌ
c. െ and ૢ െ
െ ૢ െ
ࡹൌ ൌ െ
d. and െ െ ૠ
െ െ ૠ
ࡹൌ ൌ െ െ
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e. െૡ െ and െ
െૡ െ െ ૠ
ࡹൌ ൌെ െ
f. െ and െǤ Ǥ
െ െ
ࡹൌ
െ
െ
ൌ
െ െ
ൌ
ૠ
ൌ െ
2. Let ൌ , ൌ ૡ, and suppose that is the midpoint of and and that ࡰ is the midpoint of and .
a. Find points and ࡰ.
ૡ
ൌ ൌ ൌ ൌ ૡ
ૡ
ࡰൌ ൌ ൌ ൌ
b. Find the distance between and .
ȁ െ ȁ ൌ ȁ െ െ ૡȁ
ൌ ȁെ െ ȁ
ൌ ඥሺെሻ ሺെሻ
ൌ ξ
ൌ ξ
ൌ ξ
c. Find the distance between and .
ȁ െ ȁ ൌ ȁ െ ૡ െ ȁ
ൌ ȁെ െ ȁ
ൌ ඥሺെሻ ሺെሻ
ൌ ξ
ൌ ξ
d. Find the distance between and ࡰ.
ȁ െ ࡰȁ ൌ ȁૡ െ െ ȁ
ൌ ȁ ȁ
ൌ ඥሺሻ ሺሻ
ൌ ξ
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e. Find the distance between ࡰ and .
ȁࡰ െ ȁ ൌ ȁ െ െ ૡȁ
ൌ ȁെૢ െ ȁ
ൌ ඥሺെૢሻ ሺെሻ
ൌ ξૢ
ൌ ξ
f. Find a point one-quarter of the way along the line segment connecting segment and , closer to than
to .
The point is ࡰ ൌ .
g. Terrence thinks the distance from to is the same as the distance from to . Is he correct? Explain why
or why not.
The distance from to is ξ, and the distance from to is ξ. The distances are not the same.
h. Using your answer from part (g), if ࡱ is the midpoint of and , can you find the distance fromࡱ to ?
Explain.
The distance from to is ξ, and the distance from ࡱ to should be half of this value, ξ.
i. Without doing any more work, can you find point ࡱ? Explain.
is , which is units to the right of in the real direction and unit up in the imaginary direction.
From , you should move the same amount to get to ࡱ, so ࡱ would be ૠ.
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Lesson 12: Distance and Complex Numbers
Student Outcomes
Students apply distances between complex numbers and the midpoint of a segment.
Students derive and apply a formula for finding the endpoint of a segment when given one endpoint and the
midpoint.
Lesson Notes
In Lesson 10, students learned that it is possible to interchange between points on a coordinate plane and complex
numbers. Therefore, all the work that was done in Geometry could be translated into the language of complex numbers,
and vice versa. This lesson continues exploring the midpoint between complex numbers through an Exploratory
Challenge in the form of a leapfrog game. In the Opening Exercise, students develop a formula for finding an endpoint of
a segment when given one endpoint and the midpoint. Students then use this formula in the Exploratory Challenge that
follows.
Classwork
Opening Exercise (5 minutes)
Allow students time to work on the Opening Exercise independently before discussing results as a class. The formula
derived in part (b) is used in the Exploratory Challenge.
Opening Exercise
a. Let ൌ and ൌ െ െ ૡ. Find a complex number so that is the midpoint of and .
ൌ െ െ ૢ
b. Given two complex numbers and , find a formula for a complex number in terms of and so that is
the midpoint of and .
ൌ െ
c. Verify that your formula is correct by using the result of part (a).
ൌ െ
െ െ ૢ ൌ ሺെ െ ૡሻ െ ሺ ሻ
ൌ െૡ െ െ െ
ൌ െ െ ૢ
Exercise 1 (7 minutes)
Give students time to work on the exercise in groups. Circulate the room to ensure students understand the problem.
Encourage struggling students to try a graphical approach to the problem.
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Exercise 1
Let ࢠ ൌ െ and ࢝ ൌ െ . Scaffolding:
a. Find a point one-quarter of the way along the line segment connecting ࢠand ࢝ Provide visual learners with a
closer to ࢠthan to ࢝. graph on the complex plane.
Let ࡹ be the midpoint between ࢠ and ࢝ ൌ െ .
Let ࡹᇱ be the midpoint between ࢠ and ࡹ ൌ ૠ െ ૠ.
The complex number ૠ െ ૠ represents a point on the complex plane that
is one-quarter of the way from ࢠ on the segment connecting ࢠ and ࢝Ǥ
b. Write this point in the form ࢻࢠ ࢼ࢝ for some real numbers ࢻand ࢼ. Verify
that this does in fact represent the point found in part (a).
ࢠ ࢝ ൌ ሺെ ሻ ሺ െ ሻ
ൌ ሺെૠ ሻ ሺૠ െ ሻ If students need additional
ൌ ૠ െ ૠ practice, use this example
before moving on. Have some
students find the answer using
c. Describe the location of the point ࢠ ࢝ on this line segment. the midpoint and some using
the result from part (b).
This point is located of the way from ࢠ and of the way from ࢝onࢠ࢝
തതതത.
Find a point one-quarter of the
way along the line segment
connecting segment
When debriefing, use the graph provided in the scaffold as needed.
connecting ݖൌ ͺ െ ݅and
For part (b), did anyone have an answer that did not work when you tried to ݓൌ ͳʹ ͳ݅ closer to ݖthan
verify? to ݓ.
ଵ ଷ ͳ
à ݖ ( ݓNote: This could be a very common incorrect answer. If à ͻ െ ݅
ʹ
ସ ସ
nobody offers it as an answer, perhaps suggest it.)
ଵ ଷ ଵ ଷ
Why isn’t ݖ ݓthe correct answer? After all, the point is of the way from ݖand of the way from ݓ.
ସ ସ ସ ସ
à It did not work when we tried to verify it.
ͳ ͵
ሺെͳͲͲ ͳͲͲ݅ሻ ሺͳͲͲͲ െ ͳͲͲͲ݅ሻ ൌ ሺെʹͷ ͷͲሻ ݅ሺʹͷ െ ͷͲሻ ൌ ʹͷ െ ʹͷ݅
Ͷ Ͷ
What point would this be on the segment?
ଷ ଵ
à It would be the point that is of the way from ݖand of the way from ݓ.
ସ ସ
You would think about this like a weighted average. To move the point closer to ݖ, it must be weighted more
in the calculation than ݓ.
Exploratory Challenge 1 (15 minutes)
Have students work in groups on this challenge. Each group needs a full page of graph paper. Warn students that they
need to put the three points ܣ, ܤ, and ܥfairly close together in order to stay on the page. Allow students to struggle a
little with part (g), but then provide them with help getting started if needed.
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Exploratory Challenge 1
a. Draw three points , , and in the plane.
b. തതതതതതത
Start at any position ࡼ , and leapfrog over to a new position ࡼ so that is the midpoint of ࡼ ࡼ .
c. From ࡼ , leapfrog over to a new position ࡼ so that is the midpoint തതതതതതത
ࡼ ࡼ.
d. From ࡼ , leapfrog over to a new position ࡼ so that is the midpoint തതതതതതത
ࡼ ࡼ.
e. Continue alternately leapfrogging over , then, and then .
f. What eventually happens?
At the sixth jump, you end up at the initial point ࡼ ൌ ࡼ .
g. Using the formula from the Opening Exercise, part (b), show why this happens.
ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ ࡼ
What happened on the sixth jump?
à We landed back at the starting point. (Note: Allow a couple of groups to share their graph. A sample
is provided.)
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Exploratory Challenge 2 (10 minutes)
Have students continue working in groups on this challenge. Students will need a full Scaffolding:
page of graph paper for this activity.
Provide early finishers with this
challenge.
Exploratory Challenge 2
Repeat the activity with
a. Plot a single point in the plane. two points, ܣand ܤ. But
this time, leapfrog at a ͻͲι
b. What happens when you repeatedly jump over ? angle to the left over each
You keep alternating between landing on ࡼ and landing on ࡼ . point. Will you return to
the starting point? After
how many leaps?
c. Using the formula from the Opening Exercise, part (b), show why this happens.
à You will return to the
ࡼ ൌ െ ࡼ
starting point after Ͷ
ࡼ ൌ െ ࡼ ൌ െ ࡼ ൌ ࡼ leaps.
d. Make a conjecture about what will happen if you leapfrog over two points, and , in the coordinate plane.
Answers will vary.
e. Test your conjecture by using the formula from the Opening Exercise, part (b).
Answers will vary.
f. Was your conjecture correct? If not, what is your new conjecture about what happens when you leapfrog
over two points, and , in the coordinate plane?
Answers will vary, but for the most part, students should have found that their conjecture was incorrect. In
this game, you never return to the starting position. Instead, the points continue to get farther away from
points and . This can be seen by using the formula from the Opening Exercise, part (b).
g. Test your conjecture by actually conducting the experiment.
Closing (3 minutes)
Discuss the results of Exploratory Challenge 2.
What was your initial conjecture about two points?
à Answers will vary, but most students would have predicted that at some point you would return to the
initial point.
How did the formula prove that it was incorrect?
à The formula never returned to ܲ because terms did not cancel out.
What happened when you leapfrogged over two points?
à We kept getting farther from the initial point.
Exit Ticket (5 minutes)
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Name Date
Lesson 12: Distance and Complex Numbers
Exit Ticket
1. Find the distance between the following points.
a. ሺͶǡ െͻሻ and ሺͳǡ െͷሻ
b. Ͷ െ ͻ݅ and ͳ െ ͷ݅
c. Explain why they have the same answer numerically in parts (a) and (b) but a different perspective in
geometric effect.
2. Given point ܣൌ ͵ െ ʹ݅ and point ܯൌ െʹ ݅, if ܯis the midpoint of ܣand another point ܤ, find the coordinates
of point ܤ.
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Exit Ticket Sample Solutions
1. Find the distance between the following points.
a. ሺǡ െૢሻ and ሺǡ െሻ
ൌ ඥሺ െ ሻ ሺെૢ ሻ
ൌ ξૢ
ൌ
b. ൌ െ ૢ and ൌ െ
ȁ െ ȁ ൌ ȁ െ ૢ െ ȁ
ൌ ȁ െ ȁ
ൌ ඥሺሻ ሺെሻ
ൌ ξ
ൌ
c. Explain why they have the same answer numerically in parts (a) and (b) but a different perspective in
geometric effect
The length of the line segment connecting points and is .
To find the distance between two complex numbers and , we need to calculate െ ൌ െ ૢ െ ,
which has the geometric effect of performing a translation—shifting one unit to the left and units upward
from point ൌ െ ૢ. The result is െ . And ȁ െ ȁ is the distance from the origin to the complex
number െ, which is not exactly the same as തതതത in terms of their position. However, they all have the same
numerical value in terms of distance, which is .
2. Given point ൌ െ and point ࡹ ൌ െ , if ࡹ is the midpoint of and another point , find the coordinates
of point .
ሺ െ ሻ ࡹ
െ ൌ
െ ൌ െ ࡹ
െૠ ൌ ࡹ
Problem Set Sample Solutions
1. Find the distance between the following points.
a. Point ሺǡ ሻ and point ሺǡ ሻ
ൌ ඥሺ െ ሻ ሺ െ ሻ
ൌ ξ ૢ
ൌ
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b. ൌ and ൌ
ȁ െ ȁ ൌ ȁ െ െ ȁ
ൌ ȁെ െ ȁ
ൌ ඥሺെሻ ሺെሻ
ൌ ξ ૢ
ൌ
c. ൌ െ and ൌ
ȁ െ ȁ ൌ ȁെ െ െ ȁ
ൌ ȁെ െ ȁ
ൌ ඥሺെሻ ሺെሻ
ൌ ඥሺሻ
ൌ ξ
d. ൌ െ and ൌ െ
ȁ െ ȁ ൌ ȁ െ െ ȁ
ൌ ȁ െ ȁ
ൌ ඥሺሻ ሺെሻ
ൌ ξૢ
ൌ ξ
e. ൌ െ and ൌ െ
ȁ െ ȁ ൌ ฬ െ െ ฬ
ૠ
ൌ ฬ െ ฬ
ૠ െ
ൌ ඨ൬ ൰ ൬ ൰
ૢ
ൌඨ
ሺሻ
ξૠ
ൌ
2. Given three points , , , where is the midpoint of and .
a. If ൌ െ and ൌ , find .
ൌ
ൌ െ
ൌ ሺ ሻ െ ሺെ ሻ
ൌ ૡ െ
ൌ
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b. If ൌ and ൌ െ , find .
ൌ
ൌ െ
ൌ ሺെ ሻ െ ሺ ሻ
ൌ െ െ െ
ൌ െ െ
3. Point is the midpoint between ൌ and ൌ െ െ . Find the distance between points and ࡰ for each
point ࡰ provided below.
a. ࡰ ൌ െ ૡ
െ െ
ൌ ൌ െ െ
ࡰ ൌ െ
ȁ െ ࡰȁ ൌ ȁെ െ െ ȁ
ൌ ȁെ െ ȁ
ൌ ඥሺെሻ ሺሻ
ൌ ξૢ
b. ഥ
ࡰ ൌ െ
ࡰ ൌ െ
ȁ െ ࡰȁ ൌ ȁെ െ െ ȁ
ൌ ȁെૠ ȁ
ൌ ඥሺെૠሻ ሺሻ
ൌ ξ
4. The distance between points ൌ and ൌ ࢇ ࢈ is . Find the point for each value provided below.
a. ࢇൌ
ൌ ඥሺ െ ሻ ሺ െ ࢈ሻ
ൌ ૢ ሺ െ ࢈ሻ
െ ࢈ ൌ േ
࢈ ൌ െǡ
ൌ െ or ൌ
b. ࢈ൌ
ൌ ඥሺ െ ࢇሻ ሺ െ ሻ
ൌ ሺ െ ࢇሻ
ሺ െ ࢇሻ ൌ
െࢇ ൌ
ࢇൌ
ൌ
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5. Draw five points in the plane ,,,ࡰ,ࡱ. Start at any position, ࡼ , and leapfrog over to a new position, ࡼ (so,
is the midpoint of തതതതതതത
ࡼ ࡼ). Then, leapfrog over , then , then ࡰ, then ࡱ, then , then , then , then ࡰ, then ࡱ,
then again, and so on. How many jumps will it take to get back to the starting position, ࡼ ?
It takes jumps to return to the starting position.
ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ െ ࡼ
ࡼ ൌ ࡰ െ ࡼ ൌ ࡰ െ െ ࡼ
ࡼ ൌ ࡱ െ ࡼ ൌ ࡱ െ ࡰ െ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡱ ࡰ െ െ ࡼ ൌ െࡱ ࡰ െ ࡼ
ࡼૠ ൌ െ ࡼ ൌ ࡱ െ ࡰ െ െ ࡼ ൌ ࡱ െ ࡰ െ ࡼ
ࡼૡ ൌ െ ࡼૠ ൌ െ ࡱ ࡰ െ ࡼ ൌ െࡱ ࡰ ࡼ
ࡼૢ ൌ ࡰ െ ࡼૡ ൌ ࡰ ࡱ െ ࡰ െ ࡼ ൌ ࡱ െ ࡼ
ࡼ ൌ ࡱ െ ࡼૢ ൌ ࡱ െ ࡱ ࡼ ൌ ࡼ
6. For the leapfrog puzzle problems in both Exploratory Challenge 1 and Problem 5, we are given an odd number of
points to leapfrog over. What if we leapfrog over an even number of points? Let ൌ , ൌ , and ࡼ ൌ .
Will ࡼ ever return to the starting position, ࡼ ? Explain how you know.
No, we cannot get back to the starting position. For example, if we leapfrog over two given even points, and .
ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ െ ࡼ ൌ െ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ ࡼ ൌ െ ࡼ
ࡼ ൌ െ ࡼ ൌ െ െ ࡼ ൌ െ െ ࡼ
If is even, ࡼ ൌ െ ࡼ ൌ ሺ െ ሻ ࡼ . Then, if ࡼ ൌ ࡼ , we have ൌ ሺ െ ሻ, which would mean
that ൌ ǡ which we know to be false. Thus, for even values of , ࡼ will never return to ࡼ .
If is odd, ࡼ ൌ ሺ ሻ െ ሺ െ ሻ െ ࡼ . Then, if ࡼ ൌ ࡼ we have
ࡼ ൌ ሺ ሻ െ ሺ െ ሻ
ൌ ሺ ሻ െ ሺ െ ሻሺ ሻ
ൌ െ െ ሺ െ ሻ
ൌ െ െ ሺ െ ሻ
ሺ ሻ ൌ െǤ
Since ሺ ሻ is an imaginary number and െis a real number, it is impossible forሺ ሻ to equal െ. Thus, for
odd values of , ࡼ will never return to ࡼ .
Therefore, it is not possible for ࡼ to ever coincide withࡼ for these values of , , and ࡼ .
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Lesson 13: Trigonometry and Complex Numbers
Student Outcomes
Students represent complex numbers in polar form and convert between rectangular and polar
representations.
Students explain why the rectangular and polar forms of a given complex number represent the same number.
Lesson Notes
This lesson introduces the polar form of a complex number and defines the argument of a complex number in terms of a
rotation. This definition aligns with the definitions of the sine and cosine functions introduced in Algebra II, Module 2
and ties into work with right-triangle trigonometry from Geometry. This lesson continues to emphasize the usefulness of
representing complex numbers as transformations. Analysis of the angle of rotation and the scale of the dilation brings
a return to topics in trigonometry first introduced in Geometry and expanded on in Algebra II. This lesson reinforces the
geometric interpretation of the modulus of a complex number and introduces the notion of the argument of a complex
number. When representing a complex number in polar form, it is apparent that every complex number can be thought
of simply as a rotation and dilation of the real number ͳ. In addition to representing numbers in polar form and
converting between them, be sure to provide opportunities for students to explain why polar and rectangular forms of a
given complex number represent the same number.
It may be necessary to spend some time reviewing with students the work they did in previous courses, particularly as it
relates to right-triangle trigonometry, special right triangles, and the sine and cosine functions. Sample problems have
been provided after the Problem Set for this lesson. Specific areas for additional review and practice include the
following:
Using proportional reasoning to determine the other two sides in a special right triangle when given one side
(Geometry, Module 2),
Finding the acute angle in a right triangle using the arctangent function (Geometry, Module 2),
Describing rotations in both degrees and radians (Algebra II, Module 2),
Evaluating the sine and cosine functions at special angles in both degrees and radians (Algebra II, Module 2),
and
Evaluating the sine and cosine functions at any angle using a calculator (Geometry, Module 2 and Algebra II,
Module 2).
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Classwork
Opening (3 minutes)
Ask students to recall the special right triangles they studied in Geometry and revisited in Algebra II by showing them a
diagram with the angles labeled but the side measurements missing. Have them fill in the missing side lengths and
explain their reasoning to a partner. Check for understanding by adding the side lengths to the diagrams on the board,
and then direct students to record these diagrams in their notes. Display these diagrams prominently in the classroom
for student reference. Announce that these relationships are very helpful as they work through today’s lesson.
Opening Exercise (5 minutes)
The Opening Exercise reviews two key concepts from the previous lesson: (1) Each complex number ܽ ܾ݅ corresponds
to a point ሺܽǡ ܾሻ, and (2) the modulus of a complex number is given by ξܽଶ ܾ ଶ. The last part of the Opening Exercise
asks students to think about a rotation that takes a ray from the origin initially containing the real number ͳ to its image
from the origin passing through the point ሺܽǡ ܾሻ. The measurements of the rotation for the different points representing
the different numbers should be fairly obvious to students. However, as students work, it may be necessary to remind
them of the special right triangles they just discussed. These exercises should be done with a partner or with a small
group. Use the discussion questions that follow to guide students as they work.
Scaffolding:
Describe the location of the ray from the origin containing the real number ͳ. Provide additional practice
à It lies along the positive ݔ-axis with endpoint at the origin. problems working with
How can you determine the amount of rotation for ݖଵ and ݖଶ ? special right triangles.
à The points along the axes were one-fourth and one-half a complete Project the diagram on the
rotation, which is ͵Ͳι. board, and draw in the
rays and sides of a right
How can you determine the amount of rotation for ݖଷ and ݖସ ?
triangle to help students
à The values of ܽ and ܾ formed the legs of a special right triangle. From see the geometric
there, since ݖଷ was located in the first quadrant, the rotation was just relationships for ݖଷ and ݖସ .
Ͷͷι. For ݖସ , you would need to subtract Ͳι from ͵Ͳι to give a positive
Encourage students to
counterclockwise rotation of ͵ͲͲι, or use a clockwise rotation of െͲι.
label vertical and
How did you determine the modulus? horizontal distances using
à The modulus is given by the expression ξܽଶ ܾ ଶ . the values of ܽ and ܾ.
When students are finished, have one or two of them present their solutions for each complex number. Emphasize the
use of special triangles to determine the degrees of rotation for complex numbers not located along an axis. For
additional scaffolding, it may be necessary to draw in the ray from the origin containing the real number ͳ and the
rotated ray from the origin that contains the point ሺܽǡ ܾሻ for each complex number.
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Opening Exercise
For each complex number shown below, answer the following questions. Record your answers in the table.
a. What are the coordinates ሺࢇǡ ࢈ሻ that correspond to this complex number?
b. What is the modulus of the complex number?
c. Suppose a ray from the origin that contains the real number is rotated ࣂι so it passes through the point
ሺࢇǡ ࢈ሻ. What is a value of ࣂ?
Complex Number ሺࢇǡ ࢈ሻ Modulus Degrees of Rotation ࣂι
ࢠ ൌ െ ሺെǡ ሻ ૡι
ࢠ ൌ ሺǡ ሻ ૢι
ࢠ ൌ ሺǡ ሻ ξ ι
ࢠ ൌ െ ξ ሺǡ െξሻ ι
As students present their solutions, ask if anyone has a different answer for the number of degrees of the rotation, and
lead a discussion so students understand that the degrees of rotation have more than one possible answer and, in fact,
there are infinitely many possible answers.
Another student said that a clockwise rotation of ʹͲι would work for ଶ . Do you agree or disagree? Explain.
à I agree. This rotation also takes the initial ray from the origin to a ray containing the point Ͳ ʹ݅. If a
complete rotation is ͵Ͳι, then ʹͲι clockwise would be the same as ͻͲι counterclockwise.
At this point, remind students that positive rotations are counterclockwise and that rotation in the opposite direction is
denoted using negative numbers.
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Exercises 1–2 (5 minutes)
These exercises help students understand why the values of the argument of a Scaffolding:
complex number are limited to real numbers on the interval Ͳι ߠ ൏ ͵Ͳι.
Help students to generalize the
College-level mathematics courses make a distinction between the argument of a
expression by organizing the angles
complex number between Ͳ and ͵Ͳ and the set of all possible arguments of a given
into a table.
complex number.
݊ Degrees of Rotation
Exercises 1–2 Ͳ Ͷͷ
1. Can you find at least two additional rotations that would map a ray from the origin
ͳ Ͷͷ ͵Ͳ
through the real number to a ray from the origin passing through the point ሺǡ ሻ?
This is the number ࢠ ൌ from the Opening Exercise. Additional rotations ʹ Ͷͷ ͵Ͳሺʹሻ
could be ι ι ൌ ι or ι െ ι ൌ െι.
͵ Ͷͷ ͵Ͳሺ͵ሻ
2. How are the rotations you found in Exercise 1 related?
All rotations that take the initial ray to the ray described above must be of the form ι ι for integer values
of .
After reviewing possible solutions to the questions above, pose this next question. It may be written on the board. Give
students a few minutes to think about their responses individually, and then have them discuss them with their partners
or group members before sharing responses as a whole class.
Do you think it is possible to describe a complex number in terms of its modulus and the degrees of rotation of
a ray from the origin containing the real number ͳ? Justify your reasoning.
à Student responses will vary. In general, the response should be yes, but careful students should note
the difficulty of uniquely defining degrees of rotation. The modulus will be a distance from the origin,
and if we want to be precise, we may need to limit the possible degrees of rotation to a subset of the
real numbers such as Ͳι ߠ ൏ ͵Ͳι.
Discussion (3 minutes)
Exercises 2 and 3 show that the rotation that maps a ray from the origin containing the real number ͳ to a ray containing
a given complex number is not unique. If you know one rotation, you can write an expression that represents all the
rotations of a given complex number ݖ. However, if we limit the rotations to an interval that comprises one full rotation
of the initial ray, then we can still describe every complex number in terms of its modulus and a rotation.
Introduce the modulus and argument of a complex number.
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Every complex number ࢠ ൌ ࢞ ࢟ appears as a point on the complex plane with coordinates ሺ࢞ǡ ࢟ሻ as a point in the
coordinate plane.
In the diagram above, notice that each complex number ࢠ has a distance ࢘ from the origin to the point ሺ࢞ǡ ࢟ሻ and a
rotation of ࣂι that maps the ray from the origin along the positive real axis to the ray passing through the point ሺ࢞ǡ ࢟ሻ.
ARGUMENT OF THE COMPLEX NUMBER ࢠ: The argument of the complex number ࢠ is the radian (or degree) measure of the
counterclockwise rotation of the complex plane about the origin that maps the initial ray (i.e., the ray corresponding to
the positive real axis) to the ray from the origin through the complex number ࢠ in the complex plane. The argument of ࢠ
is denoted ܚ܉ሺࢠሻ.
MODULUS OF A COMPLEX NUMBER ࢠ: The modulus of a complex number ࢠ, denoted ȁࢠȁ, is the distance from the origin to the
point corresponding to ࢠ in the complex plane. If ࢠ ൌ ࢇ ࢈, then ȁࢠȁ ൌ ξࢇ ࢈ .
Is “modulus” indeed the right word? Does ݎൌ ȁݖȁ as we defined it in previous lessons?
à Yes, since ݎis the distance from the origin to the point ሺݔǡ ݕሻ, which is ඥ ݔଶ ݕଶ , which is also how we
define the modulus of a complex number.
Why are we limiting the argument to a subset of the real numbers?
à We only need these angles to sweep through all possible points in the coordinate plane. If we allowed
the argument to be any real number, there would be many possible arguments for any given complex
number.
Example 1 (4 minutes): The Polar Form of a Complex Number
This example models how the polar form of a complex number is derived using the sine and cosine functions that
students studied in Algebra II. Use the questions on the student materials to guide the discussion. The definitions from
Algebra II are provided below for teacher reference.
SINE FUNCTION: The sine function, ǣԹ ՜ Թ, can be defined as follows:
Let ߠ be any real number. In the Cartesian plane, rotate the initial ray by ߠ radians about the origin. Intersect the
resulting terminal ray with the unit circle to get a point ሺݔఏ ǡ ݕఏ ሻ. The value of ሺߠሻ is ݕఏ .
COSINE FUNCTION: The cosine function, ǣԹ ՜ Թ, can be defined as follows:
Let ߠ be any real number. In the Cartesian plane, rotate the initial ray by ߠ radians about the origin. Intersect the
resulting terminal ray with the unit circle to get a point ሺݔఏ ǡ ݕఏ ሻ. The value of ሺߠሻ is ݔఏ .
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What do you recall about the definitions of the sine function and the cosine function from Algebra II?
à The sine function is the ݕ-coordinate of a point, and the cosine function is the ݔ-coordinate of the
intersection point of a ray rotated ߠ radians about the origin and the unit circle.
How can the sine and cosine functions help us to relate the point ሺݔǡ ݕሻ to modulus ݎand the argument ߠ?
à The coordinates ሺݔǡ ݕሻ can be expressed in terms of the cosine and sine using the definition of the sine
and cosine functions and dilating them along the terminal ray by a factor of ݎ.
Why would it make sense to use these functions to relate a complex number in ܽ ܾ݅ form to one described
by its modulus and argument?
à The modulus is a distance from the origin to the point ሺܽǡ ܾሻ, and the argument is basically the same
type of rotation described in the definitions of the sine and cosine functions.
Example 1: The Polar Form of a Complex Number
Derive a formula for a complex number in terms of its modulus ࢘ and argument ࣂ.
Suppose that ࢠ has coordinates ሺ࢞ǡ ࢟ሻ that lie on the unit circle as shown.
a. What is the value of ࢘ǡ and what are the coordinates of the point ሺ࢞ǡ ࢟ሻ in terms of ࣂ? Explain how you know.
The value of ࢘ is . The coordinates of the point are ሺܛܗ܋ሺࣂሻ ǡ ܖܑܛሺࣂሻሻ. The definition of the sine and cosine
function says that a point on the unit circle where a rotated ray intersects the unit circle has these
coordinates.
b. If ࢘ ൌ , what would be the coordinates of the point ሺ࢞ǡ ࢟ሻ? Explain how you know.
The coordinates would be ሺԝܛܗ܋ሺࣂሻ ǡ ԝܖܑܛሺࣂሻሻ because the point lies along the same ray but are just dilated
by a scale factor of two along the ray from the origin compared to when ࢘ ൌ .
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c. If ࢘ ൌ , what would be the coordinates of the point ሺ࢞ǡ ࢟ሻ? Explain how you know.
The coordinates would be ሺ ܛܗ܋ሺࣂሻ ǡ ܖܑܛሺࣂሻሻ because a circle of radius units would be similar to a
circle with radius but dilated by a factor of .
d. Use the definitions of sine and cosine to write coordinates of the point ሺ࢞ǡ ࢟ሻ in terms of cosine and sine for
any ࢘ and real number ࣂ.
࢞ ൌ ࢘ ܛܗ܋ሺࣂሻ and ࢟ ൌ ࢘ ܖܑܛሺࣂሻ
e. Use your answer to part (d) to express ࢠ ൌ ࢞ ࢟ in terms of ࢘ and ࣂ.
ࢠ ൌ ࢞ ࢟ ൌ ࢘ ܛܗ܋ሺࣂሻ ࢘ ܖܑܛሺࣂሻ ൌ ࢘ሺܛܗ܋ሺࣂሻ ܖܑܛሺࣂሻሻ
Monitor students as they work in small groups to derive the polar form of a complex number from the rectangular form.
After a few minutes, ask for a few volunteers to share their ideas, and then make sure to have students record the
derivation shown below in their notes and revise their work to be accurate and precise.
Annotate the diagram above showing that the ݔ- and ݕ-values correspond to the points on a circle of radius ݎthat is a
dilation of the unit circle. Thus, the point ሺݔǡ ݕሻ can be represented as ሺ ݎሺߠሻ ǡ ݎሺߠሻሻǤ
The diagram shown above makes us recall the definitions of sine and cosine. We see from the following
diagram:
ݔൌ ݎሺߠሻ and ݕൌ ݎሺߠሻ.
Which means that every complex number can be written in the form:
ݖൌ ݔ ݅ ݕൌ ݎሺߠሻ ݅ ݎሺߠሻ ൌ ݎሺ ሺߠሻ ݅ ሺߠሻሻ.
Review the definition shown below, and then have students work in small groups to answer Exercises 3–6.
POLAR FORM OF A COMPLEX NUMBER: The polar form of a complex number ࢠ is ࢘ሺܛܗ܋ሺࣂሻ ܖܑܛሺࣂሻሻ, where ࢘ ൌ ȁࢠȁ and
ࣂ ൌ ܚ܉ሺࢠሻ.
RECTANGULAR FORM OF A COMPLEX NUMBER: The rectangular form of a complex number ࢠ is ࢇ ࢈, where ࢠ corresponds to
the point ሺࢇǡ ࢈ሻ in the complex plane, and is the imaginary unit. The number ࢇ is called the real part of ࢇ ࢈, and the
number ࢈ is called the imaginary part of ࢇ ࢈.
Use the graphic organizer below to help students make sense of this definition. A blank version is included in the
student materials. The graphic organizer has space for up to three examples of complex numbers that can either be
completed as a class or assigned to students. Have students work with a partner to provide the polar and rectangular
forms of both numbers. Have partners take turns explaining why the polar and rectangular forms of the examples
represent the same number.
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General Form Polar Form Rectangular Form
ࢠ ൌ ࢘ሺܛܗ܋ሺࣂሻ ܖܑܛሺࣂሻሻ ࢠ ൌ ࢇ ࢈
Examples
ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ ξ
࣊ ࣊
ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
Key Features Modulus Modulus
࢘ ඥࢇ ࢈
Argument Coordinate
ࣂ ሺࢇǡ ࢈ሻ
Coordinate ࢇ ൌ ࢘ ܛܗ܋ሺࣂሻ
ሺ࢘ ܛܗ܋ሺࣂሻ ǡ ࢘ ܖܑܛሺࣂሻሻ ࢈ ൌ ࢘ ܖܑܛሺࣂሻ
Explain to students that this form of a complex number is particularly useful when considering geometric
representations of complex numbers. This form clearly shows that every complex number ݖcan be described as a
rotation of ߠι and a dilation by a factor of ݎof the real number ͳ.
Exercises 3–6 (8 minutes)
Students should complete these exercises with a partner or in small groups. Monitor progress as students work, and
offer suggestions if they are struggling to work with the new representation of a complex number.
Exercises 3–6
3. Write each complex number from the Opening Exercise in polar form.
Rectangular Polar Form
ࢠ ൌ െ ሺܛܗ܋ሺૡιሻ ܖܑܛሺૡιሻሻ
ࢠ ൌ ሺܛܗ܋ሺૢιሻ ܖܑܛሺૢιሻሻ
ࢠ ൌ ξሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ࢠ ൌ െ ξ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
4. Use a graph to help you answer these questions.
a. What is the modulus of the complex number െ ?
If you graph the point ሺǡ െሻ, then the distance between the origin and the point is given by the distance
formula, so the modulus would be ඥሺሻ ሺെሻ ൌ ξ.
b. What is the argument of the number െ ?
If you graph the point ሺǡ െሻ, then the rotation that will take the ray from the origin through the real
number to a ray containing that point will be ι because the point lies on a line from the origin in
Quadrant IV that is exactly in between the two axes. The argument would be ι. We choose that rotation
because we defined the argument to be a number between and .
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c. Write the complex number in polar form.
ξሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
d. Arguments can be measured in radians. Express your answer to part (c) using radians.
ૠ࣊
In radians, ι is ; the number would be
ૠ࣊ ૠ࣊
ξ ൬ ܛܗ܋൬ ൰ ܖܑܛ൬ ൰൰ .
e. Explain why the polar and rectangular forms of a complex number represent the same number.
െ is thought of as a point with coordinates ሺǡ െሻ in the complex plane. The point can also be located
by thinking of the ray extending from the origin rotated ι. The distance from the origin to the point along
that ray is the modulus, which is ξ units.
Debrief Exercises 3 and 4 by having one or two students volunteer their solutions. On Exercise 4, some students may use
right-triangle trigonometry while others take a more geometric approach and reason out the value of the argument from
the graph and their knowledge of special right triangles. Pause and review radian measure if students are struggling to
answer Exercise 4, part (d). When reviewing these first two exercises, be sure to emphasize why the work from
Example 1 validates that the polar and rectangular forms of a complex number represent the same number.
Next, give students a few minutes to work individually on using this new form of a complex number. They need to
approximate the location of a few of these rotations unless provided with a protractor. If the class is struggling to
evaluate trigonometric functions of special angles, students may use a calculator, a copy of the unit circle, or their
knowledge of special triangles to determine the values of ܽ and ܾ. Students need a calculator to answer Exercise 6,
part (c).
5. State the modulus and argument of each complex number, and then graph it using the modulus and argument.
a. ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
࢘ ൌ , ࣂ ൌ ι
࣊ ࣊
b. ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ , ࣂ ൌ
c. ሺܛܗ܋ሺૢιሻ ܖܑܛሺૢιሻሻ
࢘ ൌ , ࣂ ൌ ૢι
6. Evaluate the sine and cosine functions for the given values of ࣂ, and then express each complex number in
rectangular form, ࢠ ൌ ࢇ ࢈. Explain why the polar and rectangular forms represent the same number.
a. ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ξ
ቆെ ቇ ൌ െ ξ
The polar form of a complex number and the rectangular form represent the same number because they both
give you the same coordinates of a point that represents the complex number. In this example, units along
a ray from the origin rotated ι corresponds to the coordinate ሺെǡ ξሻ.
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࣊ ࣊
b. ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
ξ ξ ξ ξ
ቆ ቇൌ
࣊
The polar form and rectangular form represent the same number because the values of ܛܗ܋ቀ ቁ and
࣊ ඥ
ܖܑܛቀ ቁ are exactly .
c. ሺܛܗ܋ሺૢιሻ ܖܑܛሺૢιሻሻ
Rounded to two decimal places, the rectangular form is െǤ ૢ െ Ǥ . This form of the number is close to,
but not exactly, the same as the number expressed in polar form because the values of the trigonometric
functions are rounded to the nearest hundredth.
Review the solutions to these exercises with the entire class to check for understanding before moving on to Example 2.
Make sure students understand that in Exercise 6, they rewrote each complex number given in polar form as an
equivalent complex number written in rectangular form. Emphasize that in part (c), the rectangular form is an
approximation of the polar form.
Example 2 (8 minutes): Writing a Complex Number in Polar Form
This example gives students a way to convert any complex number in rectangular form to its polar form using the inverse
tangent function. To be consistent with work from previous grades, discussions of inverse tangent must be limited to
the work students did in Geometry where they solved problems involving right triangles only. Students develop the
inverse trigonometric functions in Module 3.
Ask students to recall what they did in the Opening Exercise and Exercise 5 to determine the argument.
How were you able to determine the argument in the Opening Exercises and in Exercise 5?
à The complex numbers were on an axis or had coordinates that corresponded to lengths of sides in
special right triangles, so we could recognize the proper degrees of rotation.
The problems in the Opening Exercise and Exercise 5 were fairly easy because of special right-triangle relationships or
the fact that the rotations coincided with an axis.
How can you express any complex number given in rectangular form in polar form?
à The modulus, ݎ, is given by ξܽଶ ܾ ଶ . To determine the angle, we would need a way to figure out the
rotation based on the location of the point ሺܽǡ ܾሻ.
Model how to construct a right triangle and use right-triangle trigonometry relationships to determine a value of an
acute angle, which can then be used to determine the argument of the complex number.
What did you learn in Geometry about finding an angle in a right triangle if you know two of the side
measures?
à We applied the arctan, arcsin, or arccos to the ratios of the known side lengths.
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Example 2: Writing a Complex Number in Polar Form
a. Convert to polar form.
ሺܛܗ܋ሺǤ ιሻ ܖܑܛሺǤ ιሻሻ
b. Convert െ to polar form.
ሺܛܗ܋ሺǤ ૢιሻ ܖܑܛሺǤ ૢιሻሻ
What is the modulus of ͵ Ͷ݅?
à The modulus is ͷ.
Draw a diagram like the one shown below, and use trigonometry ratios to help you determine the argument. Plot the
point ሺ͵ǡ Ͷሻ, and draw a line segment perpendicular to the ݔ-axis from the point to the ݔ-axis. Draw the ray from the
origin through the point ሺ͵ǡ Ͷሻ and the ray from the origin through the real number ͳ. Label the acute angle between
these rays ߠ.
The line segment from ሺͲǡ Ͳሻ to ሺ͵ǡ Ͳሻ, the line segment we just drew, and the segment from the origin to the
point form a right triangle. What is the tangent ratio of the acute angle whose vertex is at the origin?
Ͷ
à The tangent is ሺߠሻ ൌ .
͵
Use a calculator to estimate the measure of this angle. What is the argument of ͵ Ͷ݅?
Ͷ
à We can use ߠ ൌ ቀ ቁ. Rounded to the nearest hundredth, ߠ ൌ ͷ͵Ǥͳι.
͵
Write ͵ Ͷ݅ in polar form.
à The polar form is ͷሺ ሺͷ͵Ǥͳιሻ ݅ ሺͷ͵Ǥͳιሻሻ with the angle rounded to the nearest tenth.
Part (b) of this example shows how the process above needs to be tweaked when the complex number is not located in
the first quadrant.
The modulus is ͷ. When we plot the point ሺ͵ǡ െͶሻ and draw a line segment perpendicular to the ݔ-axis, we can see that
Ͷ
the acute angle at the origin in this triangle will still have a measure equal to ቀ ቁ ൌ ͷ͵Ǥͳι.
͵
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Model how to draw this diagram so students see how to use the arctangent function to find the measure of the acute
angle at the origin in the triangle they constructed.
Use your knowledge of angles to determine the argument of ͵ െ Ͷ݅. Explain your reasoning.
à An argument of ͵ െ Ͷ݅ would be ͵Ͳι െ ͷ͵Ǥͳι ൌ ͵ͲǤͻι. The positive rotation of ray from the origin
containing the real number ͳ that maps to a ray passing through this point would be ͷ͵ι less than a full
rotation of ͵Ͳι.
What is the polar form of ͵ െ Ͷ݅?
à The polar form is ͷሺ ሺ͵ͲǤͻιሻ ݅ ሺ͵ͲǤͻιሻሻ.
Why do the polar and rectangular forms of a complex number represent the same number?
à ͵ െ Ͷ݅ can be thought of as the point ሺ͵ǡ െͶሻ in the complex plane. The point can be located by
extending the ray from the origin rotated ͵ͲǤͻι. The point is a distance of ͷ units (the modulus) from
the origin along that ray.
Exercise 7 (4 minutes)
Have students practice the methods just demonstrated in Example 2. They can work individually or with a partner.
Review the solutions to these problems with the whole class before moving on to the lesson closing.
Exercise 7
7. Express each complex number in polar form. State the arguments in radians rounded to the nearest thousandth.
a.
ܚ܉ሺ ሻ ൌ ିܖ܉ܜ ൬ ൰ ൎ Ǥ ૢ
ȁ ȁ ൌ ξ ൌ ξૢ
ൎ ξૢሺܛܗ܋ሺǤ ૢሻ ܖܑܛሺǤ ૢሻሻ
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b. െ
ܚ܉ሺࢠሻ ൌ ࣊ െ ିܖ܉ܜ ൬ ൰ ൎ Ǥ ૢૠ
ȁെ ȁ ൌ ξૠ
െ ൌ ξૠሺܛܗ܋ሺǤ ૢૠሻ ܖܑܛሺǤ ૢૠሻሻ
Closing (3 minutes)
Review the Lesson Summary, and then ask students to describe to a partner the geometric meaning of the modulus and
argument of a complex number. Then, have the partner describe the steps required to convert a complex number in
rectangular form to polar form. Encourage students to refer back to their work in this lesson as they discuss what they
learned with their partners.
Lesson Summary
The polar form of a complex number ࢠ ൌ ࢘ሺܛܗ܋ሺࣂሻ ܖܑܛሺࣂሻሻ where ࣂ is the argument of ࢠ and ࢘ is the modulus
of ࢠ. The rectangular form of a complex number is ࢠ ൌ ࢇ ࢈.
The polar and rectangular forms of a complex number are related by the formulas ࢇ ൌ ࢘ ܛܗ܋ሺࣂሻ, ࢈ ൌ ࢘ ܖܑܛሺࣂሻ, and
࢘ ൌ ξࢇ ࢈ .
The notation for modulus is ȁࢠȁ, and the notation for argument is ܚ܉ሺࢠሻ.
Exit Ticket (5 minutes)
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Name Date
Lesson 13: Trigonometry and Complex Numbers
Exit Ticket
1. State the modulus and argument of each complex number. Explain how you know.
a. Ͷ Ͳ݅
b. െʹ ʹ݅
2. Write each number from Problem 1 in polar form.
ߨ ߨ ͷඥ͵ ͷ
3. Explain why ͷ ቀ ቀ ቁ ݅ ቀ ቁቁ and ݅ represent the same complex number.
ʹ ʹ
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Exit Ticket Sample Solutions
1. State the modulus and argument of each complex number. Explain how you know.
a.
The modulus is ǡ and the argument is ι. The real number is units from the origin and lies in the same
position as a ray from the origin containing the real number ǡ so the rotation is ι.
b. െ
The modulus is ξǡ and the argument is ι. The values of ࢇ and ࢈ correspond to sides of a ι–ι–ૢι
right triangle, so the modulus would be ξǡ and the rotation is ι less than ૡι.
2. Write each number from Problem 1 in polar form.
a. ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
b. ξሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
࣊ ࣊ ඥ
3. Explain why ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ and represent the same complex number.
࣊ ࣊
ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
࣊ ࣊ ඥ
If you evaluate ܛܗ܋ቀ ቁ and ܖܑܛቀ ቁ, you get and , respectively.
࣊ ࣊ ξ
ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ ൌ
ඥ ඥ
is thought of as a point with coordinates ൬ ǡ ൰ in the complex plane. The point can also be located by
࣊
thinking of the ray extending from the origin rotated radians. The distance from the origin to the point along that
ray is the modulus, which is units.
Problem Set Sample Solutions
1. Explain why the complex numbers ࢠ ൌ െ ξ, ࢠ ൌ െ ξ, and ࢠ ൌ െ ξ can all have the same
argument. Draw a diagram to support your answer.
They all lie on the same ray from the origin that represents a ι
rotation.
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2. What is the modulus of each of the complex numbers ࢠ , ࢠ , and ࢠ given in Problem 1 above?
The moduli are , , and .
3. Write the complex numbers from Exercise 1 in polar form.
ࢠ ൌ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ࢠ ൌ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ࢠ ൌ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
4. Explain why െ ξ and ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ represent the same number.
The point ൫ǡ െξ൯ lies on a ray from the origin that has been rotated ι rotation from the initial ray. The
distance of this point from the origin along this ray is units (the modulus). Using the definitions of sine and cosine,
any point along that ray will have coordinates ሺ ܛܗ܋ሺιሻ ǡ ܖܑܛሺιሻሻ.
5. Julien stated that a given modulus and a given argument uniquely determine a complex number. Confirm or refute
Julien’s reasoning.
Julien’s reasoning is correct. If you rotate a ray from the origin containing the real number and then locate a point
a fixed number of units along that ray from the origin, it will give you a unique point in the plane.
6. Identify the modulus and argument of the complex number in polar form, convert it to rectangular form, and sketch
the complex number in the complex plane. ι ܚ܉ሺࢠሻ ιor ܚ܉ሺࢠሻ ࣊(radians)
a. ࢠ ൌ ܛܗ܋ሺιሻ ܖܑܛሺιሻ
࢘ ൌ , ܚ܉ሺࢠሻ ൌ ι
ξ
ࢠൌ
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࣊ ࣊
b. ࢠ ൌ ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ , ܚ܉ሺࢠሻ ൌ radians
ࢠ ൌ ξ ξ
࣊ ࣊
c. ࢠ ൌ ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ , ܚ܉ሺࢠሻ ൌ radians
ࢠ ൌ ξ
࣊ ࣊
d. ࢠ ൌ ξ ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ ξ, ܚ܉ሺࢠሻ ൌ radians
ࢠ ൌ െ ξ
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e. ࢠ ൌ ሺܛܗ܋ሺǤ ૠሻ ܖܑܛሺǤ ૠሻሻ
࢘ ൌ , ܚ܉ሺࢠሻ ൌ Ǥ ૠ radians
ࢠ ൌ െ
f. ࢠ ൌ ሺܛܗ܋ሺǤ ૢૡሻ ܖܑܛሺǤ ૢૡሻሻ
࢘ ൌ , ܚ܉ሺࢠሻ ൌ Ǥ ૢૡ radians
ࢠ ൌ െ
g. ࢠ ൌ ξሺܛܗ܋ሺǤ ૡሻ ܖܑܛሺǤ ૡሻሻ
࢘ ൌ ξ, ܚ܉ሺࢠሻ ൌ Ǥ ૡ
ࢠ ൌ െ െ
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࣊ ࣊
h. ࢠ ൌ ξ ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ ξ, ܚ܉ሺࢠሻ ൌ
ࢠ ൌ ξ െ
࣊ ࣊ ࣊ ࣊
7. Convert the complex numbers in rectangular form to polar form. If the argument is a multiple of , , , or ,
࢈
express your answer exactly. If not, use ܖ܉ܜ܋ܚ܉ቀ ቁ to find ܚ܉ሺࢠሻ rounded to the nearest thousandth,
ࢇ
ܚ܉ሺࢠሻ ࣊(radians).
a. ࢠ ൌ ξ
ܚ܉ሺࢠሻ is in Quadrant I.
࢈
ܚ܉ሺࢠሻ ൌ ܖ܉ܜ܋ܚ܉൬ ൰
ࢇ
ൌ ܖ܉ܜ܋ܚ܉൬ ൰
ξ
࣊
ൌ
࢘ ൌ ȁࢠȁ
ൌ ට൫ξ൯ ሺሻ
ൌ
࣊ ࣊
ࢠ ൌ ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ
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b. ࢠ ൌ െ
ܚ܉ሺࢠሻ is in Quadrant II.
࢈
ܚ܉ሺࢠሻ ൌ ࣊ െ ܖ܉ܜ܋ܚ܉൬ ൰
ࢇ
ൌ ࣊ െ ܖ܉ܜ܋ܚ܉൬ ൰
࣊
ൌ ࣊െ
࣊
ൌ
࢘ ൌ ȁࢠȁ
ൌ ඥሺെሻ ሺሻ
ൌ ξ
࣊ ࣊
ࢠ ൌ ξ ൬ ܛܗ܋൬ ൰ ܖܑܛ൬ ൰൰
c. ࢠ ൌ െ ξ
ܚ܉ሺࢠሻ is in Quadrant IV.
࢈
ܚ܉ሺࢠሻ ൌ ࣊ െ ܖ܉ܜ܋ܚ܉൬ ൰
ࢇ
ξ
ൌ ࣊ െ ܖ܉ܜ܋ܚ܉ቆ ቇ
࣊
ൌ ࣊ െ
࣊
ൌ radians
࢘ ൌ ȁࢠȁ
ൌ ටሺሻ ൫െξ൯
ൌ
࣊ ࣊
ࢠ ൌ ൬ ܛܗ܋൬ ൰ ܖܑܛ൬ ൰൰
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d. ࢠ ൌ െ െ
ܚ܉ሺࢠሻ is in Quadrant III.
ܚ܉ሺࢠሻ ൌ ࣊ ܖ܉ܜ܋ܚ܉൬ ൰
ൌ ࣊ ܖ܉ܜ܋ܚ܉൬ ൰
ξ
ൎ Ǥ radians
࢘ ൌ ȁࢠȁ
ൌ ඥሺെሻ ሺെሻ
ൌ
ࢠ ൌ ሺܛܗ܋ሺǤ ሻ ܖܑܛሺǤ ሻሻ
e. ࢠ ൌ ૠ െ
ܚ܉ሺࢠሻ is in Quadrant IV.
࢈
ܚ܉ሺࢠሻ ൌ ࣊ െ ܖ܉ܜ܋ܚ܉൬ ൰
ࢇ
ൌ ࣊ െ ܖ܉ܜ܋ܚ܉൬ ൰
ૠ
ൎ Ǥ ૢૢ radians
࢘ ൌ ȁࢠȁ
ൌ ඥሺૠሻ ሺെሻ
ൌ
ࢠ ൌ ሺܛܗ܋ሺǤ ૢૢሻ ܖܑܛሺǤ ૢૢሻሻ
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8. Show that the following complex numbers have the same argument.
a. ࢠ ൌ ξ andࢠ ൌ ξ
ඥ ࣊ ࣊
ܚ܉ሺࢠ ሻ ൌ ܖ܉ܜ܋ܚ܉൬ ൰ ൌ and ܚ܉ሺࢠ ሻ ൌ ൫ξ൯ ൌ
b. ࢠ ൌ andࢠ ൌ
࣊ ࣊
ܚ܉ሺࢠ ሻ ൌ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ and ܚ܉ሺࢠ ሻ ൌ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ
9. A square with side length of one unit is shown below. Identify a complex number in polar form that corresponds to
each point on the square.
ൌ ξሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ൌ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ൌ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ࡰ ൌ ሺܛܗ܋ሺૢιሻ ܖܑܛሺૢιሻሻ
10. Determine complex numbers in polar form whose coordinates are the vertices of the square shown below.
Ԣ ൌ ሺܛܗ܋ሺૢιሻ ܖܑܛሺૢιሻሻ
Ԣ ൌ ξሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
Ԣ ൌ ሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
ࡰԢ ൌ ξሺܛܗ܋ሺιሻ ܖܑܛሺιሻሻ
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11. How do the modulus and argument of coordinate in Problem 9 correspond to the modulus and argument of point
Ԣin Problem 10? Does a similar relationship exist when you compare toԢ, to Ԣ, and ࡰ to ࡰԢ? Explain why
you think this relationship exists.
The modulus multiplied by a factor of ξ and the argument is ι more. The same is true when you compare to
Ԣ and ࡰ to ࡰԢ. The relationship could also be true for and Ԣ, although the argument of and Ԣ can really be
any number since the modulus is .
12. Describe the transformations that map ࡰ to ԢԢԢࡰԢ.
Rotate by ι counterclockwise, and then dilate by a factor of ξ.
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Trigonometry Review: Additional Resources
1. Evaluate the following.
a. ሺ͵Ͳιሻ ߨ
b. ቀ ቁ
͵
c. ሺʹʹͷιሻ ͷߨ
d. ቀ ቁ
ͷߨ f. ሺ͵͵Ͳιሻ
e. ቀ ቁ
͵
2. Solve for the acute angle ߠ, both in radians and degrees, in a right triangle if you are given the opposite side, ܱ, and
adjacent side, ܣ. Round to the nearest thousandth.
a. ܱ ൌ ͵ and ܣൌ Ͷ
ܱ
b. ܱ ൌ and ܣൌ ͳ
ߠ
ܣ
c. ܱ ൌ ͵ξ͵ and ܣൌ ʹ
3. Convert angles in degrees to radians, and convert angles in radians to degrees.
a. ͳͷͲι
ସగ
b.
ଷ
ଷగ
c.
ସ
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Trigonometry Review: Additional Resources
1. Evaluate the following.
࣊
a. ܖܑܛሺιሻ b. ܛܗ܋ቀ ቁ
࣊
c. ܖܑܛሺιሻ d. ܛܗ܋ቀ ቁ
ξ ξ
െ െ
࣊
e. ܖܑܛቀ ቁ f. ܛܗ܋ሺιሻ
ξ ξ
െ
2. Solve for the acute angle ࣂ, both in radians and degrees, in a right triangle if you are given the opposite side, ࡻ, and
adjacent side, . Round to the nearest thousandth.
a. ࡻ ൌ and ൌ
ܖ܉ܜ܋ܚ܉൬ ൰ ൎ Ǥ radians ൌ Ǥ ૡૢૡι
ܱ
b. ࡻ ൌ and ൌ
ߠ
ܖ܉ܜ܋ܚ܉൬ ൰ ൎ Ǥ radians ൌ ૡǤ ૡι
ܣ
c. ࡻ ൌ ξ and ൌ
ξ
ܖ܉ܜ܋ܚ܉ቆ ቇ ൎ Ǥ radians ൌ ૡǤ ૢૡι
3. Convert angles in degrees to radians, and convert in radians to degrees.
a. ι
࣊
࣊
b.
ι
࣊
c.
ι
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Lesson 14: Discovering the Geometric Effect of Complex
Multiplication
Student Outcomes
Students determine the geometric effects of transformations of the form ܮሺݖሻ ൌ ܽݖ, ܮሺݖሻ ൌ ሺܾ݅ሻݖ, and
ܮሺݖሻ ൌ ሺܽ ܾ݅ሻ ݖfor real numbers ܽ and ܾ.
Lesson Notes
In this lesson, students observe the geometric effect of transformations of the form ܮሺݖሻൌሺܾܽ݅ሻ ݖon a unit square
and formulate conjectures. Today’s observations are mathematically established in the following lesson. As in the
previous lessons, this lesson continues to associate points ሺܽǡܾሻ in the coordinate plane with complex numbers ܾܽ݅,
where ܽ and ܾ are real numbers. The Problem Set includes another chance to revisit the definition and the idea of a
linear transformation. Showing that these transformations are linear also provides algebraic fluency practice with
complex numbers.
Classwork
Exercises 1–5 (10 minutes)
Exercises
The vertices ሺǡ ሻ, ሺǡ ሻ, ሺǡ ሻ, and ࡰሺǡ ሻ of a unit square can be represented by the complex numbers ൌ ,
ൌ , ൌ , and ࡰ ൌ .
1. Let ࡸ ሺࢠሻ ൌ െࢠ.
a. Calculate Ԣ ൌ ࡸ ሺሻ, Ԣ ൌ ࡸ ሺሻ,Ԣ ൌ ࡸ ሺሻ,
and ࡰԢ ൌ ࡸ ሺࡰሻ. Plot these four points on the
axes.
b. Describe the geometric effect of the linear
transformation ࡸ ሺࢠሻ ൌ െࢠ on the square ࡰ.
Transformation ࡸ rotates the square ࡰ by
ૡι about the origin.
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2. Let ࡸ ሺࢠሻ ൌ ࢠ.
a. Calculate Ԣ ൌ ࡸ ሺሻ, Ԣ ൌ ࡸ ሺሻ,Ԣ ൌ ࡸ ሺሻ,
and ࡰ ൌ ࡸ ሺࡰሻ. Plot these four points on the
axes.
b. Describe the geometric effect of the linear
transformation ࡸ ሺࢠሻ ൌ ࢠ on the square ࡰ.
Transformation ࡸ dilates the square ࡰ by a
factor of.
3. Let ࡸ ሺࢠሻ ൌ ࢠ.
a. Calculate Ԣ ൌ ࡸ ሺሻ, Ԣ ൌ ࡸ ሺሻ,Ԣ ൌ ࡸ ሺሻ,
and ࡰԢ ൌ ࡸ ሺࡰሻ. Plot these four points on the
axes.
b. Describe the geometric effect of the linear
transformation ࡸ ሺࢠሻ ൌ ࢠ on the square ࡰ.
Transformation ࡸ rotates the square ࡰ by
ૢι counterclockwise about the origin.
4. Let ࡸ ሺࢠሻ ൌ ሺሻࢠ.
a. Calculate Ԣ ൌ ࡸ ሺሻ, Ԣ ൌ ࡸ ሺሻ, Ԣ ൌ ࡸ ሺሻ,
and ࡰԢ ൌ ࡸ ሺࡰሻ. Plot these four points on the
axes.
b. Describe the geometric effect of the linear
transformation ࡸ ሺࢠሻ ൌ ሺሻࢠ on the square
ࡰ.
Transformation ࡸ rotates the square ࡰ by
ૢι counterclockwise about the origin and
dilates by a factor of .
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5. Explain how transformations ࡸ , ࡸ , and ࡸ are related.
Transformation ࡸ is the result of doing transformations ࡸ and ࡸ (in either order).
Discussion (8 minutes)
What is the geometric effect of the transformation ܮሺݖሻ ൌ ܽ ݖfor a real number ܽ Ͳ?
à The effect of ܮis dilation by the factor ܽ.
What happens to a unit square in this case?
à The orientation of the square does not change; it is not reflected or rotated, but the sides of the square
are dilated by ܽ.
What is the effect on the square if ܽ ͳ?
à The sides of the square will get larger.
What is the effect on the square if Ͳ ൏ ܽ ൏ ͳ?
à The sides of the square will get smaller.
What is the geometric effect of the transformation ܮሺݖሻ ൌ ܽ ݖif ܽ ൌ Ͳ?
à If ܽ ൌ Ͳ, then ܮሺݖሻ ൌ Ͳ for every complex number ݖ. This transformation essentially shrinks the square
down to the point at the origin.
What is the geometric effect of the transformation ܮሺݖሻ ൌ ܽ ݖfor a real number ܽ ൏ Ͳ?
à If ܽ ൏ Ͳ, then ܮሺݖሻ ൌ ܽ ݖൌ െȁܽȁ ݖ, so ܮis a dilation by ȁܽȁ and a rotation by ͳͺͲι. This transformation
will dilate the original unit square and then rotate it about point ܣinto the third quadrant.
What is the geometric effect of the transformation ܮሺݖሻ ൌ ሺܾ݅ሻ ݖfor a real number ܾ Ͳ?
à The transformation ܮdilates by ܾ and rotates by ͻͲι counterclockwise.
What is the effect on the unit square if ܾ ͳ?
à The sides of the square will get larger.
What is the effect on the unit square if Ͳ ൏ ܾ ൏ ͳ?
à The sides of the square will get smaller.
What is the effect on the unit square if ܾ ൏ Ͳ?
à If ܾ ൏ Ͳ, then ܮሺݖሻ ൌ ሺܾ݅ሻ ݖൌ ݅ሺܾݖሻ, so ܮis a dilation by ȁܾȁ and a rotation by ͳͺͲι, followed by a
rotation by ͻͲι. This transformation will rotate and dilate the original unit square and then rotate it
about point ܣto the fourth quadrant.
Exercise 6 (6 minutes)
6. We will continue to use the unit square ࡰ with ൌ , ൌ , ൌ ,ࡰ ൌ for this exercise.
a. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ࢠ on the unit square?
By our work in the first five exercises and the previous discussion, we know that this transformation dilates
the unit square by a factor of .
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b. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺሻࢠ on the unit square?
By our work in the first five exercises, this transformation will dilate the unit square by a factor of and
rotate it ૢιcounterclockwise about the origin.
c. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ on the unit square?
Since ൌ െ, this transformation is ࡸሺࢠሻ ൌ െࢠ, which will dilate the unit square by and rotate it ૡι
about the origin.
d. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ on the unit square?
Since ൌ െ, this transformation is ࡸሺࢠሻ ൌ ሺെሻࢠ, which will dilate the unit square by a factor of and
rotate it ૠιcounterclockwise about the origin.
e. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ on the unit square?
Since ൌ ሺ ሻ ൌ ሺെሻ ൌ , this transformation is ࡸሺࢠሻ ൌ ࢠ, which is the same transformation as in part
(a). Thus, this transformation dilates the unit square by a factor of .
f. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ on the unit square?
Since ൌ ڄൌ , this is the same transformation as in part (b). This transformation will dilate the unit
square by a factor of and rotate it ૢι counterclockwise about the origin.
g. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ on the unit square, for some integer
?
If is a multiple of , then ࡸሺࢠሻ ൌ ሺ ሻࢠ ൌ ࢠ will dilate the unit square by a factor of .
If is one more than a multiple of , then ࡸሺࢠሻ ൌ ሺ ሻࢠ ൌ ሺሻࢠ will dilate the unit square by a factor of
and rotate it ૢιcounterclockwise about the origin.
If is two more than a multiple of , then ࡸሺࢠሻ ൌ ሺ ሻࢠ ൌ െࢠ will dilate the unit square by and rotate it
ૡι about the origin.
If is three more than a multiple of , then ࡸሺࢠሻ ൌ ሺ ሻࢠ ൌ ሺെሻࢠ will dilate the unit square by a factor of
and rotate it ૠιcounterclockwise about the origin.
Exploratory Challenge (12 minutes)
Divide students into at least eight groups of two or three students each. Assign each group to the 1-team, 2-team,
3-team, or 4-team. There should be at least two groups on each team so that students can check their answers against
another group when the results are shared at the end of the exercises. Each team explores a different transformation of
the form ܮሺݖሻ ൌ ሺܽ ܾ݅ሻݖ.
Scaffolding:
Before students begin working on the Exploratory Challenge, ask the following:
For struggling students,
What is the geometric effect of the transformation ܮሺݖሻ ൌ ͵?ݖ accompany this discussion
à This transformation will dilate by a factor of three. with a visual
What is the geometric effect of the transformation ܮሺݖሻ ൌ െ͵?ݖ representation of each
transformation on the unit
à This transformation will dilate by a factor of three and rotate by ͳͺͲι square ܦܥܤܣ.
about the origin.
Omit this discussion for
advanced students.
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What is the geometric effect of the transformation ܮሺݖሻ ൌ Ͷ݅?ݖ
à This transformation will dilate by a factor of four and rotate by ͻͲι about the origin.
What is the geometric effect of the transformation ܮሺݖሻ ൌ െͶ݅?ݖ
à This transformation will dilate by a factor of four and rotate by ʹͲι about the origin.
Exploratory Challenge
Your group has been assigned either to the 1-team, 2-team, 3-team, or 4-team. Each team will answer the questions
below for the transformation that corresponds to their team number:
ࡸ ሺࢠሻ ൌ ሺ ሻࢠ
ࡸ ሺࢠሻ ൌ ሺെ ሻࢠ
ࡸ ሺࢠሻ ൌ ሺെ െ ሻࢠ
ࡸ ሺࢠሻ ൌ ሺ െ ሻࢠǤ
The unit square ࡰ with ൌ , ൌ , ൌ ,ࡰ ൌ is shown below. Apply your transformation to the vertices of
the square ࡰǡ and plot the transformed points Ԣ, Ԣ, Ԣ, and ࡰԢ on the same coordinate axes.
The solution shown below is for transformation ܮଵ . The transformed square for ܮଶ , ܮଷ , and ܮସ will be rotated ͻͲι, ͳͺͲι,
and ʹͲι counterclockwise about the origin from the one shown, respectively.
For the 1-team:
a. Why is ᇱ ൌ ?
Because ൌ , we have ᇱ ൌ ࡸ ሺሻ ൌ ሺ ሻሺሻ ൌ .
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b. What is the argument of ?
The argument of is the amount of counterclockwise rotation between the positive ࢞-axis and the ray
connecting the origin and the point ሺǡ ሻ.
c. What is the modulus of ?
The modulus of is ȁ ȁ ൌ ξ ൌ ξ ൌ .
For the 2-team:
a. Why is ᇱ ൌ െ ?
Because ൌ , we have ᇱ ൌ ࡸ ሺሻ ൌ ሺെ ሻሺሻ ൌ െ .
b. What is the argument of െ ?
The argument of െ is the amount of counterclockwise rotation between the positive ࢞-axis and the ray
connecting the origin and the point ሺെǡ ሻ.
c. What is the modulus of െ ?
The modulus of െ is ȁെ ȁ ൌ ඥሺെሻ ൌ ξ ൌ .
For the 3-team:
a. Why is ᇱ ൌ െ െ ?
Because ൌ , we have ᇱ ൌ ࡸ ሺሻ ൌ ሺെ െ ሻሺሻ ൌ െ െ .
b. What is the argument of െ െ ?
The argument of െ െ is the amount of counterclockwise rotation between the positive ࢞-axis and the ray
connecting the origin and the point ሺെǡ െሻ.
c. What is the modulus of െ െ ?
The modulus of െ െ is ȁെ െ ȁ ൌ ඥሺെሻ ሺെሻ ൌ ξ ൌ .
For the 4-team:
a. Why is ᇱ ൌ െ ?
Because ൌ , we have ᇱ ൌ ࡸ ሺሻ ൌ ሺ െ ሻሺሻ ൌ െ .
b. What is the argument of െ ?
The argument of െ is the amount of counterclockwise rotation between the positive ࢞-axis and the ray
connecting the origin and the point ሺǡ െሻ.
c. What is the modulus of െ ?
The modulus of െ is ȁ െ ȁ ൌ ඥ ሺെሻ ൌ ξ ൌ .
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All groups should also answer the following:
a. Describe the amount the square has been rotated counterclockwise.
The square has been rotated the amount of counterclockwise rotation between the positive ࢞-axis and ray
ሬሬሬሬሬሬሬԦ
Ԣ.
b. What is the dilation factor of the square? Explain how you know.
First, we need to calculate the length of one side of the square. The length ᇱ is given by
ᇱ ൌ ඥሺ െ ሻ ሺ െ ሻ ൌ ǤThen, the dilation factor of the square is because the final square has
sides that are five times longer than the sides of the original square.
c. What is the geometric effect of your transformation ࡸ , ࡸ , ࡸ , or ࡸ on the unit square ࡰ?
(Answered for transformation ࡸ .) The transformation rotates the square counterclockwise by the argument
of ሺ ሻ and dilates it by a factor of the modulus of .
d. Make a conjecture: What do you expect to be the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ on
the unit square ࡰ?
This transformation should rotate the square counterclockwise by the argument of and dilate it by a
factor of ȁ ȁ ൌ ξ ൌ ξ.
e. Test your conjecture with the unit square on the axes below.
Closing (5 minutes)
Ask one group from each team to share their results from the Exploratory Challenge at the front of the class. Be sure
that each group has made the connection that if the transformation is given by ܮሺݖሻ ൌ ሺܽ ܾ݅ሻݖ, then the geometric
effect of the transformation is to dilate by ȁܽ ܾ݅ȁ and to rotate by ሺܽ ܾ݅ሻ.
Exit Ticket (4 minutes)
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Name Date
Lesson 14: Discovering the Geometric Effect of Complex
Multiplication
Exit Ticket
1. Identify the linear transformation ܮthat takes square
ܦܥܤܣto square ܣԢܤԢܥԢܦԢ as shown in the figure on the
right.
2. Describe the geometric effect of the transformation
ܮሺݖሻ ൌ ሺͳ െ ͵݅ሻ ݖon the unit square ܦܥܤܣ, where
ܣൌ Ͳ, ܤൌ ͳ, ܥൌ ͳ ݅, and ܦൌ ݅. Sketch the unit
square transformed by ܮon the axes on the right.
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Exit Ticket Sample Solutions
1. Identify the linear transformation ࡸ that takes square
ࡰ to square ԢԢԢࡰԢ as shown in the figure on the
right.
The transformation ࡸ takes the point ൌ to the point
ᇱ ൌ െ െ , so this transformation is given by
ࡸሺࢠሻ ൌ ሺെ െ ሻࢠ.
2. Describe the geometric effect of the transformation
ࡸሺࢠሻ ൌ ሺ െ ሻࢠ on the unit square ࡰ, where
ൌ , ൌ , ൌ , and ࡰ ൌ . Sketch the unit
square transformed by ࡸ on the axes on the right.
This transformation dilates by ȁ െ ȁ ൌ ξ ൌ ξ
and rotates counterclockwise by ܚ܉ሺ െ ሻ.
Problem Set Sample Solutions
3. Find the modulus and argumentfor each of the following complex numbers.
ඥ
a. ࢠ ൌ
ඥ ࣊
ฬ ฬ ൌ , ࢠ is in Quadrant ࡵ; thus, ܚ܉ሺࢠ ሻ ൌ ܖ܉ܜ܋ܚ܉൭ ඥ ൱ ൌ ι ൌ ܌܉ܚ.
b. ࢠ ൌ ξ
ඥ ࣊
ห ξห ൌ , ࢠ is in Quadrantࡵ; thus, ܚ܉ሺࢠ ሻ ൌ ܖ܉ܜ܋ܚ܉൬ ൰ ൌ ι ൌ ܌܉ܚ.
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c. ࢠ ൌ െ
ȁ ȁ ൌ ξ, ࢠ is in Quadrant ࡵࡵ; thus, ܚ܉ሺࢠሻ ൌ ࣊ െ ܖ܉ܜ܋ܚ܉ቀ ቁ ൎ ࣊ െ Ǥ ൎ Ǥ ܌܉ܚ.
d. ࢠ ൌ െ െ
࣊ ࣊
ȁെ െ ȁ ൌ ξ, ࢠ is in Quadrant ࡵࡵࡵ; thus, ܚ܉ሺࢠ ሻ ൌ ࣊ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ ࣊ ൌ ܌܉ܚ.
e. ࢠ ൌ െ
࣊ ૠ࣊
ȁ ȁ ൌ ξ, ࢠ is in Quadrant ࡵࢂ; thus, ܚ܉ሺࢠ ሻ ൌ ࣊ െ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ ࣊ െ ൌ ܌܉ܚ.
f. ࢠ ൌ െ
ȁ െ ȁ ൌ ξ, ࢠ is in Quadrant ࡵࢂ; thus, ܚ܉ሺࢠ ሻ ൌ ࣊ െ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ ࣊ െ Ǥ ૠ ൌ Ǥ ૠ܌܉ܚ.
4. For parts (a)–(c), determine the geometric effect of the specified transformation.
a. ࡸሺࢠሻ ൌ െࢠ
The transformation ࡸ dilates by and rotates by ૡι about the origin.
b. ࡸሺࢠሻ ൌ െࢠ
The transformation ࡸ dilates by and rotates by ૡι about the origin.
c. ࡸሺࢠሻ ൌ െ ࢠ
The transformation ࡸ dilates by and rotates by ૡι about the origin.
d. Describe the geometric effect of the transformation ࡸሺࢠሻ ൌ ࢇࢠ for any negative real number ࢇ.
The transformation ࡸ dilates by ȁࢇȁ and rotates by ૡι about the origin.
5. For parts (a)–(c), determine the geometric effect of the specified transformation.
a. ࡸሺࢠሻ ൌ ሺെሻࢠ
The transformation ࡸ dilates by and rotates counterclockwise by ૠι about the origin.
b. ࡸሺࢠሻ ൌ ሺെሻࢠ
The transformation ࡸ dilates by and rotates by ૠι about the origin.
c. ࡸሺࢠሻ ൌ ቀെ ቁ ࢠ
The transformation ࡸ dilates by and rotates counterclockwise by ૠι about the origin.
d. Describe the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ࢈ሻࢠ for any negative real number ࢈.
The transformation ࡸ dilates by ȁ࢈ȁ and rotates by ૠι counterclockwise about the origin.
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6. Suppose that we have two linear transformations, ࡸ ሺࢠሻ ൌ ࢠ and ࡸ ሺࢠሻ ൌ ሺሻࢠ.
a. What is the geometric effect of first performing transformation ࡸ and then performing transformation ࡸ ?
The transformation ࡸ dilates by , dilates by , and rotates by ૢι counterclockwise about the origin.
b. What is the geometric effect of first performing transformation ࡸ and then performing transformationࡸ ?
The transformation ࡸ dilates by , rotates by ૢι counterclockwise about the origin, and then dilates by .
c. Are your answers to parts (a) and (b) the same or different? Explain how you know.
The answers are the same.
ࡸ ሺࡸ ሺࢠሻሻ ൌ ሺሻࡸ ሺࢠሻ ൌ ሺሻሺࢠሻ ൌ ሺሻࢠ ࡸ ሺࡸ ሺࢠሻሻ ൌ ࡸ ሺࢠሻ ൌ ൫ሺሻࢠ൯ ൌ ሺሻࢠ
For example, let ࢠ ൌ െ .
ࡸ ൌ ሺ െ ሻ ൌ െ ૢ
ࡸ ൌ ሺሻሺ െ ሻ ൌ
ࡸ ሺࡸ ሻ ൌ ሺሻሺ െ ૢሻ ൌ
ࡸ ሺࡸ ሻ ൌ ሺ ሻ ൌ
7. Suppose that we have two linear transformations, ࡸ ሺࢠሻ ൌ ሺ ሻࢠ and ࡸ ሺࢠሻ ൌ െࢠ. What is the geometric effect
of first performing transformation ࡸ and then performing transformation ࡸ ?
We have ȁ ȁ ൌ , and the argument of is ܖ܉ܜ܋ܚ܉ቀ ቁ ൎ Ǥ radians, which is about Ǥ ૡૠι.
Therefore, the transformation ࡸ followed byࡸ dilates with scale factor , rotates by approximately Ǥ ૡૠι
counterclockwise, and then rotates by ૡιǤ
8. Suppose that we have two linear transformations, ࡸ ሺࢠሻ ൌ ሺ െ ሻࢠ and ࡸ ሺࢠሻ ൌ െࢠ. What is the geometric effect
of first performing transformation ࡸ and then performing transformation ࡸ ?
We see that ȁ െ ȁ ൌ , and the argument of െ is ܖ܉ܜ܋ܚ܉ቀ ቁ ൎ ࣊ െ Ǥ radians, which is about
Ǥ ૡૠι. Therefore, the transformation ࡸ followed byࡸ dilates with scale factor , rotates by approximately
Ǥ ૡૠι counterclockwise, and then rotates by ૡι.
9. Explain the geometric effect of the linear transformation ࡸሺࢠሻ ൌ ሺࢇ െ ࢈ሻࢠ, where ࢇ and ࢈ are positive real
numbers.
Note that the complex number ࢇ െ ࢈ is represented by a point in the fourth quadrant. The transformation ࡸ dilates
࢈
with scale factor ȁࢇ െ ࢈ȁ and rotates counterclockwise by ࣊ െ ܖ܉ܜ܋ܚ܉ቀ ቁ.
ࢇ
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10. In Geometry, we learned the special angles of a right triangle whose hypotenuse is unit. The figures are shown
above. Describe the geometric effect of the following transformations.
ඥ
a. ࡸ ሺࢠሻ ൌ ൬ ൰ ࢠ
ξ ࣊
ቤ ቤ ൌ , ܚ܉ሺࢠሻ ൌ ι ൌ ܌܉ܚ
The transformation ࡸ rotates counterclockwise by ι.
b. ࡸ ሺࢠሻ ൌ ൫ ξ൯ࢠ
࣊
ห ξห ൌ ,ܚ܉ሺࢠሻ ൌ ι ൌ ܌܉ܚ
The transformation ࡸ dilates with scale factor and rotates counterclockwise by ι.
ඥ ඥ
c. ࡸ ሺࢠሻ ൌ ൬ ൰ ࢠ
ξ ξ ࣊
ቤ ቤ ൌ ,ܚ܉ሺࢠሻ ൌ ι ൌ ܌܉ܚ
The transformation ࡸ dilates by and rotates counterclockwise by ι.
d. ࡸ ሺࢠሻ ൌ ሺ ሻࢠ
࣊
ȁ ȁ ൌ ξ, ܚ܉ሺࢠሻ ൌ ι ൌ ܌܉ܚ
The transformation ࡸ dilates with scale factor ξ and rotates counterclockwise by ι.
11. Recall that a function ࡸ is a linear transformation if all ࢠ and࢝ in the domain of ࡸ and all constants ࢇ meet the
following two conditions:
i. ࡸሺࢠ ࢝ሻ ൌ ࡸሺࢠሻ ࡸሺ࢝ሻ
ii. ࡸሺࢇࢠሻ ൌ ࢇࡸሺࢠሻ
Show that the following functions meet the definition of a linear transformation.
a. ࡸ ሺࢠሻ ൌ ࢠ
ࡸ ሺࢠ ࢝ሻ ൌ ሺࢠ ࢝ሻ ൌ ࢠ ࢝ ൌ ࡸ ሺࢠሻ ࡸ ሺ࢝ሻ
ࡸ ሺࢇࢠሻ ൌ ሺࢇࢠሻ ൌ ࢇࢠ ൌ ࢇሺࢠሻ ൌ ࢇࡸ ሺࢠሻ
b. ࡸ ሺࢠሻ ൌ ࢠ
ࡸ ሺࢠ ࢝ሻ ൌ ሺࢠ ࢝ሻ ൌ ࢠ ࢝ ൌ ࡸ ሺࢠሻ ࡸ ሺ࢝ሻ
ࡸ ሺࢇࢠሻ ൌ ሺࢇࢠሻ ൌ ࢇࢠ ൌ ࢇሺࢠሻ ൌ ࢇࡸ ሺࢠሻ
c. ࡸ ሺࢠሻ ൌ ሺ ሻࢠ
ࡸ ሺࢠ ࢝ሻ ൌ ሺ ሻሺࢠ ࢝ሻ ൌ ሺ ሻࢠ ሺ ሻ࢝ ൌ ࡸ ሺࢠሻ ࡸ ሺ࢝ሻ
ࡸ ሺࢇࢠሻ ൌ ሺ ሻሺࢇࢠሻ ൌ ሺ ሻࢇࢠ ൌ ࢇሺሺ ሻࢠሻ ൌ ࢇࡸ ሺࢠሻ
12. The vertices ሺǡ ሻ,ሺǡ ሻ,ሺǡ ሻ,ࡰሺǡ ሻ of a unit square can be represented by the complex numbers ൌ ,
ൌ , ൌ ,ࡰ ൌ . We learned that multiplication of those complex numbers by rotates the unit square by
ૢι counterclockwise. What do you need to multiply by so that the unit square will be rotated by ૢι clockwise?
We need to multiply by ൌ െ.
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Lesson 15: Justifying the Geometric Effect of Complex
Multiplication
Student Outcomes
Students understand why the geometric transformation effect of the linear transformation ܮሺݖሻ ൌ ݖݓis
dilation by ȁݓȁ and rotation by the argument of ݓ.
Lesson Notes (optional)
In Lesson 13, students observed that the transformation ܮሺݖሻ ൌ ሺ͵ Ͷ݅ሻ ݖhas the geometric effect of a rotation by the
argument of ͵ Ͷ݅ and a dilation by the modulus ȁ͵ Ͷ݅ȁൌ ͷ. In this lesson, this result is generalized to a linear
transformation ܮሺݖሻൌ ݖݓfor a complex number ݓ, using the geometric representation of a complex number as a point
in the complex plane. However, before they begin thinking about the transformation ܮ, students first need to
represent multiplication of complex numbers geometrically on the complex plane, so that is where this lesson begins.
This lesson covers one of nine cases for the geometric position of the complex scalar ݓin the coordinate plane, and the
remaining cases are carefully scaffolded in the Problem Set. Consider extending this to a two-day lesson and having
students work in groups on these remaining cases during the second day of class. Consider having groups present the
remaining eight cases to the rest of the class.
Classwork
Opening Exercise (8 minutes)
In the Opening Exercise, students review complex multiplication and consider it geometrically to justify the geometric
effect of a linear transformation ܮሺݖሻ ൌ ሺܽ ܾ݅ሻ ݖdiscovered in Lesson 13.
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Opening Exercise
For each exercise below, compute the product ࢝ࢠ. Then, plot the complex numbers ࢠ, ࢝, and ࢝ࢠ on the axes provided.
a. ࢠ ൌ , ࢝ ൌ
࢝ࢠ ൌ ሺ ሻሺ ሻ
ൌ
ൌ െ ૠ
ൌ ૠ
b. ࢠ ൌ , ࢝ ൌ െ
࢝ࢠ ൌ ሺ ሻሺെ ሻ
ൌ െ െ ૡ
ൌ െ െ ૡ
ൌ െૢ
c. ࢠ ൌ െ , ࢝ ൌ െ െ
࢝ࢠ ൌ ሺെ ሻሺെ െ ሻ
ൌ െ െ
ൌ െ
ൌ െ
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d. For each part (a), (b), and (c), draw line segments connecting each point ࢠ, ࢝, and ࢝ࢠ to the origin.
Determine a relationship between the arguments of the complex numbers ࢠ, ࢝, and ࢝ࢠ.
It appears that the argument of ࢝ࢠ is the sum of the arguments of ࢠ and ࢝.
Discussion (5 minutes)
This Discussion outlines the point of the lesson. It is claimed that the geometric effect of the linear transformation
ܮሺݖሻ ൌ ݖݓfor complex numbers ݓis twofold: a dilation by ȁݓȁ and a rotation by the argument of ݓ. The teacher then
leads students through the justification for why these observations hold in every case. The observation was made in
Lesson 13 using the particular examples ܮଵ ሺݖሻ ൌ ሺ͵ Ͷ݅ሻݖ, ܮଶ ሺݖሻ ൌ ሺെ͵ Ͷ݅ሻݖ, ܮଷ ሺݖሻ ൌ ሺെ͵ െ Ͷ݅ሻݖ, and
ܮସ ሺݖሻ ൌ ሺ͵ െ Ͷ݅ሻݖǤ In the lesson itself, only the case of ܮሺݖሻ ൌ ሺܽ ܾ݅ሻݖwhere ܽ Ͳ and ܾ Ͳ is addressed. The
remaining cases are included in the Problem Set.
At the end of Lesson 13, what did you discover about the geometric effects of the transformations
ܮଵ ሺݖሻ ൌ ሺ͵ Ͷ݅ሻݖ, ܮଶ ሺݖሻ ൌ ሺെ͵ Ͷ݅ሻݖ, ܮଷ ሺݖሻ ൌ ሺെ͵ െ Ͷ݅ሻݖ, and ܮସ ሺݖሻ ൌ ሺ͵ െ Ͷ݅ሻ?ݖ
à These transformations had the geometric effect of dilation by ȁ͵ Ͷ݅ȁ ൌ ͷ and rotation by the
argument of ͵ Ͷ݅ (or ͵ െ Ͷ݅, െ͵ െ Ͷ݅, െ͵ Ͷ݅, as appropriate).
Can we generalize this result to any linear transformation ܮሺݖሻ ൌ ݖݓ, for a complex number ?ݓWhy or why
not?
à Yes, it seems that we can generalize this. We tried it for ܮሺݖሻ ൌ ሺʹ ݅ሻݖ, and it worked.
For a general linear transformation ܮሺݖሻ ൌ ݖݓ, what do we need to establish in order to generalize what we
discovered in Lesson 13?
Students may struggle with stating these ideas using proper mathematical terminology. Allow them time to grapple with
the phrasing before providing the correct terminology.
à We need to show that the modulus of ܮሺݖሻ is equal to the product of the modulus of ݓand the modulus
of ݖǤ That is, we need to show that ȁܮሺݖሻȁ ൌ ȁݓȁ ڄȁݖȁǤ
à We need to show that the angle made by the ray through the origin and ݖis a rotation of the ray
through the origin and ܮሺݖሻ by ሺݓሻ. That is, we need to show that ൫ܮሺݖሻ൯ ൌ ሺݓሻ ሺݖሻ.
Exercises 1–2 (5 minutes)
Exercises
1. Let ࢝ ൌ ࢇ ࢈ and ࢠ ൌ ࢉ ࢊ.
Scaffolding:
a. Calculate the product ࢝ࢠ.
Allow struggling students
࢝ࢠ ൌ ሺࢇ ࢈ሻሺࢉ ࢊሻ
to complete these
ൌ ሺࢇࢉ െ ࢈ࢊሻ ሺࢇࢊ ࢈ࢉሻ exercises for concrete
values of ݖand ݓ, such as
ݖൌ Ͷ െ ͵݅ and
ݓൌ ͷ ͳʹ݅.
Ask advanced students to
think about the
relationship between the
arguments of ݓand ݖ.
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b. Calculate the moduli ȁ࢝ȁ, ȁࢠȁ, and ȁ࢝ࢠȁ.
ȁ࢝ȁ ൌ ඥࢇ ࢈
ȁࢠȁ ൌ ඥࢉ ࢊ
ȁ࢝ࢠȁ ൌ ඥሺࢇࢉ െ ࢈ࢊሻ ሺࢇࢊ ࢈ࢉሻ
ൌ ඥࢇ ࢉ െ ࢇ࢈ࢉࢊ ࢈ ࢊ ࢇ ࢊ ࢇ࢈ࢉࢊ ࢈ ࢉ
ൌ ඥࢇ ሺࢉ ࢊ ሻ ࢈ ሺࢉ ࢊ ሻ
ൌ ඥሺࢇ ࢈ ሻሺࢉ ࢊ ሻ
c. What can you conclude about the quantities ȁ࢝ȁ, ȁࢠȁ, and ȁ࢝ࢠȁ?
From part (b), we can see that ȁ࢝ࢠȁ ൌ ȁ࢝ȁ ڄȁࢠȁ.
2. What does the result of Exercise 1 tell us about the geometric effect of the transformation ࡸሺࢠሻ ൌ ࢝ࢠ?
We see that ȁࡸሺࢠሻȁ ൌ ȁ࢝ࢠȁ ൌ ȁ࢝ȁ ڄȁࢠȁ, so the transformation ࡸdilates by a factor of ȁ࢝ȁ.
Discussion (15 minutes)
In this Discussion, lead students through the geometric argument that ሺݖݓሻ is the sum of ሺݓሻand ሺݖሻǤ The
images presented here show one of many cases, but the mathematics is not dependent on the case. The remaining
cases are addressed in the Problem Set.
We have established half of what we need to show today, that is, that one geometric effect of the
transformation ܮሺݖሻ ൌ ݖݓis a dilation by the modulus of ݓǡ ȁݓȁ. Now, we will demonstrate that another
geometric effect of this transformation is a rotation by the argument of ݓ.
Let ݓൌ ܽ ܾ݅, where ܽ and ܾ are real numbers. Representations of the complex numbers ݖand ݓas points
in the coordinate plane are shown below.
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Then, ݖݓൌ ሺܽ ܾ݅ሻ ݖൌ ܽ ݖ ሺܾ݅ሻݖ. Recall from Lesson 13 that ܽ ݖis a dilation of ݖby ܽ, and ሺܾ݅ሻ ݖis a
dilation of ݖby ܾ and a rotation by ͻͲι. Let’s add the points ܽ ݖand ሺܾ݅ሻ ݖto the figure.
We know that ݖݓൌ ܽ ݖ ሺܾ݅ሻݖ, so we can find the location of ݖݓin the plane by adding ܽ ݖ ሺܾ݅ሻݖ
geometrically. (We do not need to find a formula for the coordinates of ݖݓ.)
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Now, we can build a triangle with vertices at the origin, ܽ ݖand ݖݓ. And we can build another triangle with
vertices at the origin, ݓand ܽ.
What do we notice about these two triangles?
à They appear to both be right triangles. They appear to be similar.
For simplicity’s sake, let’s label the vertices of these triangles. Denote the origin by ܱ, and let ܲ ൌ ݓ, ܳ ൌ ܽ,
ܴ ൌ ݖݓ, and ܵ ൌ ܽݖǤ
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What are the lengths of the sides of the small triangle, ᇞ ܱܲܳ?
à We have
ܱܲ ൌ ȁݓȁ
ܱܳ ൌ ȁܽȁ
ܲܳ ൌ ȁܾȁǤ
What are the lengths of the sides of the large triangle, ᇞ ܱܴܵ?
à We have
ܱܴ ൌ ȁݖݓȁ ൌ ȁݓȁ ڄȁݖȁ
ܱܵ ൌ ȁܽݖȁ ൌ ȁܽȁ ڄȁݖȁ
ܴܵ ൌ ȁ ݖݓെ ܽݖȁ
ൌ ȁܽ ݖ ሺܾ݅ሻ ݖെ ܽݖȁ
ൌ ȁሺܾ݅ሻݖȁ
ൌ ȁܾ݅ȁ ڄȁݖȁ
ൌ ȁܾȁ ڄȁ݅ȁ ڄȁݖȁ
ൌ ȁܾȁ ڄȁݖȁǤ
How do the side lengths of ᇞ ܱܴܵand ᇞ ܱܲܳ relate?
à We see that
ܱܴ ȁݓȁ ڄȁݖȁ
ൌ ൌ ȁݖȁǡ
ܱܲ ȁݓȁ
ܱܵ ȁܽȁ ڄȁݖȁ
ൌ ൌ ȁݖȁǡ
ܱܳ ȁܽȁ
ܴܵ ȁܾȁ ڄȁݖȁ
ൌ ൌ ȁݖȁǤ
ܲܳ ȁܾȁ
What can we conclude about ᇞ ܱܴܵand ᇞ ܱܲܳ?
à We can conclude that ᇞ ܱܴܵ ᇞ ܱܲܳ by SSS similarity.
Now that we know ᇞ ܱܴܵ ᇞ ܱܲܳ, we can conclude that ܱܴܵס؆ ܱܳܲסǤ So, how can we use this angle
congruence to help us answer the original question?
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Where are ሺݖሻ, ሺݓሻ, and ሺݖݓሻ in our diagrams? How do they relate to the angles in the triangles?
à From the diagram,ሺݓሻ ൌ ܱ݉ܳܲס, ሺݖሻ ൌ ܱ݉ܳܵסǡ and ሺݖݓሻ ൌ ܱܴ݉ܳסǤ
However, we have shown that ݉ ܱܳܲסൌ ܱܴ݉ܵס.
We see that
ሺݖݓሻ ൌ ܱܴ݉ܳס
ൌ ݉ ܱܴܵס ܱ݉ܳܵס
ൌ ሺݖሻ ሺݓሻǤ
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Then, since ൫ܮሺݖሻ൯ ൌ ሺݖݓሻ ൌ ሺݖሻ ሺݓሻ, the point ܮሺݖሻ ൌ ݓሺݖሻ is the image of ݖunder rotation
by ሺݓሻ about the origin. Thus, the transformation ܮሺݖሻ ൌ ݖݓalso has the geometric effect of rotation by
ሺݓሻǤ
While our Discussion only addressed the case where ݓis represented by a point in the first quadrant, the
result holds for any complex number ݓǤ You will consider the other cases for ݓin the Problem Set.
Exercise 3 (4 minutes)
3. If ࢠ and ࢝ are the complex numbers with the specified arguments and moduli, locate the point that represents the
product ࢝ࢠon the provided coordinate axes.
࣊
a. ȁ࢝ȁ ൌ , ܚ܉ሺ࢝ሻ ൌ
࣊
ȁࢠȁ ൌ , ܚ܉ሺࢠሻ ൌ െ
b. ȁ࢝ȁ ൌ , ܚ܉ሺ࢝ሻ ൌ ࣊
࣊
ȁࢠȁ ൌ , ܚ܉ሺࢠሻ ൌ
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࣊
c. ȁ࢝ȁ ൌ , ܚ܉ሺ࢝ሻ ൌ
࣊
ȁࢠȁ ൌ , ܚ܉ሺࢠሻ ൌ െ
Closing (4 minutes)
Ask students to write in their journals or notebooks to explain the process for geometrically describing the product of
two complex numbers. Students should mention the following key points.
For complex numbers ݖand ݓ, the modulus of the product is the product of the moduli:
ȁݖݓȁ ൌ ȁݓȁ ڄȁݖȁǤ
For complex numbers ݖandݓ, the argument of the product is the sum of the arguments:
ሺݖݓሻ ൌ ሺݓሻ ሺݖሻ Ǥ
Lesson Summary
For complex numbers ࢠ and ࢝,
The modulus of the product is the product of the moduli:
ȁ࢝ࢠȁ ൌ ȁ࢝ȁ ڄȁࢠȁ.
The argument of the product is the sum of the arguments:
ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺ࢝ሻ ܚ܉ሺࢠሻ.
Exit Ticket (4 minutes)
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Name Date
Lesson 15: Justifying the Geometric Effect of Complex
Multiplication
Exit Ticket
1. What is the geometric effect of the transformation ܮሺݖሻ ൌ ሺെ ͺ݅ሻ?ݖ
͵ ͷߨ
2. Suppose that ݓis a complex number with ȁݓȁ ൌ and ሺݓሻ ൌ , and ݖis a complex number with ȁݖȁ ൌ ʹ and
ʹ
ߨ
ሺݖሻ ൌ .
͵
a. Explain how you can geometrically locate the point that represents the product ݖݓin the coordinate plane.
b. Plot ݓ, ݖ, and ݖݓon the coordinate grid.
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Exit Ticket Sample Solutions
1. What is the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺെ ૡሻࢠ?
For this transformation, ࢝ ൌ െ ૡ, so ȁ࢝ȁ ൌ ඥሺെሻ ૡ ൌ ξ ൌ . The transformation ࡸ dilates by a
factor of and rotates counterclockwise by ܚ܉ሺെ ૡሻ.
࣊
2. Suppose that ࢝ is a complex number with ȁ࢝ȁ ൌ and ܚ܉ሺ࢝ሻ ൌ , and ࢠ is a complex number with ȁࢠȁ ൌ and
࣊
ܚ܉ሺࢠሻ ൌ .
a. Explain how you can geometrically locate the point that represents the product ࢝ࢠ in the coordinate plane.
࣊ ࣊ ૠ࣊
The product ࢝ࢠ has argument ൌ and modulus ڄ ൌ . So, we find the point that is distance
ૠ࣊
units from the origin on the ray that has been rotated radians from the positive ࢞-axis.
b. Plot ࢝, ࢠ, and ࢝ࢠ on the coordinate grid.
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Problem Set Sample Solutions
Problems 1 and 2 establish that any linear transformation of the form ܮሺݖሻ ൌ ݖݓhas the geometric effect of a rotation
by ሺݓሻ and dilation by ȁݓȁ. Problems 3 and 4 lead to the development in the next lesson in which students build new
transformations from ones they already know.
1. In the lesson, we justified our observation that the geometric effect of a transformation ࡸሺࢠሻ ൌ ࢝ࢠ is a rotation by
ܚ܉ሺ࢝ሻ and a dilation by ȁ࢝ȁ for a complex number ࢝ that is represented by a point in the first quadrant of the
coordinate plane. In this exercise, we will verify that this observation is valid for any complex number ࢝. For a
complex number ࢝ ൌ ࢇ ࢈, we only considered the case where ࢇ and ࢈ . There are eight additional
possibilities we need to consider.
a. Case 1: The point representing ࢝ is the origin. That is, ࢇ ൌ and࢈ ൌ Ǥ
In this case, explain why ࡸሺࢠሻ ൌ ሺࢇ ࢈ሻࢠ has the geometric effect of rotation by ܚ܉ሺࢇ ࢈ሻ and dilation
by ȁࢇ ࢈ȁ.
If ࢇ ࢈ ൌ ൌ , then ܚ܉ሺࢇ ࢈ሻ ൌ , and ȁࢇ ࢈ȁ ൌ . Rotating a point ࢠ by ι does not change
the location of ࢠ, and dilation by sends each point to the origin. Since ࡸሺࢠሻ ൌ ࢠ ൌ for every complex
number ࢠ, we can say that ࡸ dilates by and rotates by , so ࡸ rotates counterclockwise by ܚ܉ሺࢇ ࢈ሻ and
dilates by ȁࢇ ࢈ȁ.
b. Case 2: The point representing ࢝ lies on the positive real axis. That is, ࢇ and ࢈ ൌ .
In this case, explain why ࡸሺࢠሻ ൌ ሺࢇ ࢈ሻࢠ has the geometric effect of rotation by ܚ܉ሺࢇ ࢈ሻ and dilation
by ȁࢇ ࢈ȁǤ
If ࢈ ൌ , then ࡸሺࢠሻ ൌ ࢇࢠ, which dilates ࢠ by a factor of ࢇ and does not rotate ࢠ. Since ࢇ ࢈ lies on the
positive real axis, ܚ܉ሺࢇ ࢈ሻ ൌ . Also, ȁࢇ ࢈ȁ ൌ ȁࢇȁ ൌ ࢇ, since ࢇ . Thus, ࡸ dilates by ȁࢇ ࢈ȁ and
rotates counterclockwise by ܚ܉ሺࢇ ࢈ሻ.
c. Case 3: The point representing ࢝ lies on the negative real axis. That is, ࢇ ൏ and ࢈ ൌ .
In this case, explain why ࡸሺࢠሻ ൌ ሺࢇ ࢈ሻࢠ has the geometric effect of rotation by ܚ܉ሺࢇ ࢈ሻ and dilation
by ȁࢇ ࢈ȁ.
If ࢈ ൌ , then ࡸሺࢠሻ ൌ ࢇࢠ, which dilates ࢠ by a factor of ȁࢇȁ and rotates ࢠ by ૡι. Since ࢇ ࢈ lies on the
negative real axis, ܚ܉ሺࢇ ࢈ሻ ൌ ૡι. Also, ȁࢇ ࢈ȁ ൌ ȁࢇȁ. Thus, ࡸ dilates by ȁࢇ ࢈ȁ and rotates
counterclockwise by ܚ܉ሺࢇ ࢈ሻ.
d. Case 4: The point representing ࢝ lies on the positive imaginary axis. That is, ࢇ ൌ and ࢈ .
In this case, explain why ࡸሺࢠሻ ൌ ሺࢇ ࢈ሻࢠ has the geometric effect of rotation by ܚ܉ሺࢇ ࢈ሻ and dilation
by ȁࢇ ࢈ȁ.
If ࢇ ൌ , then ࡸሺࢠሻ ൌ ሺ࢈ሻࢠ, which dilates ࢠ by a factor of ࢈ and rotates ࢠ by ૢι counterclockwise. Since
ࢇ ࢈ lies on the positive imaginary axis, ܚ܉ሺࢇ ࢈ሻ ൌ ૢι. Also, ȁࢇ ࢈ȁ ൌ ࢈. Thus, ࡸ dilates by
ȁࢇ ࢈ȁ and rotates counterclockwise by ܚ܉ሺࢇ ࢈ሻǤ
e. Case 5: The point representing ࢝ lies on the negative imaginary axis. That is, ࢇ ൌ and ࢈ ൏ .
In this case, explain why ࡸሺࢠሻ ൌ ሺࢇ ࢈ሻࢠ has the geometric effect of rotation by ܚ܉ሺࢇ ࢈ሻ and dilation
by ȁࢇ ࢈ȁ.
If ࢇ ൌ , then ࡸሺࢠሻ ൌ ሺ࢈ሻࢠ, which dilates ࢠ by a factor of ȁ࢈ȁ and rotates ࢠ by ૠι counterclockwise. Since
ࢇ ࢈ lies on the negative imaginary axis, ܚ܉ሺࢇ ࢈ሻ ൌ ૠι. Also, ȁࢇ ࢈ȁ ൌ ȁ࢈ȁ. Thus, ࡸ dilates by
ȁࢇ ࢈ȁ and rotates counterclockwise by ܚ܉ሺࢇ ࢈ሻ.
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f. Case 6: The point representing ࢝ ൌ ࢇ ࢈ lies in the second quadrant. That is, ࢇ ൏ and ࢈ . Points
representing ࢝, ࢠ, ࢇࢠ, ሺ࢈ሻࢠ, and ࢝ࢠ ൌ ࢇࢠ ሺ࢈ሻࢠ are shown in the figure below.
For convenience, rename the origin ࡻ, and let ࡼ ൌ ࢝, ࡽ ൌ ࢇ, ࡾ ൌ ࢝ࢠ, ࡿ ൌ ࢇࢠ, and ࢀ ൌ ࢠ, as shown below.
Let ࡽࡻࡼסൌ ࣂ.
i. Argue that ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ.
The lengths of the sides of the triangles are the following:
ࡻࡼ ൌ ȁ࢝ȁ ࡻࡾ ൌ ȁ࢝ȁ ڄȁࢠȁ
ࡻࡽ ൌ ȁࢇȁ ࡻࡿ ൌ ȁࢇȁ ڄȁࢠȁ
ࡼࡽ ൌ ࢈ ࡾࡿ ൌ ࢈ ڄȁࢠȁ
ࡻࡾ ࡻࡿ ࡾࡿ
Thus, ൌ ൌ ൌ ȁࢠȁ, so ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ.
ࡻࡼ ࡻࡽ ࡼࡽ
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ii. Express the argument of ࢇࢠ in terms of ܚ܉ሺࢠሻ.
ܚ܉ሺࢇࢠሻ ൌ ૡι ܚ܉ሺࢠሻ
iii. Express ܚ܉ሺ࢝ሻ in terms of ࣂ, where ࣂ ൌ ࡽࡻࡼס.
ܚ܉ሺ࢝ሻ ൌ ૡι െ ࣂ
iv. Explain why ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ െ ࣂ.
Because ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ, ࡿࡻࡾסൌ ࡽࡻࡼסൌ ࣂ.
ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ െ ࢃࡻࡾס
ൌ ܚ܉ሺࢇࢠሻ െ ࡽࡻࡼס
ൌ ܚ܉ሺࢇࢠሻ െ ࣂ
v. Combine your responses from parts (ii), (iii), and (iv) to express ܚ܉ሺ࢝ࢠሻ in terms of ܚ܉ሺࢠሻ and
ܚ܉ሺ࢝ሻ.
ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ െ ࣂ
ൌ ൫ૡι ܚ܉ሺࢠሻ൯ െ ൫ૡι െ ܚ܉ሺ࢝ሻ൯
ൌ ܚ܉ሺࢠሻ ܚ܉ሺ࢝ሻ
g. Case 7: The point representing ࢝ ൌ ࢇ ࢈ lies in the third quadrant. That is, ࢇ ൏ and ࢈ ൏ .
Points representing ࢝, ࢠ, ࢇࢠ, ሺ࢈ሻࢠ, and ࢝ࢠ ൌ ࢇࢠ ሺ࢈ሻࢠ are shown in the figure below.
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For convenience, rename the origin ࡻ, and let ࡼ ൌ ࢝, ࡽ ൌ ࢇ, ࡾ ൌ ࢝ࢠ, ࡿ ൌ ࢇࢠ, and ࢀ ൌ ࢠ, as shown below.
Let ࡽࡻࡼסൌ ࣂ.
i. Argue that ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ.
The lengths of the sides of the triangles are as follows:
ࡻࡼ ൌ ȁ࢝ȁ ࡻࡾ ൌ ȁ࢝ȁ ڄȁࢠȁ
ࡻࡽ ൌ ȁࢇȁ ࡻࡿ ൌ ȁࢇȁ ڄȁࢠȁ
ࡼࡽ ൌ ȁ࢈ȁ ࡾࡿ ൌ ȁ࢈ȁ ڄȁࢠȁ
ࡻࡾ ࡻࡿ ࡾࡿ
Thus, ൌ ൌ ൌ ȁࢠȁ, so ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ.
ࡻࡼ ࡻࡽ ࡼࡽ
ii. Express the argument of ࢇࢠ in terms of ܚ܉ሺࢠሻ.
ܚ܉ሺࢇࢠሻ ൌ ૡι ܚ܉ሺࢠሻ
iii. Express ܚ܉ሺ࢝ሻ in terms of ࣂ, where ࣂ ൌ ࡽࡻࡼס.
ܚ܉ሺ࢝ሻ ൌ ૡι ࣂ
iv. Explain why ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ ࣂǤ
Because ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ, ࡿࡻࡾסൌ ࡽࡻࡼסൌ ࣂ.
ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ ࡿࡻࡾס
ൌ ܚ܉ሺࢇࢠሻ ࡽࡻࡼס
ൌ ܚ܉ሺࢇࢠሻ ࣂ
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v. Combine your responses from parts (ii), (iii), and (iv) to express ܚ܉ሺ࢝ࢠሻ in terms of ܚ܉ሺࢠሻ and
ܚ܉ሺ࢝ሻ.
ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ ࣂ
ൌ ൫ૡι ܚ܉ሺࢠሻ൯ ሺܚ܉ሺ࢝ሻ െ ૡιሻ
ൌ ܚ܉ሺࢠሻ ܚ܉ሺ࢝ሻ
h. Case 8: The point representing ࢝ ൌ ࢇ ࢈ lies in the fourth quadrant. That is, ࢇ and ࢈ ൏ .
Points representing ࢝, ࢠ, ࢇࢠ, ሺ࢈ሻࢠ, and ࢝ࢠ ൌ ࢇࢠ ሺ࢈ሻࢠ are shown in the figure below.
For convenience, rename the origin ࡻ, and let ࡼ ൌ ࢝, ࡽ ൌ ࢇ, ࡾ ൌ ࢝ࢠ, ࡿ ൌ ࢇࢠ, and ࢀ ൌ ࢠ, as shown below.
Let ࡽࡻࡼסൌ ࣂ.
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i. Argue that ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ.
The lengths of the sides of the triangles are the following:
ࡻࡼ ൌ ȁ࢝ȁ ࡻࡾ ൌ ȁ࢝ȁ ڄȁࢠȁ
ࡻࡽ ൌ ࢇ ࡻࡿ ൌ ࢇ ڄȁࢠȁ
ࡼࡽ ൌ ȁ࢈ȁ ࡾࡿ ൌ ȁ࢈ȁ ڄȁࢠȁ
ࡻࡾ ࡻࡿ ࡾࡿ
Thus, ൌ ൌ ൌ ȁࢠȁ, so ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ.
ࡻࡼ ࡻࡽ ࡼࡽ
ii. Express ܚ܉ሺ࢝ሻ in terms of ࣂ, where ࣂ ൌ ࡽࡻࡼס.
ܚ܉ሺ࢝ሻ ൌ ι െ ࣂ
iii. Explain why ࡾࡻࡽסൌ ࣂ െ ܚ܉ሺࢠሻ.
Because ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ, ࡾࡻࡿסൌ ࡽࡻࡼסൌ ࣂ.
ࡾࡻࡽסൌ ࡾࡻࡿסെ ࡽࡻࡿס
ൌ ࣂ െ ࡽࡻࡿס
ൌ ࣂ െ ܚ܉ሺࢠሻ
iv. Express ܚ܉ሺ࢝ࢠሻ in terms of ࡾࡻࡽס.
Because ᇞ ࡻࡼࡽ ᇞ ࡻࡾࡿ, ࡿࡻࡾסൌ ࡽࡻࡼסൌ ࣂ.
ܚ܉ሺ࢝ࢠሻ ൌ ܚ܉ሺࢇࢠሻ ࡿࡻࡾס
ൌ ܚ܉ሺࢇࢠሻ ࡽࡻࡼס
ൌ ܚ܉ሺࢇࢠሻ ࣂ
v. Combine your responses from parts (ii), (iii), and (iv) to express ܚ܉ሺ࢝ࢠሻ in terms of ܚ܉ሺࢠሻ and
ܚ܉ሺ࢝ሻ.
ܚ܉ሺ࢝ࢠሻ ൌ ι െ ࡾࡻࡽס
ൌ ι െ ൫ࣂ െ ܚ܉ሺࢠሻ൯
ൌ ሺι െ ࣂሻ ܚ܉ሺࢠሻ
ൌ ܚ܉ሺ࢝ሻ ܚ܉ሺࢠሻ
2. Summarize the results of Problem 1, parts (a)–(h), and the lesson.
For any complex number ࢝, the transformation ࡸሺࢠሻ ൌ ࢝ࢠ has the geometric effect of rotation by ܚ܉ሺ࢝ሻ and
dilation by ȁ࢝ȁ.
3. Find a linear transformation ࡸ that will have the geometric effect of rotation by the specified amount without
dilating.
a. ι counterclockwise
We need to find a complex number ࢝so thatȁ࢝ȁ ൌ and ܚ܉ሺ࢝ሻ ൌ ι. Then, ࢝ can be represented by a
point on the unit circle such that the ray through the origin and ࢝ is the terminal ray of the positive ࢞-axis
rotated by ι. Then, the ࢞-coordinate of࢝ isܛܗ܋ሺιሻ, and the ࢟-coordinate of࢝isܖܑܛሺιሻ, so we
ඥ ඥ ඥ ඥ ඥ
have࢝ ൌ ܛܗ܋ሺιሻ ܖܑܛሺιሻ ൌ . Then, ࡸሺࢠሻ ൌ ൬ ൰ ࢠ ൌ ሺ ሻࢠ.
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b. ιcounterclockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺιሻ ܖܑܛሺιሻ൯ࢠ
ξ
ൌቆ ቇࢠ
c. ૡι counterclockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺૡιሻ ܖܑܛሺૡιሻ൯ࢠ
ൌ െࢠ
d. 1ι counterclockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺιሻ ܖܑܛሺιሻ൯ࢠ
ξ
ൌ ቆെ ቇࢠ
e. ι clockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺെιሻ ܖܑܛሺെιሻ൯ࢠ
ξ
ൌ ቆ െ ቇ ࢠ
f. ૢι clockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺെૢιሻ ܖܑܛሺെૢιሻ൯ࢠ
ൌ െࢠ
g. ૡι clockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺെૡιሻ ܖܑܛሺെૡιሻ൯ࢠ
ൌ െࢠ
h. ιclockwise
ࡸሺࢠሻ ൌ ൫ܛܗ܋ሺെιሻ ܖܑܛሺെιሻ൯ࢠ
ξ ξ
ൌ ቆെ െ ቇ ࢠ
ξ
ൌെ ሺ ሻࢠ
4. Suppose that we have linear transformations ࡸ and ࡸ as specified below. Find a formula for ࡸ ൫ࡸ ሺࢠሻ൯ for
complex numbers ࢠ.
a. ࡸ ሺࢠሻ ൌ ሺ ሻࢠ and ࡸ ሺࢠሻ ൌ ሺ െ ሻࢠ
ࡸ ൫ࡸ ሺࢠሻ൯ ൌ ࡸ ൫ሺ ሻࢠ൯
ൌ ሺ െ ሻ൫ሺ ሻࢠ൯
ൌ ሺ െ ሻሺ ሻࢠ
ൌ ࢠ
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b. ࡸ ሺࢠሻ ൌ ሺ െ ሻࢠ and ࡸ ሺࢠሻ ൌ ሺ ሻࢠ
ࡸ ൫ࡸ ሺࢠሻ൯ ൌ ࡸ ൫ሺ െ ሻࢠ൯
ൌ ሺ ሻ൫ሺ െ ሻࢠ൯
ൌ ሺ ሻሺ െ ሻࢠ
ൌ ሺ ሻࢠ
c. ࡸ ሺࢠሻ ൌ ሺെ ሻࢠ and ࡸ ሺࢠሻ ൌ ሺെ െ ሻࢠ
ࡸ ൫ࡸ ሺࢠሻ൯ ൌ ࡸ ൫ሺെ ሻࢠ൯
ൌ ሺെ െ ሻ൫ሺെ ሻࢠ൯
ൌ ሺെ െ ሻሺെ ሻࢠ
ൌ ሺ െ ሻࢠ
d. ࡸ ሺࢠሻ ൌ ሺࢇ ࢈ሻࢠ and ࡸ ሺࢠሻ ൌ ሺࢉ ࢊሻࢠ for real numbers ࢇ, ࢈, ࢉ, and ࢊ.
ࡸ ൫ࡸ ሺࢠሻ൯ ൌ ࡸ ൫ሺࢇ ࢈ሻࢠ൯
ൌ ሺࢉ ࢊሻ൫ሺࢇ ࢈ሻࢠ൯
ൌ ሺࢇ ࢈ሻሺࢉ ࢊሻࢠ
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A STORY OF FUNCTIONS Lesson 16 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 16: Representing Reflections with Transformations
Student Outcomes
Students create a sequence of transformations that produce the geometric effect of reflection across a given
line through the origin.
Lesson Notes
In this lesson, students apply complex multiplication from Lesson 14 to construct a transformation of the plane that
reflects across a given line. So far, students have only looked at linear transformations of the form ܮǣԧ ՜ ԧby
ܮሺݖሻ ൌ ݖݓfor a complex number ݓ, and all such linear transformations have the geometric effect of rotation by ሺݓሻ
and dilation by ȁݓȁ. In later lessons, when matrices are used to define transformations, students see that reflection can
ݔ ݔ
be represented by a transformation of the form ܮቀቂݕቃቁ ൌ ܣቂݕቃ for a matrix ܣ, which better fits the form that they are
used to for linear transformations. Students may need to be reminded of the following ŶŽƚĂƚŝŽŶƐĨŽƌƚƌĂŶƐĨŽƌŵĂƚŝŽŶƐ
ŽĨƚŚĞƉůĂŶĞĨƌŽŵ'ĞŽŵĞƚƌLJ͗
A rotation by ߠ degrees about the origin is denoted byܴሺǡఏιሻ .
A reflection across line κ is denoted by ݎκ .
Classwork
Opening Exercise (6 minutes)
Students should work in pairs or small groups for these exercises. Students did problems identical or nearly identical to
parts (a) and (b) in the Problem Set for Lesson 14, and they learned in Lesson 6 that taking the conjugate of ݖproduces
the reflection of ݖacross the real axis.
Opening Exercise
a. Find a transformation ࡾሺǡιሻ ǣ ԧ ՜ ԧ that rotates a point represented by the complex number ࢠ by ι
counterclockwise in the coordinate plane but does not produce a dilation.
ξ ξ
ࡸሺࢠሻ ൌ ቆ ቇ ࢠ
b. Find a transformation ࡾሺǡିιሻ ǣ ԧ ՜ ԧ that rotates a point represented by the complex number ࢠ by ι
clockwise in the coordinate plane but does not produce a dilation.
ξ ξ
ࡸሺࢠሻ ൌ ቆ െ ቇ ࢠ
c. Find a transformation ࢘࢞-axis ǣ ԧ ՜ ԧ that reflects a point represented by the complex number ࢠ across the
࢞-axis.
ࡸሺࢠሻ ൌ ࢠത, the conjugate of ࢠ
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Discussion (15 minutes)
This Discussion sets up the problem for the day, which is finding a linear transformation that reflects across a line
through the origin. For familiarity and ease of calculation, students begin with a reflection across the line with equation
ݕൌ ݔ. Students need to know the results of the Opening Exercise, so be sure to verify that all groups got the correct
answers before proceeding with the Discussion. The circle with radius ݖis shown lightly in the figure to help with
performing transformations accurately.
Discussion
We want to find a transformation ࢘र ǣ ԧ ՜ ԧ that reflects a point
representing a complex number ࢠacross the diagonal line र with
equation࢟ ൌ ࢞.
Recall from Algebra II, Module 3 that the transformation ሺݔǡ ݕሻ ՜ ሺݕǡ ݔሻ accomplishes this reflection across the diagonal
line in the coordinate plane, but students are now looking for a formula that produces this result for the complex
number ݔ ݅ݕ. If students mention this transformation, praise them for making the connection to past work, and ask
them to keep this response in mind for verifying the answer they get with the new approach. The steps outlined below
demonstrate that the reflection across a line other than the ݔ-axis or ݕ-axis can be accomplished by a sequence of
rotations so that the line of reflection aligns with the ݔ-axis, reflects across the ݔ-axis, and rotates so that the line is back
in its original position.
Display or reproduce the image above to guide students through this Discussion as they Scaffolding:
take notes. Ask students to draw a point ݎκ ሺݖሻ where they think the reflection of ݖacross
For struggling students, use
line κ is. Draw it on the teacher version also. Walk through the sequence of
transparency sheets to model
transformations geometrically before introducing the analytical formulas.
the sequence of rotating by
We know how to find transformations that produce the effect of rotating by a certain െͶͷι, reflecting across the real
amount around the origin, dilating by a certain scale factor, and reflecting across the axis, and then rotating by ͶͷιǤ
ݔ-axis or the ݕ-axis. Which of these transformations help us to reflect across the diagonal
line? Allow students to make suggestions or conjectures.
How much do we have to rotate around the origin to have line κ align
with the positive ݔ-axis?
à We need to rotate െͶͷι.
Draw the image of after rotation by െͶͷι about the origin. Label the new point
ଵ . Give students a quick minute to draw ଵ on their version before displaying
the teacher version.
Where is the original line κ now?
à It coincides with the positive ݔ-axis.
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We know how to reflect across the ݔ-axis. Draw the reflection of point ݖଵ across the ݔ-axis, and label
it ݖଶ .
Now that we have done a reflection, we need to
rotate back to where we started. How much do we
have to rotate around the origin to put line κ back
where it originally was?
à We need to rotate Ͷͷι.
Draw the image of ݖଶ under rotation by Ͷͷι about
the origin. The image should coincide with the
original estimate of ݎκ ሺݖሻ.
Now that students have had a chance to think about the
geometric steps involved in reflecting ݖacross diagonal line κ,
repeat the process using the formulas for the three
transformations.
What is the transformation that accomplishes
rotation by െͶͷι? Students answered this question
in the Opening Exercise.
ඥʹ ඥʹ
à The transformation is ܮሺݖሻ ൌ ൬ െ݅ ൰ ݖ.
ʹ ʹ
We will refer to this transformation using the notation we used in Geometry. We can also factor out the
ξଶ ඥʹ
constant : ܴሺǡିସହιሻ ሺݖሻ ൌ ሺͳ െ ݅ሻݖ.
ଶ ʹ
After we rotate the plane so that line κ lies along the ݔ-axis, we reflect the new point ݖacross the ݔ-axis.
What is the formula for the transformation ݎ௫-axis we use to accomplish the reflection?
à We use the conjugate of ݖ, so we have ݎ௫-axis ሺݖሻ ൌ ݖҧ.
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Now, we can rotate the plane back to its original position by rotating by Ͷͷι counterclockwise around the
origin. What is the formula for this rotation?
ඥʹ
à From the Opening Exercise, using the notation from Geometry, we have ܴሺǡସହιሻ ሺݖሻ ൌ ሺͳ ݅ሻݖ.
ʹ
We then have
ξʹ
ݖଵ ൌ ܴሺǡିସହιሻ ሺݖሻ ൌ ሺͳ െ ݅ሻݖ
ʹ
ݖଶ ൌ ݎ௫-axis ሺݖଵ ሻ ൌ ݖഥଵ
ξʹ
ݖଷ ൌ ܴሺǡସହιሻ ሺݖଶ ሻ ൌ ሺͳ ݅ሻݖଶ Ǥ
ʹ
Putting the formulas together, we have
ݖଷ ൌ ܴሺǡସହιሻ ሺݖଶ ሻ
ൌ ܴሺǡସହιሻ ൫ݎ௫-axis ሺݖଵ ሻ൯
ൌ ܴሺǡସହιሻ ൬ݎ௫-axis ቀܴሺǡିସହιሻ ሺݖሻቁ൰Ǥ
Stop here before going forward with the analytic equations, and ensure that all students understand that this formula
means that they are first rotating point ݖbyെͶͷι about the origin, then reflecting across the ݔ-axis, and then rotating
byͶͷι about the origin. Remind students that the innermost transformations happen first.
Applying the formulas, we have
ݖଷ ൌ ܴሺǡସହιሻ ൬ݎ௫-axis ቀܴሺǡିସହιሻ ሺݖሻቁ൰
ξʹ
ൌ ܴሺǡସହιሻ ቌݎ௫-axis ቆ ሺͳ െ ݅ሻݖቇቍ
ʹ
തതതതതതതതതതതതതത
ξʹ
ൌ ܴሺǡସହιሻ ൭ ሺͳ െ ଓሻݖ൱
ʹ
തതതതതതതതതതതതത
ξʹ
ൌ ܴሺǡସହιሻ ൭ ሺͳ െ ଓሻ ݖ ڄҧ൱
ʹ
ξʹ
ൌ ܴሺǡସହιሻ ቆ ሺͳ ݅ሻݖҧቇ
ʹ
ξʹ ξʹ
ൌ ሺͳ ݅ሻ ቆ ሺͳ ݅ሻݖҧቇ
ʹ ʹ
ͳ
ൌ ሺͳ ݅ሻଶ ݖҧ
ʹ
ൌ ݅ݖҧǤ
Then, the transformation ݎκ ሺݖሻ ൌ ݅ݖҧ has the geometric effect of reflection across the diagonal line κ with
equation ݕൌ ݔ.
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Exercises 1–2 (5 minutes)
Exercises
1. The number ࢠin the figure used in the Discussion above is the complex number . Compute ࢘र ሺ ሻ, and
plot it below.
2. We know from previous courses that the reflection of a point ሺ࢞ǡ ࢟ሻ across the line with equation࢟ ൌ ࢞is the point
ሺ࢟ǡ ࢞ሻ. Does this agree with our result from the previous discussion?
Yes, we can represent the point ሺ࢞ǡ ࢟ሻ by ࢠ ൌ ࢞ ࢟. Then,
࢘र ሺࢠሻ ൌ ࢘र ሺ࢞ ࢟ሻ
തതതതതതതതതതത
ൌ ሺ࢞ ଙ࢟ሻ
ൌ ሺ࢞ െ ࢟ሻ
ൌ ࢟ ࢞ǡ
which corresponds to the point ሺ࢟ǡ ࢞ሻ.
Exercise 3 (10 minutes)
In this exercise, students repeat the previous calculations to find an analytic formula for
Scaffolding:
reflection across the line κthat makes aͲι angle with the positive ݔ-axis.
Ask struggling students the
following questions to guide
3. We now want to find a formula for the transformation of reflection across the line र that
their work in Exercise 3.
makes a ι angle with the positive ࢞-axis. Find formulas to represent each component of
the transformation, and use them to find one formula that represents the overall What transformation will
transformation. rotate line κso that it
The transformation consists of rotating line रso that it coincides with the࢞-axis, reflecting coincides with theݔ-axis?
across the ࢞-axis, and rotating the ࢞-axis back to the original position of line र. The
à െͲι
components of the transformation can be represented by these formulas:
What transformation will
ξ
ࢠ ൌ ࡾሺǡିιሻ ሺࢠሻ ൌ ቆ െ ቇ ࢠ reflect across the ݔ-axis?
ࢠ
ࢠ ൌ ࢘࢞-axis ሺࢠ ሻ ൌ തതത à The conjugate
ξ What transformation will
ࢠ ൌ ࡾሺǡιሻ ሺࢠ ሻ ൌ ቆ ቇ ࢠ Ǥ
rotate the ݔ-axis back to
Putting the formulas together, we have
the original position of line
κ?
ࢠ ൌ ࡾሺǡιሻ ሺࢠ ሻ
à Ͳι
ൌ ࡾሺǡιሻ ൫࢘࢞-axis ሺࢠ ሻ൯
ൌ ࡾሺǡιሻ ൬࢘࢞-axis ቀࡾሺǡିιሻ ሺࢠሻቁ൰Ǥ
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Stop here before going forward with the analytic equations, and ensure that all students
understand that this formula means that we are first rotating point ࢠbyെι about the
Scaffolding:
origin, then reflecting across the ࢞-axis, and then rotating byι about the origin. Remind
students that the innermost transformations happen first. Have early finishers repeat
Applying the formulas, we have
Exercise 3 for the line κ that
makes a െ͵Ͳι angle with the
࢘र ሺࢠሻ ൌ ࡾሺǡιሻ ൬࢘࢞-axis ቀࡾሺǡିιሻ ሺࢠሻቁ൰ positive ݔ-axis.
ξ
ൌ ࡾሺǡιሻ ൮࢘࢞-axis ቌቆ െ ቇ ࢠቍ൲
തതതതതതതതതതതതതതതത
ξ
ൌ ࡾሺǡιሻ ൭ቆ െ ଙቇ ࢠ൱
തതതതതതതതതതതതതത
ξ
ൌ ࡾሺǡιሻ ൭ቆ െ ଙቇ ࢠ ڄത൱
ξ
ൌ ࡾሺǡιሻ ቌቆ ቇ ࢠതቍ
ξ ξ
ൌቆ ቇ ቌቆ ቇ ࢠതቍ
ξ
ൌ ቆെ ቇ ࢠതǤ
ඥ
Then, the transformation ࢘र ሺࢠሻ ൌ ൬െ ൰ ࢠത has the geometric effect of reflection across
the line रthat makes a ι angle with the positive ࢞-axis.
Closing (4 minutes)
Ask students to write in their journals or notebooks to explain the sequence of transformation that produces reflection
across a line κ through the origin that contains the terminal ray of a rotation of theݔ-axis by ߠ. Key points are
summarized in the box below.
Lesson Summary
Let र be a line through the origin that contains the terminal ray of a rotation of the ࢞-axis by ࣂ. Then, reflection
across line र can be done by the following sequence of transformations:
Rotation by െࣂ about the origin.
Reflection across the ࢞-axis.
Rotation by ࣂ about the origin.
Exit Ticket (5 minutes)
Lesson 16: Representing Reflections with Transformations 207
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Name Date
Lesson 16: Representing Reflections with Transformations
Exit Ticket
Explain the process used in the lesson to locate the reflection of a point ݖacross the diagonal line with equation ݕൌ ݔ.
Include figures in your explanation.
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A STORY OF FUNCTIONS Lesson 16 M1
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Exit Ticket Sample Solutions
Explain the process used in the lesson to locate the reflection of a point ࢠ across the diagonal line with equation ࢟ ൌ ࢞.
Include figures in your explanation.
First, we rotated the point ࢠby െι to align the diagonal line with equation ࢟ ൌ ࢞ with the ࢞-axis to get a new point ࢠ .
Then, we reflected the point ࢠ across the real axis to find point ࢠ .
Finally, we rotated everything back by ι to find the final point ࢠ ൌ ࢘र ሺࢠሻ.
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Problem Set Sample Solutions
1. Find a formula for the transformation of reflection across the line र with equation ࢟ ൌ െ࢞.
ඥ ඥ
ࢠ ൌ ࡾ൫ǡିι൯ ሺࢠሻ ൌ ൬െ െ ൰ ࢠ; if students cannot see it, you can say that:
ξ ξ ξ ξ
ࡾ൫ǡିι൯ ሺࢠሻ ൌ ࡾ൫ǡିι൯ ൬ࡾ൫ǡିι൯ ቀࡾ൫ǡିι൯ ࢠቁ൰ ൌ ቆ െ ቇ ࢠ ൌ ቆെ െ ቇ ࢠ
തതതതതതതതതതതതതതതതതതതതത തതതതതതതതതതതതതതതത
ξ ξ ξ ξ ξ ξ
ࢠ ൌ ࢘࢞ି ܛܑܠ܉ሺࢠ ሻ ൌ ቆെ െ ଙቇ ࢠ ൌ ൭െ െ ଙ൱ ࢠത ൌ ቆെ ቇ ࢠത
ξ ξ
ࢠ ൌ ࡾ൫ǡι൯ ሺࢠ ሻ ൌ ቆെ ቇ ሺࢠ ሻ
ξ ξ
ࢠ ൌ ࡾ൫ǡι൯ ൫࢘࢞ି ܛܑܠ܉ሺࢠ ሻ൯ ൌ ࡾ൫ǡι൯ ቆ࢘࢞ି ܛܑܠ܉൬ࡾ൫ǡିι൯ ሺࢠሻ൰ቇ ൌ ࡾ൫ǡι൯ ൮ ܛܑܠ܉ି࢞ܚቌቆെ െ ቇ ࢠቍ൲
തതതതതതതതതതതതതതതതതതതതത
ξ ξ ξ ξ ξ ξ ξ ξ
ൌ ࡾ൫ǡι൯ ൭ቆെ െ ଙቇ ࢠ൱ ൌ ࡾ൫ǡι൯ ቌቆെ ቇ ࢠതቍ ൌ ቆെ ቇ ቆെ ቇ ࢠത
ൌ െࢠത
2. Find the formula for the sequence of transformations comprising reflection across the line with equation ࢟ ൌ ࢞ and
then rotation by ૡι about the origin.
ξ ξ
ࢠ ൌ ࡾ൫ǡିι൯ ሺࢠሻ ൌ ቆ െ ቇ ܢ
തതതതതതതതതതതതതതതതതത തതതതതതതതതതതതത
ξ ξ ξ ξ ξ ξ
ࢠ ൌ ࢘࢞ି ܛܑܠ܉ሺࢠ ሻ ൌ ቆ െ ଙቇ ࢠ ൌ ൭ െ ଙ൱ ࢠത ൌ ቆ ቇ ࢠത
ξ ξ
ࢠ ൌ ࡾ൫ǡι൯ ሺࢠ ሻ ൌ ቆ ቇ ሺࢠ ሻ
ξ ξ ξ ξ ξ ξ
ࢠ ൌ ࡾ൫ǡι൯ ሺࢠ ሻ ൌ ቆ ቇ ሺࢠ ሻ ൌ ቆ ቇ ቆ ቇ ࢠത ൌ ࢠത
ࢠ ൌ െࢠ ൌ െࢠത
3. Compare your answers to Problems 1 and 2. Explain what you find.
They have the same answer/formula that will produce the same transformation of ࢠ.
4. Find a formula for the transformation of reflection across the line र that makes a െι angle with the positive
࢞-axis.
ξ
ࢠ ൌ ࡾ൫ǡι൯ ሺࢠሻ ൌ ቆ ቇ ࢠ
തതതതതതതതതതതതതതതതതത തതതതതതതതതതതത
ξ ξ ξ
ࢠ ൌ ࢘࢞ି ܛܑܠ܉ሺࢠ ሻ ൌ ቆ ଙቇ ࢠ ൌ ൭ ଙ൱ ࢠത ൌ ቆ െ ቇ ࢠത
ξ
ࢠ ൌ ࡾ൫ǡିι൯ ሺࢠ ሻ ൌ ቆ െ ቇ ሺࢠ ሻ
ξ ξ ξ ξ
ࢠ ൌ ࡾ൫ǡିι൯ ሺࢠ ሻ ൌ ቆ െ ቇ ሺࢠ ሻ ൌ ቆ െ ቇ ቆ െ ቇ ࢠത ൌ ቆ െ ቇ ࢠത
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5. Max observed that when reflecting a complex number, ࢠ ൌ ࢇ ࢈ about the line ࢟ ൌ ࢞, ࢇ and ࢈ are reversed,
which is similar to how we learned to find an inverse function. Will Max’s observation also be true when the line
࢟ ൌ െ࢞ is used, where ࢇ ൌ െ࢈ and ࢈ ൌ െࢇ? Give an example to show his assumption is either correct or incorrect.
Yes, to reflect a complex number ࢠ ൌ ࢇ ࢈ about the line ࢟ ൌ െ࢞, we need to do ࡾǡିι ǡ ࢘࢞ି ܛܑܠ܉, and then ࡾǡι ,
which will produce the answer to be ࢠ ൌ ࢈ ࢇ.
The examples vary. This example will work: ࢠ ൌ .
ࢠ ൌ ࢇ ࢈
ξ ξ ξ
ࢠ ൌ ࡾǡι ሺ ܛܑܠ܉ି࢞ܚቀࡾǡିι ሺࢠሻቁ ൌ ࡾǡι ሺ࢘࢞ି ܛܑܠ܉ቌቆെ െ ቇ ࢠቍ ൌ ࡾǡι ቆ തതതതതതതതതതതതത
ሺെ െ ଙሻࢠቇ
ξ ξ ξ
ൌ ࡾǡι ቆ ሺെ ሻࢠതቇ ൌ ሺെ ሻ ሺെ ሻࢠത ൌ ሺെሻࢠത ൌ െࢠത ൌ െሺࢇ ࢈ሻ ൌ ࢈ െ ࢇ
ࢠ ൌ ǡ ࢠ ൌ െሺ െ ሻ ൌ െ െ
6. For reflecting a complex number, ࢠ ൌ ࢇ ࢈ about the line ࢟ ൌ ࢞, will Max’s idea work if he makes ࢈ ൌ ࢇ and
࢈
ࢇ ൌ ? Use ࢠ ൌ as an example to show whether or not it works.
࢈
No, it will not work based on the example shown. ࢠ ൌ ࢇ ൌ ൈ ൌ Since the angle ࢇ ് ૢι,
this is not a reflection.
7. What would the formula look like if you want to reflect a complex number about the line ࢟ ൌ ࢞, where ?
For reflecting a complex number or a point about the line going through the origin, we need to know the angle of
the line with respect to the positive ࢞-axis to do rotations. So, we can use the slope of the line to find the angle that
we need to rotate, which is ܖ܉ܜ܋ܚ܉ሺሻ. Now we can come up with a general formula that can be applied to
reflecting about the line ࢟ ൌ ࢞, where .
ࢠ ൌ ࡾǡܖ܉ܜ܋ܚ܉ሺሻ ሺ࢘࢞ି ܛܑܠ܉ቀࡾǡܖ܉ܜ܋ܚ܉ሺିሻ ሺࢠሻቁ,
where ࡾǡܖ܉ܜ܋ܚ܉ሺିሻ ൌ ܛܗ܋൫ܖ܉ܜ܋ܚ܉ሺെሻ൯ ή ܖܑܛ൫ܖ܉ܜ܋ܚ܉ሺെሻ൯
ࡾǡܖ܉ܜ܋ܚ܉ሺሻ ൌ ܛܗ܋൫ܖ܉ܜ܋ܚ܉ሺሻ൯ ή ܖܑܛ൫ܖ܉ܜ܋ܚ܉ሺሻ൯
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A STORY OF FUNCTIONS Lesson 17 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 17: The Geometric Effect of Multiplying by a
Reciprocal
Student Outcomes
Students apply their knowledge to understand that multiplication by the reciprocal provides the inverse
geometric operation to a rotation and dilation.
Students understand the geometric effects of multiple operations with complex numbers.
Lesson Notes
This lesson explores the geometric effect of multiplication by the reciprocal to construct a transformation that undoes
multiplication. In this lesson, students verify that the transformation of multiplication by the reciprocal produces the
same result geometrically as it does algebraically. This lesson ties together the work of Lessons 13–15 on linear
transformations of the form ܮǣ ԧ ՜ ԧ by ܮሺݖሻ ൌ ݖݓfor a complex number ݓand all such linear transformations having
the geometric effect of rotation by ሺݓሻ and dilation by ȁݓȁ to the work done in Lessons 6 and 7 on complex number
division. In later lessons, when matrices are used to define transformations, these topics are revisited and extended.
This lesson is structured as a series of exploratory challenges that are scaffolded to allow
Scaffolding:
students to make sense of the connections between algebraic operations with complex
numbers and the corresponding transformations. The lesson concludes with students Use these concrete examples
to scaffold the opening as
considering all the operations (and their related transformations) together and working
needed for students:
with combinations of operations and describing them as a series of transformations of a
complex number. In the Problem Set, students connect the work of this module back to In ݔ ʹ, how do you undo
linear transformations that they studied in Lessons 1 and 2. adding ʹ?
à You would subtract ʹ.
ݔʹെʹൌݔ
Classwork In ͷݔ, how do you undo
Opening (2 minutes) multiplication by ͷ?
à You would divide by ͷ.
Ask students to brainstorm real-world operations that undo each other, for example,
putting their shoes on and taking them off. Have students briefly share ideas with their ͷݔ
ൌݔ
group mates. Next, have them think of mathematical operations that undo each other. ͷ
For example, division by ͵ undoes multiplication by ͵. Remind students that the word In ݔଷ , how do you undo
inverse is often used when talking about operations that undo each other. During this the operation of cubing?
lesson, be sure to correct students who confuse the words opposite, reciprocal, and à You would take the
inverse. cube root.
య
ඥ ݔଷ ൌ ݔ
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Opening Exercise (3 minutes)
Students work with this complex number in the subsequent Exploratory Challenge. If Scaffolding:
students struggle to find the argument and modulus of ͳ ݅ in this exercise, take time to Throughout this lesson,
review notation and methods for determining the argument of a complex number written students work with friendly
in rectangular form. All the actual complex numbers in this lesson correspond to friendly rotations and use their
గ గ గ knowledge of special right
rotations such as , , , etc.
ସ ଷ ଶ triangles and proportional
reasoning. In the classroom,
Opening Exercise display prominent visual
Given ࢝ ൌ , what is ܚ܉ሺ࢝ሻ and ȁ࢝ȁ? Explain how you got your answer. reminders such as drawings of
࣊ special triangles (see Lesson 12
ܚ܉ሺ࢝ሻ ൌ and ȁ࢝ȁ ൌ ξ. I used the formula ȁ࢝ȁ ൌ ξࢇ ࢈ to determine the modulus, and
of this module), a unit circle
since the point ሺǡ ሻ lies along a ray from the origin that has been rotated ι from the ray
࣊ with benchmark rotations
through the origin that contains the real number , the argument must be .
labeled in degrees and radians
(see Algebra II, Module 2), etc.
Exploratory Challenge 1/Exercises 1–9 (10 minutes)
Students should complete the next nine exercises in small groups of three to four students. As teams work on these
problems, circulate around the room to monitor progress. Some groups may get stuck on Exercise 3. Since the
argument of a complex number on an interval Ͳ ሺݖሻ ൏ ʹߨ has been defined, students need to figure a positive
rotation on this interval that is equivalent to െሺݖሻ. Lead a whole-class discussion at this point if needed before
moving the groups on to complete the rest of the exercises in this Exploratory Challenge. After Exercise 6, have one or
two students report their group’s response to this item.
Exploratory Challenge 1/Exercises 1–9
1. Describe the geometric effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ. Scaffolding:
The transformation rotates the complex number about the origin through ι and dilates Work with specific angle
the number by a scale factor of ξ. measures to help struggling
students understand the
2. Describe a way to undo the effect of the transformation ࡸሺࢠሻ ൌ ሺ ሻࢠ.
answer to Exercise 3.
Name a positive and
Geometrically, we need to rotate in the opposite direction, െι, and dilate by a factor of
negative rotation that take
. a ray from the origin
ξ
containing the real
number ͳ through each
3. Given that ܚ܉ሺࢠሻ ൏ ࣊ for any complex number, how could you describe any
clockwise rotation of ࣂ as an argument of a complex number? point.
ܚ܉ሺࢠሻ ൌ ࣊ െ ࣂ would result in the same rotation as a clockwise rotation of ࣂ.
ሺͳǡͳሻ
ሺͲǡʹሻ
4. Write a complex number in polar form that describes a rotation and dilation that will undo ሺെͳǡ ξ͵ሻ
multiplication by ሺ ሻ, and then convert it to rectangular form. ሺെ͵ǡͲሻ
ሺܛܗ܋ሺιሻ ԝܖܑܛሺιሻሻ ൌ
ή ή
ήെ
ൌ െ
ሺെʹǡ െʹሻ
ξ ξ ξ ξ ξ
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5. In a previous lesson, you learned that to undo multiplication by , you would multiply by the reciprocal .
ା
Write the complex number in rectangular form ࢠ ൌ ࢇ ࢈ where ࢇ and ࢈ are real numbers.
ା
െ െ
ൌ ൌ ൌ െ
ሺ ሻሺ െ ሻ
6. How do your answers to Exercises 4 and 5 compare? What does that tell you about the geometric effect of
multiplication by the reciprocal of a complex number?
The geometric effect of rotation by ࣊ െ ܚ܉ሺࢠሻ and dilation by appears to be the same as multiplication by the
ȁࢠȁ
reciprocal when the problem is solved algebraically.
7. Jimmy states the following:
Multiplication by has the reverse geometric effect of multiplication by ࢈.
ࢇା࢈
Do you agree or disagree? Use your work on the previous exercises to support your reasoning.
Geometrically undoing the effect of multiplication by ࢇ ࢈ by rotating in the opposite direction by the argument
and dilating by the reciprocal of the modulus gave us the same results as when we rewrote in rectangular
ࢇା࢈
form. This statement appears to be true. In each case, we got the same complex number.
8. Show that the following statement is true when ࢠ ൌ െ ξ:
The reciprocal of a complex number ࢠ with modulus ࢘ and argument ࣂ is with modulus and argument ࣊ െ ࣂ.
ࢠ ࢘
࣊
Since െ ξ has modulus and argument , we must have
࣊ ࣊ ξ ξ
ൌ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ ൌ ቆ ቇ ൌ Ǥ
െ ξ ૡ ૡ
Multiplying by using the conjugate, we have
ሺ ξሻ ξ ξ
ൌ ൌ ൌ Ǥ
െ ξ ሺ െ ξሻሺ ξሻ ૡ ૡ
Since both methods produce equivalent complex numbers, this statement is true when ࢠ ൌ െ ξ.
9. Explain using transformations why ࢠ ή ൌ .
ࢠ
The complex number ࢠ can be thought of as a rotation of the real number by ܚ܉ሺࢠሻ and a dilation by ȁࢠȁ. If we
multiply this number by its reciprocal, then we rotate ܚ܉ሺࢠሻ in the opposite direction and dilate by a factor of .
ȁࢠȁ
This will return the rotation to and the modulus to , which describes the real number .
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Debrief this section by making sure that students are clear on the geometric effect of
Scaffolding:
multiplication by the reciprocal of a complex number. Explain that this allows us to
understand division of complex numbers as transformations as well. A proof that the As an alternative to presenting
ଵ this proof, have students verify
geometric effect of multiplication by the reciprocal is the same as is provided below. the geometric effect of
ା
multiplying by the reciprocal of
ଵ
Let ܽ ܾ݅ ൌ ݎሺ ሺߠሻ ݅ԝሺߠሻሻ. Then, a complex number whose modulus is and a complex number with specific
examples.
ଵ
whose argument is ʹߨ െ ߠ would be ሺ ሺʹߨ െ ߠሻ ݅ԝሺʹߨ െ ߠሻሻ. We need to show Let ݖൌ ʹ ʹ݅. For each
௭
that number below, find .
௪
ͳ ͳ ݓൌ ͳെ݅
ൌ
ܽ ܾ݅ ݎሺ ሺߠሻ ݅ԝሺߠሻሻ ݓൌ ͵݅
ଵ ݓൌ െͶ െ Ͷ݅
is equivalent to ሺ ሺʹߨ െ ߠሻ ݅ԝሺʹߨ െ ߠሻሻ.
ݓൌ െͷ
ͳ ͳ
ൌ
ܽ ܾ݅ ݎሺ ሺߠሻ ݅ԝሺߠሻሻ
ͳ ݎሺ ሺߠሻ െ ݅ԝሺߠሻሻ
ൌ ή
ݎሺ ሺߠሻ ݅ԝሺߠሻሻ ݎሺ ሺߠሻ െ ݅ԝሺߠሻሻ
ݎሺ ሺߠሻ െ ݅ԝሺߠሻሻ
ൌ
ݎଶሺ ଶ ሺߠሻ െ ݅ ଶ ԝଶ ሺߠሻሻ
ͳ ሺ ሺߠሻ െ ݅ԝሺߠሻሻ
ൌ ή
ݎଶ ሺߠሻ ଶ ሺߠሻ
By the Pythagorean identity,
ͳ ͳ
ൌ ሺ ሺߠሻ െ ݅ԝሺߠሻሻǤ
ܽ ܾ݅ ݎ
By using identities ሺെߠሻ ൌ ሺߠሻ and ሺെߠሻ ൌ െሺߠሻ and ሺʹߨ െ ߠሻ ൌ ሺെߠሻ and
ሺʹߨ െ ߠሻ ൌ ሺെߠሻ, we substitute to get
ͳ ͳ
ൌ ሺ ሺʹߨ െ ߠሻ ݅ԝሺʹɎ െ ɅሻሻǤ
ܽ ܾ݅ ݎ
Exploratory Challenge 2/Exercise 10 (15 minutes)
This second challenge is a culminating activity that gives students the opportunity to review all the transformations and
operations on complex numbers studied in this module. Students should continue working in groups on this activity.
After groups have completed the graphic organizer, invite representatives from the different groups to summarize their
findings one row per group. If time is an issue, each group may work on only one row. Depending on the size of the
class, more than one group may be assigned the same row. Have students prepare and present a poster summarizing
their work on their assigned operation. While each group is presenting, students can take notes.
In this section, students are making connections between the algebraic structure of complex numbers and the related
geometric representations and transformations.
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Exploratory Challenge 2/Exercise 10
10. Complete the graphic organizer below to summarize your work with complex numbers so far.
Operation Geometric Example. Illustrate algebraically and geometrically.
Transformation Let ࢠ ൌ െ and ࢝ ൌ െǤ
Addition Translation of ࢠ െ ሺെሻ ൌ െ
by ࢝
ࢠ࢝ You can see that the point ሺǡ െሻ has been translated down units.
Subtraction Translation of ࢠ െ െ ሺെሻ ൌ െ
by ࢝
ࢠെ࢝ You can see that the point ሺǡ െሻ has been translated up units.
Conjugate of ࢠ Reflection of ࢠ ࢠത ൌ . The point ሺǡ െሻ is reflected across the real axis to the point ሺǡ ሻ.
across the real
axis
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Multiplication Rotation of ࢠ ࣊ ࣊
ܚ܉ሺ࢝ሻ ൌ and ȁ࢝ȁ ൌ . Thus, ࢝ࢠ is െ rotated and dilated by a
by ܚ܉ሺ࢝ሻ
࢝ήࢠ
followed by factor of .
dilation by a
factor of ȁ࢝ȁ ૠ࣊ ૠ࣊
െ ൌ ξ ቀ ܛܗ܋ቀ ቁ ܖܑܛቀ ቁቁ, so the new modulus will be ξ, and
the new argument will be a number between and ࣊ that corresponds to a
ૠ࣊ ࣊ ࣊ ࣊
rotation of ൌ . The argument would be .
െሺ െ ሻ ൌ െ െ ൌ െ െ ; that does indeed have modulus ξ and
࣊
argument .
Reciprocal of ࢠ Rotates the real ૠ࣊
െ is a rotation of and a dilation by ξ of the real number . The
number the
opposite ૠ࣊
reciprocal would be a rotation of െ and a dilation by . The argument of
rotation of ࢠ ξ
and a dilation ࣊ ξ
by the the reciprocal would be ǡ and the modulus would be .
reciprocal of
the modulus of ା ା ࣊
ൌ ൌ ൌ with ܚ܉ሺࢠሻ ൌ and ȁࢠȁ ൌ ξ
ࢠ ି ሺିሻሺାሻ ૡ
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Division Rotates ࢝ by ࣊ ξ
࣊ െ ࢇ࢘ࢍሺࢠሻ Rotation of ࢝ by ࣊ െ ܚ܉ሺࢠሻ ൌ and dilation by would result in a complex
࢝
and followed ࣊ ࣊ ૠ࣊
ࢠ ξ ξ
by a dilation of number whose argument was ൌ and a modulus of ήൌ Ǥ
ȁࢠȁ ࢝ ି ିሺାሻ ିା
ൌ ൌ ൌ ൌ െ
ࢠ ି ሺିሻሺାሻ ૡ
࢝ ࢝ ૠ࣊
ቚ ቚ ൌ ξ and ܚ܉ቀ ቁ ൌ
ࢠ ࢠ
Exercises 11–13 (7 minutes)
Students should work on these problems with a partner. Ask each student to explain one problem to their partners to
check for understanding. Then, invite one or two students to share their results on the board.
Exercises 11–13
Let ࢠ ൌ െ , and let ࢝ ൌ . Describe each complex number as a transformation of ࢠǡ and then write the number in
rectangular form.
11. ࢝ࢠത
ࢠ is reflected across the real axis; then, that number is rotated by ܚ܉ሺ࢝ሻ and dilated by ȁ࢝ȁ.
࣊ ࣊ ࣊
ܚ܉ሺࢠሻ ൌ and ȁࢠȁ ൌ ξ. ܚ܉ሺࢠതሻ ൌ with the same modulus as ࢠ. Rotation by ܚ܉ሺ࢝ሻ ൌ and dilation by
࣊ ࣊ ૠ࣊
would give a complex number with argument of ൌ and modulus of ξ, which is the modulus and
argument of the number shown below.
࢝ࢠത ൌ ሺെ െ ሻ ൌ െ
12.
ࢠത
ࢠ is reflected across the real axis and then rotated ࣊ െ ܚ܉ሺࢠതሻ and dilated by . The result is a dilation of ࢠ.
ȁࢠതȁ
࣊
Reflection of ࢠ across the real axis results in a complex number with argument and modulus of ξ. The
࣊ ࣊ ξ
reciprocal has argument ࣊ െ ൌ and modulus ൌ . This number has the same argument as ࢠ and is a
ξ
dilation by a factor of .
ିା ିା
ൌ ൌ ൌ ൌെ
ࢠത ିି ሺିିሻሺିାሻ
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13. തതതതതതതത
࢝ࢠ
ࢠ is translated by ࢝ vertically units up since the real part of ࢝ is and the imaginary part is . This new number is
reflected across the real axis.
࢝ ࢠ ൌ െ ൌ െ
തതതതതതതത
࢝ ࢠ ൌ െ െ
Closing (3 minutes)
The graphic organizer students made in Exploratory Challenge 2 function as a summary for
Scaffolding:
this lesson. Invite students to answer the following questions in writing or to discuss them
If needed, make the closing
with a partner.
questions more concrete by
What is the geometric effect of complex number division specifying specific complex
ଵ numbers for ݖand ݓ.
(multiplication of ݖby )?
௪
ଵ
à The number ݖis rotated ʹߨ െ ሺݓሻ and dilated by .
ȁ௪ȁ
ଵ
How are the modulus and argument of the complex number related to the complex number ?ݖ
௭
ଵ ଵ ଵ
à The modulus of is , and the argument of is ʹߨ െ ሺݖሻ, which is the same as rotation of െሺݖሻ.
௭ ௭
Exit Ticket (5 minutes)
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Name Date
Lesson 17: The Geometric Effect of Multiplying by a Reciprocal
Exit Ticket
Let ݖൌ ͳ ξ͵݅ and ݓൌ ξ͵ െ ݅. Describe each complex number as a transformation of ݖ, and then write the number
in rectangular form, and identify its modulus and argument.
௭
1.
௪
ଵ
2.
௪௭
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Exit Ticket Sample Solutions
Let ࢠ ൌ ξ and ࢝ ൌ ξ െ . Describe each complex number as a transformation of ࢠ, and then write the number in
rectangular form, and identify its modulus and argument.
ࢠ
1.
࢝
ࢠ is rotated by െܚ܉ሺ࢝ሻ and dilated by .
ȁ࢝ȁ
࣊ ࣊ ࣊ ࣊
ܚ܉ሺࢠሻ ൌ and ࣊ െ ܚ܉ሺ࢝ሻ ൌ ࣊ െ ൌ . So, division by ࢝ should rotate ࢠ to . ȁࢠȁ ൌ ǡ and ൌ , so
ȁ࢝ȁ
ࢠ
the modulus of should be ή ൌ . This rotation and dilation describe the complex number . Algebraically, we
࢝
get the same number.
ࢠ ξ ξ ξ
ൌ ൌ ή ൌ ൌ
࢝ ξ െ ξ െ ξ
2.
࢝ࢠ
ࢠ is rotated ܚ܉ሺ࢝ሻ and dilated by ȁ࢝ȁ and then rotated െܚ܉ሺ࢝ࢠሻ and dilated by . For the given values of ࢠ
ȁ࢝ࢠȁ
and ࢝, this transformation results in a dilation of ࢝ by a factor of .
࣊ ࣊
ܚ܉ሺࢠሻ ൌ and ܚ܉ሺ࢝ሻ ൌ . Adding these arguments and finding an equivalent rotation between and ࣊
࣊
gives a rotation of and ȁ࢝ࢠȁ ൌ ή ൌ . This describes the complex number ξ . The reciprocal of this
࣊ ξ
number has argument and modulus , which describes the complex number െ .
ૡ ૡ
ξ െ ξ െ ξ
ൌ ൌ ൌ ή ൌ ൌ െ
࢝ࢠ ሺξ െ ሻሺ ξሻ ξ ξ ξ െ ૡ ૡ
Problem Set Sample Solutions
1. Describe the geometric effect of multiplying ࢠ by the reciprocal of each complex number listed below.
a. ࢝ ൌ
࣊
ܚ܉ሺ࢝ ሻ ൌ and ȁ࢝ ȁ ൌ
࣊ ࣊
ࢠ is rotated by ࣊ െ ܚ܉ሺ࢝ ሻ, which is ࣊ െ ൌ , and dilated by .
b. ࢝ ൌ െ
ܚ܉ሺ࢝ ሻ ൌ ࣊ and ȁ࢝ ȁ ൌ
ࢠ is rotated by ࣊ െ ܚ܉ሺ࢝ ሻ, which is ࣊ െ ࣊ ൌ ࣊, and dilated by .
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c. ࢝ ൌ ξ
࣊
ܚ܉ሺ࢝ ሻ ൌ and ȁ࢝ ȁ ൌ
ૈ ࣊
ࢠ is rotated by ࣊ െ ܚ܉ሺ࢝ ሻ, which is ࣊ െ ൌ , and dilated by .
d. ࢝ ൌ െ ξ
࣊
ܚ܉ሺ࢝ ሻ ൌ and ȁ࢝ ȁ ൌ
࣊ ࣊
ࢠ is rotated by ࣊ െ ܚ܉ሺ࢝ ሻ, which is ࣊ െ ቀ ቁ ൌ , and dilated by .
2. Let ࢠ ൌ െ െ ξ. Show that the geometric transformations you described in Problem 1 really produce the
correct complex number by performing the indicated operation and determining the argument and modulus of each
number.
ିିξ
a.
࢝
ࢠ െ െ ξ െ െ ξ െ െξ ξ ࢠ ࢠ ࣊
ൌ ൌ ή ൌ ൌെ ǡ ฬ ฬ ൌ ǡ ܚ܉൬ ൰ ൌ
࢝ െ ૢ ࢝ ࢝
ȁࢠȁ ൌ , so the result of division is a complex number whose modulus is of .
࣊
ܚ܉ሺࢠሻ ൌ , so the result of division by a complex number is whose argument represents the same rotation
࣊ ࣊ ૠ࣊ ࣊
as ൌ , which would be .
ିିξ
b.
࢝
ࢠ െ െ ξ ࢠ ࢠ ࣊
ൌ ൌ ξǡ ฬ ฬ ൌ ǡ ܚ܉൬ ൰ ൌ
࢝ െ ࢝ ࢝
ȁࢠȁ ൌ and of is .
࣊ ࣊ ࣊ ࣊ ࣊
ܚ܉ሺࢠሻ ൌ , so the result of division will rotate ࢠ by െ and െ ൌ .
ିିξ
c.
࢝
ࢠ െ െ ξ െξ െ െ ξ ࢠ ࢠ ૠ࣊
ൌ ൌ ൌ െξ െ ǡ ฬ ฬ ൌ ǡ ܚ܉൬ ൰ ൌ
࢝ ξ ࢝ ࢝
ȁࢠȁ ൌ and of is .
࣊ ࣊ ࣊ ࣊ ૠ࣊
ܚ܉ሺࢠሻ ൌ , so the result of division will rotate ࢠ by െ and െ ൌ .
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ିିξ
d.
࢝
ࢠ െ െ ξ െ െ ξ െ ξ ࢠ ࢠ ࣊
ൌ ൌ ൌ െ ξǡ ฬ ฬ ൌ ǡ ܚ܉൬ ൰ ൌ
࢝ െ ξ ࢝ ࢝
ȁࢠȁ ൌ ǡ and of is .
࣊ ࣊ ࣊ ࣊ ࣊ ࣊
ܚ܉ሺࢠሻ ൌ , so the result of division will rotate ࢠ by െ , and െ ൌ െ . A rotation of െ will be
࣊
the same as a rotation of , which is the argument of the quotient.
3. In Exercise 12 of this lesson, you described the complex number as a transformation of ࢠ for a specific complex
ࢠത
number ࢠ. Show that this transformation always produces a dilation of ࢠ ൌ ࢇ ࢈.
ࢇା࢈ ࢇା࢈
ࢠത ൌ ࢇ െ ࢈, and ൌ ή ൌ ൌ ሺࢇ ࢈ሻ. This complex number is the product of a real
ࢇି࢈ ࢇି࢈ ࢇା࢈ ࢇ ା࢈ ࢇ ା࢈
number and the original complex number ࢠǡ so it will have the same argument as ࢇ ࢈, but the modulus will be a
different number.
4. Does ࡸሺࢠሻ ൌ satisfy the conditions that ࡸሺࢠ ࢝ሻ ൌ ࡸሺࢠሻ ࡸሺ࢝ሻ and ࡸሺࢠሻ ൌ ࡸሺࢠሻǡ which makes it a linear
ࢠ
transformation? Justify your answer.
ࢇି࢈ ࢇ ࢈
ൌ ൌ , which is a complex number whose real part is and whose imaginary part is െ .
ࢠ ࢇା࢈ ࢇ ା࢈ ࢇ ା࢈ ࢇ ࢈
Since all complex numbers satisfy the conditions that make them a linear transformation, and is a complex
ࢠ
number, it will also be a linear transformation.
തതതത
5. Show that ࡸሺࢠሻ ൌ ࢝ ൬ ࢠ൰ describes a reflection of ࢠ about the line containing the origin and ࢝ for ࢠ ൌ and
࢝
࢝ ൌ .
തതതതതതതതതതത
െ െሺሻሺሻ
ࡸሺࢠሻ ൌ ሺ ሻ ൬ ሺଙሻ൰ ൌ ሺ ሻ ቀ ቁൌ ൌ ൌ , which is the image of the transformation z
െ ሺെሻሺሻ
that is reflected about the line containing the origin and ࢝.
6. Describe the geometric effect of each transformation function on ࢠ where ࢠ, ࢝, and ࢇ are complex numbers.
ࢠെ࢝
a. ࡸ ሺࢠሻ ൌ
ࢇ
ࢠ is translated by ࢝, rotated by ࣊ െ ܚ܉ሺࢇሻ, and dilated by .
ȁࢇȁ
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തതതതതതതതത
ࢠെ࢝
b. ࡸ ሺࢠሻ ൌ ቀ ቁ
ࢇ
ࢠ is translated by ࢝, reflected about the real axis, rotated by ࣊ െ ܚ܉ሺࢇሻ, and dilated by , and reflected
ȁࢇȁ
about the real axis.
തതതതതതതതത
ࢠെ࢝
c. ࡸ ሺࢠሻ ൌ ࢇ ቀ ቁ
ࢇ
ࢠ is translated by ࢝, reflected about the real axis, rotated by ࣊ െ ܚ܉ሺࢇሻ, and dilated by , and reflected
ȁࢇȁ
about the real axis, rotated by ܚ܉ሺࢇሻ, and dilated by ȁࢇȁ.
തതതതതതതതത
ࢠെ࢝
d. ࡸ ሺࢠሻ ൌ ࢇ ቀ ቁ࢝
ࢇ
ࢠ is translated by ࢝, reflected about the real axis, rotated by ࣊ െ ܚ܉ሺࢇሻ, and dilated by , and reflected
ȁࢇȁ
about the real axis, rotated by ܚ܉ሺࢇሻ, dilated by ȁࢇȁ, and translated by ࢝.
7. Verify your answers to Problem 6 if ࢠ ൌ െ , ࢝ ൌ , and ࢇ ൌ െ െ .
ࢠെ࢝
a. ࡸ ሺࢠሻ ൌ
ࢇ
ૠ࣊ ࣊
ȁࢠȁ ൌ ξǡ ܚ܉ሺࢠሻ ൌ ǡ ȁ࢝ȁ ൌ ǡ ܚ܉ሺ࢝ሻ ൌ ࣊ǡ ȁࢇȁ ൌ ξǡ ܚ܉ሺࢇሻ ൌ ൌ Ǥ ૢૠ
ࢠ െ ࢝ െ െ െ ࢠെ࢝ ࢠെ࢝
ൌ ൌ ൌ ǡ ቚ ቚ ൌ ξǡ ܚ܉ቀ ቁ ൌ Ǥ ૠ
ࢇ െ െ െ െ ࢇ ࢇ
ࢠ െ ࢝ ൌ െ ǡ ȁࢠ െ ࢝ȁ ൌ ξǡ ܚ܉ሺࢠ െ ࢝ሻ ൌ Ǥ
ࢠെ࢝
ൈ ȁࢠ െ ࢝ȁ ൌ ൈ ξ ൌ ξ ൌ ቚ ቚǡ
ȁࢇȁ ξ ࢇ
ࢠെ࢝
ܚ܉ቀ ቁ ൌ ܚ܉ሺࢠ െ ࢝ሻ ൫࣊ െ ܚ܉ሺࢇሻ൯ ൌ Ǥ ࣊ െ Ǥ ૢૠ ൌ Ǥ ૠ ࣊ ൌ Ǥ ૠ
ࢇ
തതതതതതതതത
ࢠെ࢝
b. ࡸ ሺࢠሻ ൌ ቀ ቁ
ࢇ
തതതതതതതതതതത
ࢠെ࢝ തതതതതതതതതതതത
െ ଙ തതതതതതതതതതത
ࢠെ࢝ തതതതതതതതതതത
ࢠെ࢝
ቀ ቁൌ൬ തതതതതതതതതതത
൰ ൌ ሺ ଙሻ ൌ െ ǡ ฬቀ ቁฬ ൌ ξǡ ܚ܉൬ቀ ቁ൰ ൌ െǤ ૠ
ࢇ െ െ ଙ ࢇ ࢇ
തതതതതതത
തതതതതതതതതതത
ࢠെ࢝
൬ ൰ ൈ ȁࢠതതതതതതതതതത
െ ࢝ȁ ൌ ൈ ξ ൌ ξ ൌ ฬቀ ቁฬ
ȁࢇȁ ξ ࢇ
തതതതതതതതതതത
ࢠെ࢝
ܚ܉൬ቀ ቁ൰ ൌ ࣊ െ ቀܚ܉ሺࢠ െ ࢝ሻ ൫࣊ െ ܚ܉ሺࢇሻ൯ቁ ൌ ࣊ െ Ǥ െ ࣊ Ǥ ૢૠ ൌ െǤ ૠ
ࢇ
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തതതതതതതതത
ࢠെ࢝
c. ࡸ ሺࢠሻ ൌ ࢇ ቀ ቁ
ࢇ
തതതതതതതതതതത
ࢠെ࢝ തതതതതതതതതതതത
െ ଙ
ࢇൈቀ ቁ ൌ ሺെ െ ሻ ൬ തതതതതതതതതതത
൰ ൌ ሺെ െ ሻሺ ଙሻ ൌ ሺെ െ ሻሺ െ ሻ ൌ െ ǡ
ࢇ െ െ ଙ
തതതതതതതതതതത
ࢠെ࢝ തതതതതതതതതതത
ࢠെ࢝
ฬࢇ ൈ ቀ ቁฬ ൌ ξǡ ܚ܉ቆሺെ െ ሻ ቀ ቁቇ ൌ ࣊ െ Ǥ ൌ Ǥ ૡ
ࢇ ࢇ
തതതതതതത
തതതതതതതതതതത
ࢠെ࢝
ȁࢇȁ ൈ ൬ ൰ ൈ ȁࢠതതതതതതതതതത
െ ࢝ȁ ൌ ξ ൈ ൈ ξ ൌ ξ ൌ ฬሺെ െ ሻ ቀ ቁฬ
ȁࢇȁ ξ ࢇ
തതതതതതതതതതത
ࢠെ࢝
ܚ܉൬ࢇ ൈ ቀ ቁ൰ ൌ ܚ܉ሺࢇሻ ቀܚ܉ሺࢠ െ ࢝ሻ ൫࣊ െ ܚ܉ሺࢇሻ൯ቁ
ࢇ
ൌ Ǥ ૢૠ ࣊ െ Ǥ െ ࣊ Ǥ ૢૠ ൌ Ǥ ૡ
തതതതതതതതത
ࢠെ࢝
d. ࡸ ሺࢠሻ ൌ ࢇ ቀ ቁ࢝
ࢇ
തതതതതതതതതതത
ࢠെ࢝ തതതതതതതതതതതത
െ ଙ
ࢇൈቀ ቁ ࢝ ൌ ሺെ െ ሻ ൬ ൰ ൌ െ ൌ െ ǡ
ࢇ െ െ ଙ
തതതതതതതതതതത
ࢠെ࢝ തതതതതതതതതതത
ࢠെ࢝ ࣊
ฬࢇ ൈ ቀ ቁ ࢝ฬ ൌ ξǡ ܚ܉ቆሺെ െ ሻ ቀ ቁ ࢝ቇ ൌ ൌ Ǥ
ࢇ ࢇ
ࢠെ࢝
തതതതതതതതതതത
ฬࢇ ൈ ቀ ቁ ࢝ฬ ൌ ȁെ ȁ ൌ ξ ൌǡ
ࢇ
തതതതതതതതതതത
ࢠെ࢝
ܚ܉൬ࢇ ൈ ቀ ቁ ࢝൰ ൌ ܚ܉ሺെ ሻ ൌ ܚ܉ሺെ ሻ ൌ Ǥ
ࢇ
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Name Date
1. Given ݖൌ ͵ െ Ͷ݅ and ݓൌ െͳ ͷ݅:
a. Find the distance between ݖand ݓ.
b. Find the midpoint of the segment joining ݖand ݓ.
2. Let ݖଵ ൌ ʹ െ ʹ݅ and ݖଶ ൌ ሺͳ െ ݅ሻ ξ͵ሺͳ ݅ሻ.
a. What is the modulus and argument of ݖଵ ?
b. Write ݖଵ in polar form. Explain why the polar and rectangular forms of a given complex number
represent the same number.
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PRECALCULUS AND ADVANCED TOPICS
c. Find a complex number ݓ, written in the form ݓൌ ܽ ܾ݅, such that ݖݓଵ ൌ ݖଶ .
d. What is the modulus and argument of ?ݓ
e. Write ݓin polar form.
f. When the points ݖଵ and ݖଶ are plotted in the complex plane, explain why the angle between ݖଵ and
ݖଶ measures ሺݓሻ.
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g. What type of triangle is formed by the origin and the two points represented by the complex
numbers ݖଵ and ݖଶ ? Explain how you know.
h. Find the complex number, ݒ, closest to the origin that lies on the line segment connecting ݖଵ and ݖଶ .
Write ݒin rectangular form.
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3. Let ݖbe the complex number ʹ ͵݅ lying in the complex plane.
a. What is the conjugate of ?ݖExplain how it is related geometrically to ݖ.
b. Write down the complex number that is the reflection of ݖacross the vertical axis. Explain how you
determined your answer.
ଵ
Let ݉ be the line through the origin of slope in the complex plane.
ଶ
c. Write down a complex number, ݓ, of modulus ͳ that lies on ݉ in the first quadrant in rectangular
form.
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d. What is the modulus of ?ݖݓ
e. Explain the relationship between ݖݓand ݖ. First, use the properties of modulus to answer this
question, and then give an explanation involving transformations.
f. When asked,
ଵ
“What is the argument of ”?ݖ
௪
͵ ͳ
Paul gave the answer ቀ ቁ െ ቀ ቁ, which he then computed to two decimal places.
ʹ ʹ
Provide a geometric explanation that yields Paul’s answer.
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g. When asked,
ଵ
“What is the argument of ”?ݖ
௪
Mable did the complex number arithmetic and computed ݖൊ ݓ.
ܽ
She then gave an answer in the form ቀܾቁ for some fraction . What fraction did Mable find?
Up to two decimal places, is Mable’s final answer the same as Paul’s?
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A Progression Toward Mastery
STEP 1 STEP 2 STEP 3 STEP 4
Assessment Missing or incorrect Missing or incorrect A correct answer A correct answer
Task Item answer and little answer but with some evidence supported by
evidence of evidence of some of reasoning or substantial
reasoning or reasoning or application of evidence of solid
application of application of mathematics to reasoning or
mathematics to mathematics to solve the problem, application of
solve the problem. solve the problem. OR an incorrect mathematics to
answer with solve the problem.
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem.
1 a Student provides an Student shows some Student uses the formula Student computes the
incorrect response, and knowledge of the correctly but makes distance correctly with
there is no evidence to distance formula but minor mathematical supporting work shown.
support that student does not use the formula mistakes.
understands how to correctly. OR
compute the distance. Student computes
distance correctly with
no supporting work.
b Student provides an Student shows some Student uses the formula Student computes the
incorrect response, and knowledge of the correctly but makes midpoint correctly with
there is no evidence to midpoint formula but minor mathematical supporting work shown.
support that student does not use the formula mistakes.
understands how to correctly. OR
compute the midpoint. Student computes the
midpoint correctly with
no supporting work.
2 a Student provides an Student uses the correct Student uses the correct Student computes
incorrect response, and method and answer for methods for both, but modulus and argument
there is no evidence to either the modulus or either the modulus or correctly. Work is shown
support that student the argument. argument is incorrect to support the answer.
understands how to OR due to a minor error.
compute the modulus Student uses the correct OR
and argument. method for both but Student gives correct
makes minor errors. answers for both with no
supporting work shown.
b Student shows no Student attempts to put Student writes the Student writes the
knowledge of polar form. the complex number correct polar form but correct polar form and
into polar form but does not explain why correctly explains why
makes errors. polar and rectangular polar and rectangular
forms represent the forms represent the
same number. same number.
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c Student does not Student attempts to Student applies the Student applies the
attempt to divide ݖଶ by divide ݖଶ by ݖଵ without division algorithm division algorithm
ݖଵ to find ݓ. applying the correct correctly and shows correctly and gives a
algorithm involving work but has minor correct answer with
multiplication by ͳ in the mathematical errors sufficient work shown to
form of the conjugate ݖଵ leading to an incorrect demonstrate
divided by the conjugate final answer. understanding of the
of ݖଵ . process.
d Student does not Student computes the Student uses correct Student computes
compute either answer modulus or argument methods for both, but modulus and argument
correctly, nor is there correctly. either the modulus or correctly for the answer
evidence to support that OR the argument is incorrect to part (b), and work is
student understands Student uses correct due to a minor error. shown to support
how to compute the methods for both but OR answer.
modulus and argument. arrives at incorrect Student gives correct Note: Student can earn
answers due to minor answers for both with no full points for this even if
errors. supporting work shown. the answer to part (b) is
incorrect.
e Student shows no Student shows some Student shows Student writes the
knowledge of polar form. knowledge of polar form knowledge of polar form correct polar form of the
but does make major but makes minor number.
mathematical errors. mathematical errors.
f Student makes little or Student cannot Student may not indicate Student explanation
no attempt to identify determine the angle the requested angle clearly connects
the angle measure. between the two measurement or gives an multiplication with the
complex numbers whose incorrect angle correct transformations
vertex is at the origin. measurement. Student by explaining that ݖଶ is
Student provides an provides an explanation the image of ݖଵ achieved
explanation that fails to that does not fully by rotating ݖଵ by the
connect multiplication address the connection ሺݓሻ and no dilation
with transformations. between multiplication since ȁݓȁ ൌ ͳ. The
The explanation may and transformations but answer may be
include a comparison of may include a supported with a sketch.
the moduli of ݖଵ and ݖଶ . comparison of the
The answer may include moduli of ݖଵ and ݖଶ .
a sketch to support the The answer may be
answer. supported with a sketch.
g Student makes little or Student may identify the Student identifies the Student identifies the
no attempt to identify triangle as isosceles but triangle as isosceles or triangle as equilateral
the triangle. with little or no equilateral but gives an and gives a complete
explanation. incomplete or incorrect explanation.
explanation.
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h Student makes little or Student may attempt to Student finds ݒcorrectly, Student averages ݖଵ and
no attempt to find ݒ. sketch the situation, but but there is little or no ݖଶ to find ݒand clearly
more than one explanation or work explains why using a
misconception or explaining why ݒis the geometric argument
mathematical error leads midpoint of the line regarding ’ݒs location on
to an incorrect or segment. the perpendicular
incomplete solution. OR bisector of the triangle.
Student identifies that ݒ
would be at the midpoint
but fails to compute it
correctly.
3 a Student incorrectly Student incorrectly Student gives the correct Student gives the correct
answers both the real answers either the real answer but does not give answer and includes the
and the imaginary part. or the imaginary part. an explanation. correct explanation of
how it is geometrically
related to ݖ.
b Student incorrectly Student incorrectly Student gives the correct Student gives the correct
answers both the real answers either the real answer but does not give answer and explains that
and the imaginary part. or the imaginary part. an explanation. the real part is the
opposite but that the
imaginary part stays the
same.
c Student provides an Student gives an Student gives the correct Student gives a correct
answer that is not a incorrect answer with answer with limited answer with work shown
complex number. little evidence of correct reasoning or work to to support approach.
OR reasoning (solution may support the answer. Student reasoning could
Student provides an fail to address modulus OR use the polar form of a
answer that is a complex of ͳ but be a complex Student uses correct complex number or
number with both an number on the line ݉ or reasoning, but minor apply proportional
incorrect modulus and may have a modulus of ͳ mathematical errors lead reasoning to find ݓwith
argument. There is little but not be a complex to an incorrect solution. the correct argument
or no supporting work number on the line ݉). and modulus.
shown.
d Student shows no Student shows some Student makes minor Student gives a correct
knowledge of ݖݓor the knowledge of calculating mathematical errors in answer with supporting
modulus. the modulus, but the calculating the modulus. work shown clearly.
answer is incorrect with
little supporting work
shown.
e Student does not explain Student gives minimal Student explains Student gives clear and
the relationship between explanation with some connection but does not correct explanation that
ݖand ݖݓ. mistakes. include transformations includes transformations
that occur. that occur.
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f Student shows little or Student provides an Student identifies ሺݖሻ Student explains both
no work. The explanation that may and ሺݓሻ and explains the rotation and lack of
explanation fails to include references to ଵ dilation correctly since
that multiplication by
address transformations rotations and dilations ௪ ȁݓȁ ൌ ͳ.
in a meaningful way. but does not clearly would create a clockwise In the explanation,
address the fact that rotation, but the student identifies ሺݖሻ
ଵ explanation lacks a clear and ሺݓሻ and supports
multiplication by reason why the two
௪ the reason for the
represents a clockwise should be subtracted or difference. A sketch may
rotation of ሺݓሻ. contains other minor be included.
errors.
௭ Student answers either Student correctly Student correctly
g Student answers both
௪ ௭ ݖ ௭ ݖ computes both
ݖ or ቀ ቁ incorrectly computes and ቀ ቁ
and ቀ ቁ incorrectly ௪ ݓ ௪ ݓ arguments to two
ݓ due to mathematical but fails to compare the
due to major decimal places. Work
errors. Student indicates arguments in parts (d) shown uses appropriate
misconceptions or
that both arguments and (e) or explain why notation and sufficient
calculation errors.
should be the same. they should be the same. steps to follow the
Student may argue
incorrectly that the solution.
arguments in parts (e)
and (f) should be
different.
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Name Date
1. Given ݖൌ ͵ െ Ͷ݅ and ݓൌ െͳ ͷ݅:
a. Find the distance between ݖand ݓ.
2 2
Distance ൌ ට൫3 െ ሺെ1ሻ൯ െ ( െ 1) ൫ሺെ4ሻ െ 5൯
2
ൌ ට42 ( െ 9)
ൌ ξ16 81
ൌ ξ97
b. Find the midpoint of the segment joining ݖand ݓ.
3 ሺെ1ሻ ሺെ4ሻ+5
Midpoint ൌ i
2 2
2 1
ൌ i
2 2
1
ൌ 1 i
2
2. Let ݖଵ ൌ ʹ െ ʹ݅ and ݖଶ ൌ ሺͳ െ ݅ሻ ξ͵ሺͳ ݅ሻ.
a. What is the modulus and argument of ݖଵ ?
ȁz1 ȁൌඥሺ2ሻ2 ሺ-2ሻ2 ൌξ8ൌ2ξ2
െ2 െü
argሺz1 ሻൌ tanି1 ൬ ൰ ൌ
2 4
b. Write ݖଵ in polar form. Explain why the polar and rectangular forms of a given complex number
represent the same number.
െü െü
z1 ൌ2ξ2 ቂcos ቀ ቁ i sin ቀ ቁቃ
4 4
The modulus represents the distance from the origin to the point. The degree of
rotation is the angle from the x-axis. When the polar form is expanded, the result is
the rectangular form of a complex number.
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c. Find a complex number ݓ, written in the form ݓൌ ܽ ܾ݅, such that ݖݓଵ ൌ ݖଶ .
z2 ൌ1 െ i ξ3 iξ3ൌ൫ξ3 1൯ ൫ξ3 െ 1൯i
wz1 ൌ w2 implies that
z2 ൣ൫ξ3+1൯ ൫ξ3 െ 1൯i൧ ሺ2 2iሻ
wൌ ൌ ൈ
z1 ሺ2 െ 2iሻ ሺ2 2iሻ
2ξ3 2 2iξ3 2i 2iξ3 െ 2i െ 2ξ3 2
ൌ
44
4 4iξ3
ൌ
8
1 ξ3
ൌ iǤ
2 2
d. What is the modulus and argument of ?ݓ
2
1 2 ξ3 1 3
ȁwȁ ൌ ඨ൬ ൰ ቆ ቇ ൌ ඨ ൌ 1
2 2 4 4
ξ3
ü
argሺwሻ ൌ tan -1
൮ 2 ൲ൌ
1 3
2
e. Write ݓin polar form.
ü ü
w ൌ 1 ቂcos ቀ ቁ i sin ቀ ቁቃ
3 3
f. When the points ݖଵ and ݖଶ are plotted in the complex plane, explain why the angle between ݖଵ and
ݖଶ measures ሺݓሻ.
Since z2 =wz1 , then z2 is the transformation of z1 rotated counterclockwise by argሺwሻ,
ü
which is .
3
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g. What type of triangle is formed by the origin and the two points represented by the complex
numbers ݖଵ and ݖଶ ? Explain how you know.
ü
Since ȁwȁ=1 and argሺwሻ= , the triangle formed by the origin and the points
3
representing z1 and z2 will be equilateral. All of the angles are 60° in this triangle.
h. Find the complex number, ݒ, closest to the origin that lies on the line segment connecting ݖଵ and ݖଶ .
Write ݒin rectangular form.
The point that represents v is the midpoint of the segment connecting z1 and z2 since it
must be on the perpendicular bisector of the triangle with vertex at the origin.
To find the midpoint, average z1 and z2 .
ξ3 1 2 ξ3 െ 1 െ 2 ξ3 3 ξ3 െ 3
vൌ i ൌ i
2 2 2 2
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3. Let ݖbe the complex number ʹ ͵݅ lying in the complex plane.
a. What is the conjugate of ?ݖExplain how it is related geometrically to ݖ.
zത ൌ2 െ 3i
This number is the conjugate of z and the reflection of z across the horizontal axis.
b. Write down the complex number that is the reflection of ݖacross the vertical axis. Explain how you
determined your answer.
This number is െ23i. The real coordinate has the opposite sign, but the imaginary
part keeps the same sign. This means a reflection across the imaginary (vertical) axis.
ଵ
Let ݉ be the line through the origin of slope in the complex plane.
ଶ
c. Write down a complex number, ݓ, of modulus ͳ that lies on ݉ in the first quadrant in rectangular
form.
1 1
Because the slope of m is , the argument of w is tan-1 ቀ ቁ.
2 2
Using the polar form of w,
1 1
wൌ1 cos ൬tan-1 ቀ ቁ൰ i sin ൬tan-1 ቀ ቁ൰൨. From the triangle
2 2
2 1
shown below, wൌ i.
ඥ5 ඥ5
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d. What is the modulus of ?ݖݓ
The modulus of wz is ξ13.
e. Explain the relationship between ݖݓand ݖ. First, use the properties of modulus to answer this
question, and then give an explanation involving transformations.
The modulus of wz is ξ13 and is the same as ȁzȁ.
Using the properties of modulus,
ȁwzȁൌȁwȁൈȁzȁൌ1ൈȁzȁൌ1ൈඥ22 32 ൌξ13.
Geometrically, multiplying by w will rotate z by the argሺwሻ and dilate z by ȁwȁ. Since
ȁwȁ=1, the transformation is a rotation only, so both w and z are the same distance
from the origin.
f. When asked,
ଵ
“What is the argument of ”?ݖ
௪
͵ ͳ
Paul gave the answer ቀʹቁ െ ቀʹቁ, which he then computed to two decimal places.
Provide a geometric explanation that yields Paul’s answer.
1
The product z would result in a clockwise rotation of z by the argሺwሻ. There would be
w
no dilation since ȁwȁ=1.
3
argሺzሻ ൌ tan-1 ൬ ൰
2
1
argሺwሻ ൌ tan-1 ൬ ൰
2
1 3 1
arg ൬ z൰ ൌ tan-1 ൬ ൰ െ tan-1 ൬ ൰
w 2 2
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A STORY OF FUNCTIONS Mid-Module Assessment Task M1
PRECALCULUS AND ADVANCED TOPICS
g. When asked,
ଵ
“What is the argument of ”?ݖ
௪
Mable did the complex number arithmetic and computed ݖൊ ݓ. She then gave an answer in the
ܽ
form ቀܾቁ for some fraction . What fraction did Mable find? Up to two decimal places, is
Mable’s final answer the same as Paul’s?
2 1
ሺ23iሻ ൈ ൬ െ i൰
z zw
ഥ zw
ഥ ξ5 ξ5
ൌ ൌ ൌ
w ww
ഥ ȁwȁ 1
4 6i 2i 3
ൌ െ
ξ5 ξ5 ξ5 ξ5
7 4
ൌ i
ξ5 ξ5
Comparing these angles shows they are the same.
4
tan-1 ൬ ൰ ൎ 0.52
7
3 1
tan-1 ൬ ൰ - tan-1 ൬ ൰ ൎ 0.52
2 2
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^dKZzK&&hEd/KE^
DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ
PRECALCULUS AND ADVANCED TOPICS ͻDKh> 1
Topic C
dŚĞWŽǁĞƌŽĨƚŚĞZŝŐŚƚEŽƚĂƚŝŽŶ
&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
(+) Represent addition, subtraction, multiplication, and conjugation of complex
numbers geometrically on the complex plane; use properties of this representation for
computation. For example, ሺെͳ ξ͵݅ሻଷ ൌ ͺ because ሺെͳ ξ͵݅ሻ has modulus ʹ and
argument ͳʹͲι.
(+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of Ͳ and ͳ in the real numbers. The determinant of a
square matrix is nonzero if and only if the matrix has a multiplicative inverse.
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable
dimensions to produce another vector. Work with matrices as transformations of
vectors.
(+) Work with ʹ ൈ ʹ matrices as transformations of the plane, and interpret the
absolute value of the determinant in terms of area.
/ŶƐƚƌƵĐƚŝŽŶĂůĂLJƐ͗ 13
>ĞƐƐŽŶƐ 18–19: Exploiting the Connection to Trigonometry (E, P) 1
>ĞƐƐŽŶϮϬ: Exploiting the Connection to Cartesian Coordinates (S)
>ĞƐƐŽŶϮϭ: The Hunt for Better Notation (P)
>ĞƐƐŽŶƐϮϮ–23: Modeling Video Game Motion with Matrices (P, P)
>ĞƐƐŽŶϮϰ͗ Matrix Notation Encompasses New Transformations! (P)
>ĞƐƐŽŶϮϱ͗ Matrix Multiplication and Addition (P)
>ĞƐƐŽŶƐϮϲ–27: Getting a Handle on New Transformations (E, P)
>ĞƐƐŽŶƐϮϴ–30: When Can We Reverse a Transformation? (P, E, P)
1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
242 dŽƉŝĐ: The Power of the Right Notation
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A STORY OF FUNCTIONS dŽƉŝĐC Dϭ
PRECALCULUS AND ADVANCED TOPICS
The theme of Topic C is to highlight the effectiveness of changing notations and the power provided by
certain notations such as matrices. Lessons 18 and 19 exploit the connection to trigonometry, as students see
how much of complex arithmetic is simplified. Students use the connection to trigonometry to solve
problems such as find the three cube roots of െ1. In Lesson 20, complex numbers are regarded as points in
ܾ
the Cartesian plane. If ܽ = ݓ+ ܾ݅, then the modulus is = ݎξܽଶ + ܾ ଶ , and the argument is ߙ = arctan ቀ ቁ.
ܽ
Students begin to write analytic formulas for translations, rotations, and dilations in the plane and revisit the
ideas of Geometry in this light. In Lesson 21, students discover a better notation (i.e., matrices) and develop
the 2 × 2 matrix notation for planar transformations represented by complex number arithmetic. This work
leads to Lessons 22 and 23 as students discover how geometry software and video games efficiently perform
rigid motion calculations. Students discover the flexibility of 2 × 2 matrix notation in Lessons 24 and 25 as
they add matrices and multiply by the identity matrix and the zero matrix. Students understand that
multiplying matrix ܣby the identity matrix results in matrix ܣand connect the multiplicative identity matrix
to the role of 1, the multiplicative identity, in the real number system. This is extended as students see that
the identity matrix does not transform the unit square. Students then add matrices and conclude that the
zero matrix added to matrix ܣresults in matrix ܣand is similar to 0 in the real number system. They extend
this concept to transformations on the unit square and see that adding the zero matrix has no effect, but
multiplying by the zero matrix collapses the unit square to zero. This allows for the study of additional matrix
transformations (shears, for example) in Lessons 26 and 27, multiplying matrices, and the meaning of the
determinant of a 2 × 2 matrix. Lessons 28–30 conclude Topic C and Module 1 as students discover the
inverse matrix (matrix ܣis called an inverse matrix to a matrix ܤif ܫ = ܤܣand )ܫ = ܣܤand determine when
matrices do not have inverses. Students begin to think and reason abstractly about the geometric effects of
the operations of complex numbers as they see the connection to trigonometry and the Cartesian plane .
The study of vectors and matrices is only introduced in Module 1 through a coherent connection to
transformations and complex numbers. Further and more formal study of multiplication of matrices occurs in
Module 2.
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A STORY OF FUNCTIONS Lesson 18 M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 18: Exploiting the Connection to Trigonometry
Student Outcomes
Students derive the formula for ݖ ൌ ݎ ሺ ሺ݊ߠሻ ݅ԝሺ݊ߠሻሻ and use it to calculate powers of a complex
number.
Lesson Notes
This lesson builds on the concepts from Topic B by asking students to extend their thinking about the geometric effect
of multiplication of two complex numbers to the geometric effect of raising a complex number to an integer exponent͘
This lesson is part of a two-day lesson that gives students another opportunity to work with the polar form of a complex
number, to see its usefulness in certain situations, and to exploit that form to quickly calculate powers of a complex
number. Students compare and convert between polar and rectangular form and graph complex numbers represented
both ways. On the second day, students examine graphs of powers of complex numbers in a polar grid and then
reverse the process from Day 1 to calculate ݊th roots of a complex number. Throughout the lesson, students construct
and justify arguments, looking for patterns in repeated reasoning, and use the structure of expressions and visual
representations to make sense of the mathematics.
Classwork
Opening (5 minutes)
ߨ ߨ
Display two complex numbers on the board: ͳ ݅ and ξʹ ቀ ቀ ቁ ݅ ቀ ቁቁ.
Ͷ Ͷ
Do these represent the same number? Explain why or why not.
ඥʹ ඥʹ ߨ ߨ
à ξʹ ൬ ʹ ݅ ʹ ൰ ൌ ͳ ݅; yes, they are the same number. When you expand ξʹ ቀ ቀͶቁ ݅ ቀͶቁቁ,
you get ͳ ݅.
What are the advantages of writing a complex number in polar form? What are the disadvantages?
à In polar form, you can see the modulus and argument. It is easy to multiply the numbers because you
multiply the modulus and add the arguments. It can be difficult to graph the numbers because you
have to use a compass and protractor to graph them accurately. If you are unfamiliar with the
rotations and evaluating sine and cosine functions, then converting to rectangular is difficult. It is not
so easy to add complex numbers in polar form unless you have a calculator and convert them to
rectangular form.
What are the advantages of writing a complex number in rectangular form? What are the disadvantages?
à They are easy to graph; addition and multiplication are not too difficult either. It is difficult to
understand the geometric effect of multiplication when written in rectangular form. It is not so easy to
calculate the argument of the number, and you have to use a formula to calculate the modulus.
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Opening Exercise (5 minutes)
Tell students that in this lesson they are going to begin to exploit the advantages of writing a number in polar form, and
have them quickly do the Opening Exercises. Students should work these problems individually. These exercises also
serve as a check for understanding. If students are struggling to complete these exercises quickly and accurately,
consider providing some additional practice in the form of Sprints.
Opening Exercise
a. Identify the modulus and argument of each complex number, and then rewrite it in rectangular form.
࣊ ࣊
i. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ Scaffolding:
The modulus is , and the argument is . The number is ξ ξ.
For struggling students,
encourage them to work
ii. ቀ ܛܗ܋ቀ
࣊ ࣊
ቁ ԝ ܖܑܛቀ ቁቁ
from a copy of a unit circle
to quickly identify the sine
࣊ ඥ and cosine function
The modulus is , and the argument is . The number is െ .
values.
On Opening Exercise part
ૠ࣊ ૠ࣊
iii. ξ ቀ ܛܗ܋ቀ
ቁ ԝ ܖܑܛቀ ቁቁ
(b), help students recall
ૠ࣊
how to graph complex
The modulus is ξ, and the argument is . The number is െ . numbers, construct a
triangle, and use special
triangle ratios to
ૠ࣊ ૠ࣊
iv. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ determine the argument.
ૠ࣊ ඥ
The modulus is , and the argument is . The number is െ െ .
v. ሺܛܗ܋ሺ࣊ሻ ԝܖܑܛሺૈሻሻ
The modulus is , and the argument is ࣊. The number is െ.
b. What is the argument and modulus of each complex number? Explain how you know.
i. െ
ૠ࣊
We have ȁ െ ȁ ൌ ξ, and ܚ܉ሺ െ ሻ ൌ . The point ሺǡ െሻ is located in the fourth quadrant.
ૠ࣊
The ray from the origin containing the point is a rotation of from the ray through the origin
containing the real number .
ii. ξ
࣊
We haveหξ ห ൌ , and ܚ܉൫ξ ൯ ൌ . The point ሺξǡ ሻ is located in the first quadrant.
࣊
The ray from the origin containing the point is a rotation of from the ray through the origin
containing the real number .
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iii. െ െ ξ
࣊
We have หെ െ ξห ൌ , and ܚ܉൫െ െ ξ൯ ൌ . The point ሺെǡ െξሻ is located in the third
࣊
quadrant. The ray from the origin containing the point is a rotation of from the ray through the
origin containing the real number .
iv. െ
࣊
We have ȁെȁ ൌ , and ܚ܉ሺെሻ ൌ . The point ሺǡ െሻ is located on the imaginary axis. The ray
࣊
from the origin containing the point is a rotation of from the ray through the origin containing the
real number .
v.
We have ȁȁ ൌ , and ܚ܉ሺሻ ൌ . This is the real number .
Exploratory Challenge/Exercises 1–12 (20 minutes)
Students investigate and ultimately generalize a formula for quickly calculating the value of ݖ . The class should work on
these problems in teams of three to four students each. Use the discussion questions to help move individual groups
forward as they work through the exercises in this exploration. Each group should have graph paper for each group
member and access to a calculator to check calculations if needed.
In Exercise 3, most groups probably expand the number and perform the calculation in rectangular form. Here polar
form offers little advantage. Perhaps when the exponent is a Ͷ, a case could be made that polar form is more efficient
for calculating a power of a complex number.
Be sure to pause and debrief with the entire class after Exercise 5. All students need to have observed the patterns in
the table in order to continue to make progress discovering the relationships about powers of a complex number.
Exploratory Challenge/Exercises 1–12
1. Rewrite each expression as a complex number in rectangular form.
a. ሺ ሻ
ሺ ሻሺ ሻ ൌ ൌ െ ൌ
b. ሺ ሻ
ሺ ሻ ൌ ሺ ሻ ሺ ሻ ൌ ሺ ሻ ൌ ൌ െ
c. ሺ ሻ
ሺ ሻ ൌ ሺ ሻ ሺ ሻ ൌ ή ൌ ൌ െ
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2. Complete the table below showing the rectangular form of each number and its modulus and argument.
Power of ሺ ሻ Rectangular Form Modulus Argument
ሺ ሻ
࣊
ሺ ሻ ξ
࣊
ሺ ሻ
࣊
ሺ ሻ െ ξ
ሺ ሻ െ ࣊
3. What patterns do you notice each time you multiply by another factor of ሺ ሻ?
࣊
The argument increases by . The modulus is multiplied by ξ.
Before proceeding to the rest of the exercises in this Exploratory Challenge, check to make sure each group observed the
patterns in the table required for them to make the connection that repeatedly multiplying by the same complex
number causes repeated rotation by the argument, dilation, and modulus of the number.
Debrief the first five exercises by having one or two groups present their findings on the board or document camera.
4. Graph each power of shown in the table on the same coordinate grid. Describe the location of these numbers
in relation to one another using transformations.
Starting with ሺ ሻ , each subsequent complex number is a ι rotation and a dilation by a factor of ξ of the
previous one. The graph shows the graphs of ࢠ ൌ ሺ ሻ for ൌ ǡ ǡ ǡ ǡ ǡ .
Predict what the modulus and argument of ሺ ሻ would be without actually performing the multiplication.
5.
Explain how you made your prediction.
࣊ ࣊
The modulus would be ξ, and the argument would be ࣊ ൌ .
Graph ሺ ሻ in the complex plane using the transformations you described in Exercise 5.
6.
See the solution to Exercises 4 and 5.
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7. Write each number in polar form using the modulus and argument you calculated in Exercise 4.
ሺ ሻ ൌ ሺܛܗ܋ሺሻ ԝܖܑܛሺሻሻ
࣊ ࣊
ሺ ሻ ൌ ξ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ ࣊
ሺ ሻ ൌ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ ࣊
ሺ ሻ ൌ ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
ሺ ሻ ൌ ሺܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻሻ
Use the patterns you have observed to write ሺ ሻ in polar form, and then convert it to rectangular form.
8.
࣊ ࣊
ሺ ሻ ൌ ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
What is the polar form of ሺ ሻ ? What is the modulus of ሺ ሻ ? What is its argument? Explain why
9.
ሺ ሻ
is a real number.
࣊ ࣊
In polar form, the number would be ൫ξ൯ ቀ ܛܗ܋ቀ ή ቁ ԝ ܖܑܛቀ ή ቁቁ. The modulus is ൫ξ൯ ൌ ൌ .
࣊
The argument is the rotation between and ࣊ that corresponds to a rotation of ή ൌ ࣊. The argument is ࣊.
This rotation takes the number to the negative real-axis and dilates it by a factor of ǡ , resulting in the
number െǡ , which is a real number.
Pause here to discuss the advantages of considering the geometric effect of multiplication by a complex number when
raising a complex number to a large integer exponent. Lead a discussion so students understand that the polar form of a
complex number makes this type of multiplication very efficient.
How do you represent multiplication by a complex number when written in polar form?
à The product of two complex numbers has a modulus that is the product of the two factors’ moduli and
an argument that is the sum of the two factors’ arguments.
How does understanding the geometric effect of multiplication by a complex number make solving Exercises
10 and 11 easier than repeatedly multiplying by the rectangular form of the number?
à If you know the modulus and argument of the complex number, and you want to calculate ݖ , then the
argument will be ݊ times the argument, and the modulus will be the modulus raised to the ݊.
In these exercises, you worked with powers of ͳ ݅. Do you think the patterns you observed can be
generalized to any complex number raised to a positive integer exponent? Explain your reasoning.
à Since the patterns we observed are based on repeatedly multiplying by the same complex number, and
since the geometric effect of multiplication always involves a rotation and dilation, this process should
apply to all complex numbers.
How can you quickly raise any complex number of a large integer exponent?
à Determine the modulus and argument of the complex number. Then, multiply the argument by the
exponent, and raise the modulus to the exponent. Then, you can write the number easily in polar and
then rectangular form.
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This exploration largely relies on students using inductive reasoning to observe patterns in powers of complex numbers.
The formula they write in Exercise 11 is known as DeMoivre’s formula (or DeMoivre’s theorem). More information and a
proof by mathematical induction that this relationship holds can be found at
http://en.wikipedia.org/wiki/De Moivre’s formula.
If students have been struggling with this exploration, lead a whole-class discussion on the next several exercises, or
groups can proceed to work through the rest of this Exploratory Challenge on their own. Be sure to monitor groups, and
keep referring them back to the patterns they observed in the tables and graphs as they make their generalizations.
Before students begin, announce that they are generalizing the patterns they observed in the previous exercises. Make
sure they understand that the goal is a formula or process for quickly raising a complex number to an integer exponent.
Observe groups, and encourage students to explain to one another how they are seeing the formula as they work
through these exercises.
10. If ࢠ has modulus ࢘ and argument ࣂ, what are the modulus and argument of ࢠ ? Write the number ࢠ in polar form.
The modulus would be ࢘ , and the argument would be a rotation between and ࣊ that is equivalent to ࣂ.
ࢠ ൌ ࢘ ሺܛܗ܋ሺࣂሻ ԝܖܑܛሺࣂሻሻ
11. If ࢠ has modulus ࢘ and argument ࣂ, what are the modulus and argument of ࢠ where is a nonnegative integer?
Write the number ࢠ in polar form. Explain how you got your answer.
The modulus would be ࢘ , and the argument would be a rotation between and ࣊ that is equivalent to ࣂ.
ࢠ ൌ ࢘ ሺܛܗ܋ሺࣂሻ ԝܖܑܛሺࣂሻ
12. Recall that ൌ ሺܛܗ܋ሺെࣂሻ ԝܖܑܛሺെࣂሻሻ. Explain why it would make sense that the formula holds for all integer
ࢠ ࢘
values of .
Since ൌ ࢠെ , it would make sense that the formula would hold for negative integers as well. If you plot , ,
ࢠ ࢠ ࢠ
etc., you can see the pattern holds.
In Exercise 14, students must consider why this formula holds for negative integers as well. Ask them how they could
verify graphically or algebraically that these formulae could be extended to include negative integer exponents.
Consider demonstrating this using graphing software such as Geogebra or Desmos.
Close this section by recording the formula shown below on the board. Ask students to summarize to a partner how to
use this formula with the number ሺͳ ݅ሻଵ and to record it in their notes.
Given a complex number ݖwith modulus ݎand argument ߠ, the ݊th power of ݖis given by
ݖ ൌ ݎ ሺ ሺ݊ߠሻ ݅ԝሺ݊ߠሻሻ where ݊ is an integer.
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Exercises 13–14 (5 minutes)
Students should work these exercises in their small groups or with a partner. After a few minutes, review the solutions,
and discuss any problems students had with their calculations.
Exercises 13–14
െ ૠ
13. Compute ቀ ቁ , and write it as a complex number in the form ࢇ ࢈ where ࢇ and ࢈ are real numbers.
ඥ
ି ૠ࣊
The modulus of is , and the argument is . The polar form of the number is
ξ
ૠ࣊ ૠ࣊
ૠ ൬ ܛܗ܋൬ૠ ή ൰ ԝ ܖܑܛ൬ૠ ή ൰൰Ǥ
ξ ξ
Converting this number to rectangular form by evaluating the sine and cosine values produces .
14. Write ൫ ξ൯ , and write it as a complex number in the form ࢇ ࢈ where ࢇ and ࢈ are real numbers.
࣊
The modulus of ξ is , and the argument is . The polar form of the number is
࣊ ࣊
ቀ ܛܗ܋ቀ ή ቁ ԝ ܖܑܛቀ ή ቁቁ .
Converting this number to rectangular form by evaluating the sine and cosine values produces
ሺ ή ሻ ൌ .
Closing (5 minutes)
Revisit one of the questions from the beginning of the lesson. Students can write their responses or discuss them with a
partner.
Describe an additional advantage to polar form that we discovered during this lesson.
à When raising a complex number to an integer exponent, the polar form gives a quick way to express
the repeated transformations of the number and quickly determine its location in the complex plane.
This then leads to quick conversion to rectangular form.
Review the relationship that students discovered in this lesson.
Lesson Summary
Given a complex number ࢠ with modulus ࢘ and argument ࣂ, the th power of ࢠ is given by
ࢠ ൌ ࢘ ሺܛܗ܋ሺࣂሻ ԝܖܑܛሺࣂሻሻ where is an integer.
Exit Ticket (5 minutes)
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Name Date
Lesson 18: Exploiting the Connection to Trigonometry
Exit Ticket
1. Write ሺʹ ʹ݅ሻ଼ as a complex number in the form ܽ ܾ݅ where ܽ and ܾ are real numbers.
2. Explain why a complex number of the form ሺܽ ܽ݅ሻ will either be a pure imaginary or a real number when ݊ is an
even number.
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Exit Ticket Sample Solutions
1. Write ሺ ሻૡ as a complex number in the form ࢇ ࢈ where ࢇ and ࢈ are real numbers.
࣊
We have ȁ ȁ ൌ ξ and ܚ܉ሺ ሻ ൌ .
ૡ ࣊ ࣊
Thus, ሺ ሻૡ ൌ ൫ξ൯ ቀ ܛܗ܋ቀૡ ή ቁ ԝ ܖܑܛቀૡ ή ቁቁ ൌ ሺܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻሻ ൌ ሺ ሻ ൌ .
2. Explain why a complex number of the form ሺࢇ ࢇሻ where ࢇ is a positive real number will either be a pure
imaginary or a real number when is an even number.
࣊ ࣊
Since the argument will always be , any even number multiplied by this number will be a multiple of . This will
result in a rotation to one of the axes, which means the complex number will either be a real number or a pure
imaginary number.
Problem Set Sample Solutions
1. Write the complex number in ࢇ ࢈ form where ࢇ and ࢈ are real numbers.
࣊ ࣊
a. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ ࣊ ξ
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ቆ െ ቇ
ൌ െ ξ
b. ሺܛܗ܋ሺιሻ ԝܖܑܛሺιሻሻ
ξ
ሺܛܗ܋ሺιሻ ԝܖܑܛሺιሻሻ ൌ ቆെ െ ቇ
ξ
ൌെ െ
࣊ ࣊
c. ൫ξ൯ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ ࣊ ૠ࣊ ૠ࣊
൫ξ൯ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ቆ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰ቇ
ξ ξ
ൌ ቆ െ ቇ
ൌ ξ െ ξ
d. ܛܗ܋ሺૢ࣊ሻ ԝܖܑܛሺૢ࣊ሻ
ܛܗ܋ሺૢ࣊ሻ ԝܖܑܛሺૢ࣊ሻ ൌ ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ
ൌ െ
ൌ െ
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࣊ ࣊
e. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ ࣊ ξ ξ
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ቆെ ቇ
ൌ െξ ξ
f. ሺܛܗ܋ሺૡιሻ ԝܖܑܛሺૡιሻሻ
ሺܛܗ܋ሺૡιሻ ԝܖܑܛሺૡιሻሻ ൌ ൫ܛܗ܋ሺιሻ ԝܖܑܛሺιሻ൯
ξ
ൌ ቆെ ቇ
ൌ െ ξ
2. Use the formula discovered in this lesson to compute each power of ࢠ. Verify that the formula works by expanding
and multiplying the rectangular form and rewriting it in the form ࢇ ࢈ where ࢇ and ࢈ are real numbers.
a. ൫ ξ൯
࣊
Since ࢠ ൌ ξ, we have ȁࢠȁ ൌ ξ ൌ , and ࣂ ൌ . Then,
࣊ ࣊
൫ ξ൯ ൌ ቆ ܛܗ܋ቀ ڄቁ ԝ ܖܑܛቀ ڄቁቇ ൌ ૡ൫ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ൯ ൌ െૡ
ሺ ξሻ ൌ ൫ ξ൯൫ ξ െ ൯ ൌ ൫ ξ൯൫െ ξ൯ ൌ െ ξ െ ξ െ ൌ െૡ
b. ሺെ ሻ
࣊
Since ࢠ ൌ െ , we have ȁࢠȁ ൌ ξ ൌ ξ, and ࣂ ൌ . Then,
࣊ ࣊
ሺെ ሻ ൌ ൫ξ൯ ቆ ܛܗ܋൬ ڄ ൰ ԝ ܖܑܛ൬ ڄ ൰ቇ ൌ ൫ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ൯ ൌ െ
ሺെ ሻ ൌ ሺെ ሻ ሺെ ሻ ൌ ሺ െ െ ሻሺ െ െ ሻ ൌ ሺെሻሺെሻ ൌ െ
c. ሺ ሻ
࣊
Since ࢠ ൌ , we have ȁࢠȁ ൌ ξ ൌ ξ, and ࣂ ൌ . Then,
࣊ ࣊ ξ ξ
ሺ ሻ ൌ ൫ξ൯ ቆ ܛܗ܋ቀ ڄቁ ԝ ܖܑܛቀ ڄቁቇ ൌ ૡξ ቆെ െ ቇ ൌ െૡ െ ૡ
ሺ ሻ ൌ ሺ ሻ ሺ ሻ ሺ ሻ ൌ ሺ ૡ െ ሻሺ ૡ െ ሻሺ ሻ ൌ ሺૡሻሺૡሻሺ ሻ
ൌ െሺ ሻ ൌ െૡ െ ૡ
d. ሺ െ ሻି
ૠ࣊
Since ࢠ ൌ െ , we have ȁࢠȁ ൌ ξ ൌ ξ, and ࣂ ൌ . Then,
ି ૠ࣊ ૠ࣊
ሺ െ ሻି ൌ ൫ξ൯ ቆ ܛܗ܋൬െ ڄ ൰ ԝ ܖܑܛ൬െ ڄ ൰ቇ ൌ ሺ ሻ ൌ
ૡ ૡ
ሺ െ ሻି ൌ ൌ ൌ ή ൌ ൌ
ሺ െ ሻ െ ૡ െ െૡ ૡ ૡ
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e. ൫ξ െ ൯
࣊
Since ࢠ ൌ ξ െ , we have ȁࢠȁ ൌ ටξ ൌ , and ࣂ ൌ . Then,
࣊ ࣊ ࣊ ࣊ ξ
൫ξ െ ൯ ൌ ቆ ܛܗ܋൬ ڄ ൰ ԝ ܖܑܛ൬ ڄ ൰ቇ ൌ ቆ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰ቇ ൌ ቆെ െ ቇ
ൌ െૡ െ ૡξ
൫ξ െ ൯ ൌ ൫ξ െ ൯ ൫ξ െ ൯ ൌ ൫ െ ξ െ ൯൫ െ ξ െ ൯ ൌ ൫ െ ξ൯൫ െ ξ൯
ൌ െ ૡξ െ ൌ െૡ െ ૡξ
f. ൫ξ െ ൯
࣊
Since ࢠ ൌ ξ െ , we have ȁࢠȁ ൌ ටሺξሻ ൌ ξ ൌ , and ࣂ ൌ . Then,
࣊ ࣊
൫ξ െ ൯ ൌ ቆ ܛܗ܋൬ ڄ ൰ ԝ ܖܑܛ൬ ڄ ൰ቇ ൌ ൫ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ൯ ൌ െ
൫ξ െ ൯ ൌ ൫ξ െ ൯ ൫ξ െ ൯ ൫ξ െ ൯
ൌ ൫ૠ െ ૡξ െ ૢ൯൫ૠ െ ૡξ െ ૢ൯൫ૠ െ ૡξ െ ૢ൯
ൌ ൫ૡ െ ૡξ൯൫ૡ െ ૡξ൯൫ૡ െ ૡξ൯
ൌ ൫ െ ૡξ െ ૢૠ൯൫ૡ െ ૡξ൯ ൌ ൫െૡ െ ૡξ൯൫ૡ െ ૡξ൯
ൌ െ ξ െ ξ െ ૢૢ ൌ െ
3. Given ࢠ ൌ െ െ , graph the first five powers of ࢠ by applying your knowledge of the geometric effect of
multiplication by a complex number. Explain how you determined the location of each in the coordinate plane.
࣊
Multiplication by Ȃ െ will dilate by หȂ െ ห ൌ ξ ൌ ξ and rotate by ܚ܉ሺെ െ ሻ ൌ . Then, the graph
below shows ࢠ ൌ െ െ , ࢠ ൌ ሺെ െ ሻ , ࢠ ൌ ሺെ െ ሻ , ࢠ ൌ ሺെ െ ሻ , and ࢠ ൌ ሺെ െ ሻ Ǥ
To locate each point, multiply the distance from the previous point to the origin by the modulus (ξ), and rotate
࣊
counterclockwise .
4. Use your work from Problem 3 to determine three values of for which ሺെ െ ሻ is a multiple of െ െ .
࣊
Since multiplication by െ െ rotates the point by radians, the point ሺെ െ ሻ is a multiple of the original
ࢠ ൌ െ െ every ૡ iterations. Thus, ሺെ െ ሻૢ , ሺെ െ ሻૠ ǡ ሺെ െ ሻ are all multiples of ሺ െ ሻ. Answers will
vary.
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5. Find the indicated power of the complex number, and write your answer in form ࢇ ࢈ where ࢇ and ࢈ are real
numbers.
࣊ ࣊
a. ቂ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁቃ
࣊ ࣊ ࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰൨ ൌ ቆ ܛܗ܋൬ ڄ൰ ԝ ܖܑܛ൬ ڄ ൰ቇ
ૢ࣊ ૢ࣊
ൌ ૡ ቆ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰ቇ
ξ ξ
ൌ ૡቆ ቇ
ൌ ξ ξ
࣊ ࣊
b. ቂξ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁቃ
࣊ ࣊ ࣊ ࣊
ቂξ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁቃ ൌ ൫ξ൯ ቆ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰ቇ
ൌ ሺ ሻ
ൌ
࣊ ࣊
c. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊ ࣊ ࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰
ൌ ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ
ൌ െ
࣊ ࣊
d. ቂ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁቃ
࣊ ࣊ ࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰൨ ൌ ൬ ൰ ቆ ܛܗ܋൬ ڄ൰ ԝ ܖܑܛ൬ ڄ ൰ቇ
ൌ ൫ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ൯
ૡ
ൌ
ૡ
࣊ ࣊ ି
e. ቂ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁቃ
࣊ ࣊ ି ࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰൨ ൌ ି ቆ ܛܗ܋൬െ ڄ ൰ ԝ ܖܑܛ൬ ڄ൰ቇ
࣊ ࣊
ൌ ቆ ܛܗ܋൬െ ൰ ԝ ܖܑܛ൬െ ൰ቇ
ξ
ൌ ቆെ ቇ
ξ
ൌെ
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Lesson 19: Exploiting the Connection to Trigonometry
Student Outcomes
Students understand how a formula for the ݊th roots of a complex number is related to powers of a complex
number.
Students calculate the ݊th roots of a complex number.
Lesson Notes
This lesson builds on the work from Topic B by asking students to extend their thinking about the geometric effect of
multiplication of two complex numbers to the geometric effect of raising a complex number to an integer exponent. It
is part of a two-day lesson that gives students another opportunity to work with the polar form of a complex number,
to see its usefulness in certain situations, and to exploit that to quickly calculate powers of a complex number. In this
lesson, students continue to work with polar and rectangular form and graph complex numbers represented both
ways. They examine graphs of powers of complex numbers in a polar grid and then write the ݊th root as a fractional
exponent and reverse the process from Day 1 to calculate the ݊th roots of a complex number. Throughout the lesson,
students are constructing and justifying arguments, using precise language, and using the structure of expressions
and visual representations to make sense of the mathematics.
Classwork
Opening (4 minutes)
Introduce the notion of a polar grid. Representing
complex numbers in polar form on a polar grid
makes this lesson seem easier for students and
emphasizes the geometric effect of the roots of a
complex number.
Display a copy of the polar grid on the right, and
model how to plot a few complex numbers in
polar form to illustrate that the concentric circles
make it easy to measure the modulus and the rays
at equal intervals and make representing the
rotation of the complex number easy as well.
Plot a point with the given modulus and argument.
A: modulus ൌ ͳ, argument ൌ Ͳι
B: modulus ൌ ͵, argument ൌ ͻͲι
C: modulus ൌ ͷ, argument ൌ ͵Ͳι
D: modulus ൌ , argument ൌ ͳʹͲι
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Explain to students that each circle represents a distance from the origin (the modulus). Each line represents an angle
measure. To plot a point, find the angle of rotation, and then move out to the circle that represents the distance from
the origin given by the modulus.
Opening Exercise (7 minutes)
These exercises give students an opportunity to practice working with a polar grid and to
Scaffolding:
review their work from the previous day’s lesson. Students should work individually or
with a partner on these exercises. Monitor student progress to check for understanding, For struggling students,
encourage them to label
and provide additional support as needed.
the rays in the polar grid
with the degrees of
Opening Exercise
rotation.
A polar grid is shown below. The grid is formed by rays from the origin at equal rotation intervals Provide additional practice
and concentric circles centered at the origin. The complex number ࢠ ൌ ξ is graphed on this
plotting complex numbers
polar grid.
in polar form. Some
students may find working
with degrees easier than
working with radians.
a. Use the polar grid to identify the modulus and argument of ࢠ.
࣊
The argument is , and the modulus is .
b. Graph the next three powers of ࢠ on the polar grid. Explain how you got your answers.
Each power of ࢠ is another ι rotation and a dilation by a factor of from the previous number.
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c. Write the polar form of the number in the table below, and then rewrite it in rectangular form.
Power of ࢠ Polar Form Rectangular Form
࣊ ࣊
ξ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ ξ
࣊ ࣊
൫ξ ൯ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ ξ
࣊ ࣊
൫ξ ൯ ૡ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ ૡ
࣊ ࣊
൫ξ ൯ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ૡ െ ૡξ
Have early finishers check their work by calculating one or two powers of ݖby expanding and then multiplying the
rectangular form. Examples are shown below.
ଶ
൫ξ͵ ݅൯ ൌ ͵ ʹξ͵݅ ݅ ଶ ൌ ʹ ʹξ͵݅
ଷ
൫ξ͵ ݅൯ ൌ ൫ξ͵ ݅൯൫ʹ ʹξ͵݅൯ ൌ ʹξ͵ ݅ ʹ݅ ʹξ͵݅ ଶ ൌ ͺ݅
Debrief by having one or two students explain their process to the class. Remind them again of the efficiency of working
with complex numbers in polar form and the patterns that emerge when they graph powers of a complex number.
Which way of expanding a power of a complex number would be quicker if you were going to expand
ଵ
൫ξ͵ ݅൯ ?
ͷߨ ͷߨ
Using the polar form would be far easier. It would be ʹଵ ቀ ቀ ቁ ݅ԝ ቀ ቁቁow could you describe the
͵ ͵
pattern of the numbers if we continued graphing the powers of ?ݖ
à The numbers are spiraling outward as each number is on a ray rotated ͵Ͳι from the previous one and
farther from the origin by a factor of ʹ.
Next, transition to the main focus of this lesson by giving students time to consider the next question. Have them
respond in writing and discuss their answers with a partner.
How do you think we could reverse this process, in other words, undo squaring a complex number or undo
cubing a complex number?
à That would be like taking a square root or cube root. We would have to consider how to undo the
rotation and dilation effects.
Exercises 1–3 (7 minutes)
In these exercises, students explore one of the square roots of a complex number. Later in the lesson, show students
that complex numbers have multiple roots just like a real number has two square roots (e.g., the square roots of Ͷ are ʹ
and െʹ). Students should work these exercises with a partner. If the class is struggling to make sense of Exercise 3,
work that one as a whole class.
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Exercises 1–3
The complex numbers ࢠ ൌ ൫െ ξ൯ and ࢠ are graphed below.
1. Use the graph to help you write the numbers in polar and rectangular form.
࣊ ૈ
ࢠ ൌ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ ξ
࣊ ࣊
ࢠ ൌ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ െ ξ
2. Describe how the modulus and argument of ࢠ ൌ െ ξ are related to the modulus and argument of
ࢠ ൌ ൫െ ξ൯ .
The modulus and argument are both cut in half.
3. Why could we call െ ξ a square root of െ െ ξ?
Clearly, ൫െ ξ൯ ൌ െ െ ξ. We can demonstrate this using the rectangular or polar form and verify it using
transformations of the numbers when they are plotted in the complex plane. So, it would make sense then that
raising both sides of this equation to the power should give us the desired result.
Start with the equation, ൫െ ξ൯ ൌ െ െ ξ.
ቀ൫െ ξ൯ ቁ ൌ ൫െ െ ξ൯
െ ξ ൌ ටെ െ ξ
Alternately, using the formula from Lesson 17, replace with .
ࢠ ൌ ࢘ ሺܛܗ܋ሺࣂሻ ԝܖܑܛሺࣂሻሻ
ࢠ ൌ ࢘ ൬ ܛܗ܋൬ ࣂ൰ ԝ ܖܑܛ൬ ࣂ൰൰
So, a square root of െ െ ξ would be
࣊ ࣊ ࣊ ࣊
൬ ܛܗ܋൬ ή ൰ ԝ ܖܑܛ൬ ή ൰൰ ൌ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ ξǤ
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After giving students a few minutes to work these exercises with a partner, make sure they understand that the modulus
is cut in half because ʹ ή ʹ ൌ Ͷ shows a repeated multiplication by ʹ.
How would this problem change if the modulus was ͻ
instead of Ͷ? How would this problem change if the
modulus was ͵ instead of Ͷ?
à The new modulus would have to be a number that
when squared equals ͻǡ so we would need the new
modulus to be the square root of the original
modulus.
If students seem to think that the modulus would always be divided
by ʹ instead of it being the square root of the original modulus,
model this using Geogebra. A sample screenshot is provided below
showing a complex number with the same argument and a modulus
of ͻ. Notice that the modulus of ݖଵ is ͵ while the argument is still
cut in half.
Discussion (7 minutes): The th Roots of a Complex Number
In this Discussion, model how to derive a formula to find all the ݊th roots of a complex number. Begin by reminding
students of the definition of square roots learned in Grade 8 and Algebra I.
Recall that each real number has two square roots. For example, what are the two square roots of Ͷ? The two
square roots of ͳͲ? How do you know?
à They are ʹ and െʹ because ሺʹሻଶ ൌ Ͷ, and ሺെʹሻଶ ൌ Ͷ. The two square roots of ͳͲ are ξͳͲ and െξͳͲ.
How many square roots do you think a complex number has? How many cube roots? Fourth roots, etc.?
à Since all real numbers are complex numbers, complex numbers should have two square roots as well.
Since roots are solutions to an equation ݔ ൌ ݎ, it would make sense that if our solution set is the
complex numbers, then there would be three cube roots when ݊ ൌ ͵ and four fourth roots when ݊ ൌ Ͷ.
Thus, complex numbers have multiple ݊th roots when ݊ is a positive integer. In fact, every complex number has ʹ square
roots, ͵ cube roots, Ͷ fourth roots, and so on. This work relates back to Modules 1 and 3 in Algebra II where students
learned that a degree ݊ polynomial equation has ݊ complex zeros and to previous work extending the properties of
exponents to the real number exponents. Students should take notes as the work shown below is presented.
Using the formula from Lesson 17, suppose we have an ݊th root of ݖ, ݓൌ ݏሺ ሺߙሻ ݅ԝሺߙሻሻ. Then, for ݎ Ͳ, ݏ Ͳǡ
we have
ݓ ൌ ݖ
ݏ ሺ ሺ݊ߙሻ ݅ԝሺ݊ߙሻሻ ൌ ݎሺ ሺߠሻ ݅ԝሺߠሻሻ
Equating the moduli, ݏ ൌ ݎ, which implies that ݏൌ ξݎ.
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Equating the arguments, ݊ߙ ൌ ߠ. However, since the sine and cosine functions are periodic functions with period ʹߨ,
this equation does not have a unique solution for ߙ. We know that ሺߠ ʹߨ݇ሻ ൌ ሺߠሻ and ሺߠ ʹߨ݇ሻ ൌ ሺߠሻ
for integer values of ݇ and real numbers ߠ.
Therefore,
݊ߙ ൌ ߠ ʹߨ݇
ߠ ʹߨ݇
Or, ߙ ൌ for values of ݇ up to ݊ െ ͳ. When ݇ ൌ ݊ or greater, we start repeating values for ߙ.
݊ ݊
Going back to your work in Example 1 and Exercise 6, we can find both roots of െʹ െ ʹξ͵݅ and all three cube roots of
this number.
Example 1 (5 minutes): Find the Two Square Roots of a Complex Number
Example: Find the Two Square Roots of a Complex Number
Find both of the square roots of െ െ ξ.
࣊ ࣊
ቁ ԝ ܖܑܛቀ ቁቁ. The square roots of this number will have modulus ξ ൌ
The polar form of this number is ቀ ܛܗ܋ቀ
ࣂ ࣊ ࣊
and arguments given by ࢻ ൌ for ൌ ǡ where ࣂ ൌ . Thus,
࣊ ࣊
ࢻൌ ൬ ࣊ ή ൰ ൌ
and
࣊ ࣊
ࢻൌ ൬ ࣊ ή ൰ ൌ
The square roots are
࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ ξ
and
࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ ξ.
Have students go back and add the graph of the second square root to the graph at the beginning of Exercise 1.
Exercises 4–6 (7 minutes)
Students work with the formula developed in the Discussion and presented in the Lesson Summary. Students can work
individually or with a partner. If time is running short, assign these as Problem Set exercises as well.
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Exercises 4–6
4. Find the cube roots of െ ൌ ξ.
࣊ ࣊
െ െ ξ ൌ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
ࣂ ࣊
The modulus of the cube roots will be ξ. The arguments for ൌ , , and are given by ࢻ ൌ where
࣊ ࣊ ࣊ ࣊
ࣂൌ and ൌ . Using this formula, the arguments are , , and . The three cube roots of െ െ ξ
ૢ ૢ ૢ
are
࣊ ࣊
ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
ૢ ૢ
࣊ ࣊
ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
ૢ ૢ
࣊ ࣊
ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
ૢ ૢ
5. Find the square roots of .
࣊ ࣊
ൌ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
ࣂ ࣊ ࣊
The modulus of the square roots is ξ ൌ . The arguments for ൌ and are given by ࢻ ൌ where ࣂ ൌ
࣊ ࣊
and ൌ . Using this formula, the arguments are and . The two square roots of are
࣊ ࣊
ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ ൌ ξ ξ
and
࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െξ ξ.
6. Find the cube roots of ૡ.
In polar form,
ૡ ൌ ૡሺܛܗ܋ሺሻ ԝܖܑܛሺሻሻ.
ࣂ ࣊
The modulus of the cube roots is ξૡ ൌ . The arguments for ൌ , , and are given by ࢻ ൌ where
࣊ ࣊
ࣂ ൌ and ൌ . Using this formula, the arguments are , , and . The three cube roots of ૡ are
ሺܛܗ܋ሺሻ ԝܖܑܛሺሻሻ ൌ
࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ ξ
࣊ ࣊
൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ െ െ ξ
Consider pointing out to students that the answers to Exercise 9 are the solutions to the equation ݔଷ െ ͺ ൌ Ͳ.
ݔଷ െ ͺ ൌ Ͳ
ሺ ݔെ ʹሻሺ ݔଶ ʹ ݔ Ͷሻ ൌ Ͳ
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One solution is െʹ, and the other two are solutions to ݔଶ ʹ ݔ Ͷ ൌ Ͳ. Using the quadratic formula, the other two
solutions are given by
ʹ
െʹേටʹ െͶ൫ͳ൯൫Ͷ൯
ݔൌ .
ʹ
This expression gives the solutions െͳ ξ͵݅ and െͳ െ ξ͵݅. This connection with the work in Algebra II, Module 2 is
revisited in the last few exercises in the Problem Set.
Closing (3 minutes)
Ask students to respond to this question either in writing or with a partner. They can use one of the exercises above to
explain the process. Then, review the formula that was derived during the Discussion portion of this lesson.
How do you find the ݊th roots of a complex number?
à Determine the argument and the modulus of the original number. Then, the modulus of the roots is the
ߠ ʹߨ݇
݊th root of the original modulus. The arguments are found using the formula ߙ ൌ for ݇ is the
݊ ݊
integers from Ͳ to ݊ െ ͳ. Write the roots in polar form. If you are finding the cube roots, there will be
three of them; if you are finding fourth roots, there will be four, and so on.
Review the formula students can use to find the ݊th roots of a complex number.
Lesson Summary
Given a complex number ࢠ with modulus ࢘ and argument ࣂ, the th roots of ࢠ are given by
ࣂ ࣊ ࣂ ࣊
ξ࢘ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰
for integers and such that and ൏ .
Exit Ticket (5 minutes)
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Name Date
Lesson 19: Exploiting the Connection to Trigonometry
Exit Ticket
Find the fourth roots of െʹ െ ʹξ͵݅.
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Exit Ticket Sample Solutions
Find the fourth roots of െ െ ξ.
࣊
The modulus is , and the argument is . Using the formula, the modulus of the fourth roots will be ξ, and the
࣊
arguments will be
ቀ ቁ ሺ࣊ሻ for ൌ ǡ ǡ ǡ . This gives the following complex numbers as the fourth roots of
െ െ ξǤ
࣊ ࣊ ξ
ξ ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ ൌ ξ ቆ ቇ
࣊ ࣊ ξ
ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ξ ቆെ ቇ
࣊ ࣊ ξ
ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ξ ቆെ െ ቇ
࣊ ࣊ ξ
ξ ൬ ܛܗ܋൬ ൰ ԝ ܖܑܛ൬ ൰൰ ൌ ξ ቆ െ ቇ
Problem Set Sample Solutions
1. For each complex number, what is ࢠ ?
a. ξ
െ ξ
b. െ
െૡ
c.
െ
ඥ
d. െ
ξ
െ
e.
ૢ ૢ
ૡ
f. െ
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2. For each complex number, what are the square roots of ࢠ?
a. ξ
࣊
ࢠ ൌ ξǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊ ૠ࣊
ࢻൌ ቀ ࣊ቁ ǡ ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
࣊ ࣊ ξ
࢝ ൌ ξ ቀܛܗ܋ ή ܖܑܛቁ ൌ ξ ቆ ቇ
ૠ࣊ ૠ࣊ െξ
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ξ ቆ െ ቇ
b. െ
ૠ࣊
ࢠ ൌ െ ǡ ࢘ ൌ ξૡǡ ܚ܉ሺࢠሻ ൌ ǡ
ૠ࣊ ૠ࣊ ࣊
ࢻൌ ቀ ࣊ቁ ǡ ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
ૡ ૡ
ૠ࣊ ૠ࣊
࢝ ൌ ξૡ ൬ܛܗ܋ ή ܖܑܛ൰
ૡ ૡ
࣊ ࣊
࢝ ൌ ξૡ ൬ܛܗ܋ ή ܖܑܛ ൰
ૡ ૡ
c.
࣊
ࢠ ൌ ǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊ ࣊
ࢻ ൌ ቀ ࣊ቁ ǡ ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
࣊ ࣊ ξ ξ
࢝ ൌ ξ ቀܛܗ܋ ή ܖܑܛቁ ൌ ቆ ቇ ൌ ξ ξ
࣊ ࣊ െξ െξ
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ቆ ቇ ൌ െξ െ ξ
ඥ
d. െ
െξ ࣊
ࢠൌ ǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊ ૠ࣊
ࢻൌ ቀ ࣊ቁ ǡ ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
࣊ ࣊ ࣊ ࣊
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ܛܗ܋ ή ܖܑܛ
࣊ ࣊ ૠ࣊ ૠ࣊
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ܛܗ܋ ή ܖܑܛ
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e.
ૢ ૢ
ξ ࣊
ࢠൌ ǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ǡ
ૢ ૢ ૢ
࣊ ࣊ ૢ࣊
ࢻ ൌ ቀ ࣊ቁ ǡ ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
ૡ ૡ
ξ ࣊ ૢ࣊ ξ ࣊ ૢ࣊
࢝ ൌ ඨ ൬ ܛܗ܋ ή ܖܑܛ൰ ൌ ൬ ܛܗ܋ ή ܖܑܛ൰
ૢ ૡ ૡ ૡ ૡ
ξ ࣊ ࣊ ξ ࣊ ࣊
࢝ ൌ ඨ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ൬ܛܗ܋ ή ܖܑܛ൰
ૢ
f. െ
ࢠ ൌ െ ,࢘ ൌ ,ܚ܉ሺࢠሻ ൌ ࣊
࣊ ࣊
ࢻ ൌ ሺ࣊ ࣊ሻ, ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
࣊ ࣊
࢝ ൌ ቀܛܗ܋ ή ܖܑܛቁ ൌ
࣊ ࣊
࢝ ൌ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ െ
3. Graph each complex number on a polar grid.
࣊ ࣊
a. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ ǡ ࣂ ൌ
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b. ሺܛܗ܋ሺιሻ ԝܖܑܛሺιሻሻ
࢘ ൌ ǡ ࣂ ൌ ι
࣊ ࣊
c. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ ǡ ࣂ ൌ
d. ܛܗ܋ሺ࣊ሻ ԝܖܑܛሺ࣊ሻ
࢘ ൌ ǡ ࣂ ൌ ࣊
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࣊ ࣊
e. ቀ ܛܗ܋ቀ ቁ ԝ ܖܑܛቀ ቁቁ
࣊
࢘ ൌ ǡ ࣂ ൌ
f. ሺܛܗ܋ሺιሻ ԝܖܑܛሺιሻሻ
࢘ൌ ǡ ࣂ ൌ ι
4. What are the cube roots of Ȃ ?
࣊
ࢠ ൌ െ ǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊ ૠ࣊ ࣊
ࢻൌ ቀ ࣊ቁ ǡ ൌ ǡ ǡor ࢻ ൌ ǡ or Let the cube roots of ࢠ be ࢝ , ࢝ , and ࢝ .
࣊ ࣊
࢝ ൌ ξ ቀܛܗ܋ ή ܖܑܛቁ ൌ ξሺ ሻ ൌ ξ ή
ૠ࣊ ૠ࣊ െξ
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ξ ቆ െ ቇ
࣊ ࣊ ξ
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ ൰ ൌ ξ ቆ െ ቇ
5. What are the fourth roots of ?
ࢠ ൌ ǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊
ࢻൌ ሺ ࣊ሻǡ ൌ ǡ ǡ ǡ or ࢻ ൌ ǡ ǡ ࣊ǡ or Let the fourth roots of ࢠ be ࢝ , ࢝ , ࢝ , and ࢝ .
࢝ ൌ ξሺܛܗ܋ ή ܖܑܛሻ ൌ ξሺ ሻ ൌ ξ
࣊ ࣊
࢝ ൌ ξ ቀ ܛܗ܋ǡ ή ܖܑܛቁ ൌ ξሺ ሻ ൌ ξ ή
࢝ ൌ ξሺ ࣊ܛܗ܋ ή ࣊ܖܑܛሻ ൌ ξሺെ ሻ ൌ െξ
࣊ ࣊
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ξሺ െ ሻ ൌ െξ ή
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6. What are the square roots of െ െ ?
࣊
ࢠ ൌ െ െ ǡ ࢘ ൌ ξǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊ ࣊
ࢻൌ ቀ ࣊ቁ ǡ ൌ or ࢻ ൌ ǡ ࣊ǡ or Let the square roots of ࢠ be ࢝ and ࢝ .
ૡ ૡ
࣊ ࣊
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰
ૡ ૡ
࣊ ࣊
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ ൰
ૡ ૡ
7. Find the square roots of െ. Show that the square roots satisfy the equation ࢞ ൌ .
ࢠ ൌ െ ǡ ࢘ ൌ ǡ ܚ܉ሺࢠሻ ൌ ࣊ǡ
࣊ ࣊
ࢻ ൌ ሺ࣊ ࣊ሻǡ ൌ or ࢻ ൌ or Let the square roots of ࢠ be ࢝ and ࢝ .
࣊ ࣊
࢝ ൌ ξ ቀܛܗ܋ ή ܖܑܛቁ ൌ ξሺ ሻ ൌ ξ ή
࣊ ࣊
࢝ ൌ ξ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ξሺ െ ሻ ൌ െξ ή
൫ξ ή ൯ ൌ െ ൌ
൫െξ ή ൯ ൌ െ ൌ
8. Find the cube roots of ૠ. Show that the cube roots satisfy the equation࢞ െ ૠ ൌ .
ࢠ ൌ ૠ ǡ ࢘ ൌ ૠǡ ܚ܉ሺࢠሻ ൌ ǡ
࣊ ࣊
ࢻ ൌ ሺ ࣊ሻǡ ൌ ǡ ǡ or ࢻ ൌ ǡ or Let the cube roots of ࢠ be ࢝ , ࢝ , and ࢝ .
࢝ ൌ ξૠሺܛܗ܋ ή ܖܑܛሻ ൌ ሺ ሻ ൌ
࣊ ࣊ ξ ξ
࢝ ൌ ξૠ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ቆെ ቇ ൌ െ
࣊ ࣊ ξ ξ
࢝ ൌ ξૠ ൬ܛܗ܋ ή ܖܑܛ൰ ൌ ቆെ െ ቇ ൌ െ െ
ሺሻ െ ૠ ൌ
ξ െ ξ ૢ ξ
ቌ ቆെ ቇቍ െ ૠ ൌ െૠ ቆ െ ቇ െ ૠ ൌ െૠሺሻ െ ૠ ൌ
ૡ ૡ ૡ ૡ
ξ ξ ૢ ξ
ቌ ቆെ െ ቇቍ െ ૠ ൌ െૠ ቆ െ െ ቇ െ ૠ ൌ െૠሺെሻ െ ૠ ൌ
ૡ ૡ ૡ ૡ
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Lesson 20: Exploiting the Connection to Cartesian
Coordinates
Student Outcomes
Students interpret complex multiplication as the corresponding function of two real variables.
Students calculate the amount of rotation and the scale factor of dilation in a transformation of the form
ܮሺݔǡ ݕሻ ൌ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ.
Lesson Notes
This lesson leads into the introduction of matrix notation in the next lesson. The primary purpose of this lesson is to
formalize the idea that when the complex number ݔ ݅ ݕis identified with the point ሺݔǡ ݕሻ in the coordinate plane,
multiplication by a complex number performs a rotation and dilation in the plane. All dilations throughout this lesson
and module are centered at the origin. When the formulas are written out for such rotation and dilation in terms of the
real components ݔand ݕof ݖൌ ݔ ݅ݕ, the formulas are rather cumbersome, leading to the need for a new notation
using matrices in the next lesson. This lesson serves to solidify many of the ideas introduced in Topic B and link them to
matrices.
Classwork
Opening Exercise (6 minutes) Scaffolding:
For struggling students, scaffold
Opening Exercise part (a) by first asking them to
a. Find a complex number ࢝ so that the transformation ࡸ ሺࢠሻ ൌ ࢝ࢠ produces write a complex number with
a clockwise rotation by ι about the origin with no dilation. modulus ͳ and argument ͳι, and
Because there is no dilation, we need ȁ࢝ȁ ൌ , and because there is then ask the question stated. This
rotation by ι, we need ܚ܉ሺ࢝ሻ ൌ ι. Thus, we need to find the point helps students see the connection.
where the terminal ray of a ι rotation intersects the unit circle. Do the same for part (b).
From Algebra II, we know the coordinates of the point are
Ask advanced students to find
ሺ࢞ǡ ࢟ሻ ൌ ൫ܛܗ܋ሺιሻǡ ܖܑܛሺιሻ൯ǡ
complex numbers ݓso that:
so that the complex number ࢝ is
a) the transformation
࢝ ൌ ࢞ ࢟ ൌ ܛܗ܋ሺιሻ ԝܖܑܛሺιሻ. ܮଵ ሺݖሻ ൌ ݖݓproduces a clockwise
(Students may use a calculator to find the approximation rotation by ߙι about the origin with
࢝ ൌ Ǥ ૢૢૢૢૡ Ǥ ૠ.) no dilation, and
b) the transformation
ܮଶ ሺݖሻ ൌ ݖݓproduces a dilation
with scale factor ݎwith no rotation.
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b. Find a complex number ࢝ so that the transformation ࡸ ሺࢠሻ ൌ ࢝ࢠ produces a dilation with scale factor Ǥ
with no rotation.
In this case, there is no rotation, so the argument of ࢝ must be . This means that the complex number ࢝
corresponds to a point on the positive real axis, so ࢝ has no imaginary part; this means that ࢝ is a real
number, and ࢝ ൌ ࢇ ࢈ ൌ ࢇ. Thus, ȁ࢝ȁ ൌ ࢇ ൌ Ǥ , so ࢝ ൌ Ǥ .
Discussion (8 minutes)
This teacher-led Discussion provides justification for why new notation needs to be developed.
We have seen that we can use complex multiplication to perform dilation and rotation in the coordinate plane.
By identifying the point ሺݔǡ ݕሻ with the complex number ሺ ݔ ݅ݕሻ, we can think of ܮሺݖሻ ൌ ݖݓas a
transformation in the coordinate plane. Then, complex multiplication gives us a way of finding formulas for
rotation and dilation in two-dimensional geometry.
Video game creators are very interested in the mathematics of rotation and dilation. In a first-person video
game, you are centered at the origin. When you move forward in the game, the images on the screen need to
undergo a translation to mimic what you see as you walk past them. As you walk closer to objects, they look
larger, requiring dilation. If you turn, then the images on the screen need to rotate.
We have established the necessary mathematics for representing rotation and dilation in two-dimensional
geometry, but in video games we need to use three-dimensional geometry to mimic our three-dimensional
world. Eventually, we’ll need to translate our work from two dimensions into three dimensions.
Complex numbers are inherently two-dimensional, with our association ݔ ݅ ݕ՞ ሺݔǡ ݕሻ. We will need some
way to represent points ሺݔǡ ݕǡ ݖሻ in three dimensions.
Before we can jump to three-dimensional geometry, we need to better understand the mathematics of two-
dimensional geometry.
1. First, we will rewrite all of our work about rotation and dilation of complex numbers ݔ ݅ ݕin terms of
points ሺݔǡ ݕሻ in the coordinate plane and see what rotation and dilation looks like from that
perspective.
2. Then, we will see if we can generalize the mathematics of rotation and dilation of two-dimensional
points ሺݔǡ ݕሻ to three-dimensional points ሺݔǡ ݕǡ ݖሻ.
We address point (1) in this lesson and the ones that follow and leave point (2) until the next module.
Using the notation of complex numbers, if ݓൌ ܽ ܾ݅, then ȁݓȁ ൌ ξܽଶ ܾ ଶ , and ሺݓሻ ൌ ሺܽ ܾ݅ሻ.
Then, how can we describe the geometric effect of multiplication by ݓon a complex number ?ݖ
à The geometric effect of multiplication ݖݓis dilation by ȁݓȁ and counterclockwise rotation by ሺݓሻ
about the origin.
Now, let’s rephrase this more explicitly as follows: Multiplying ݔ ݅ݕby ܽ ܾ݅ rotates ݔ ݅ݕabout the origin
through ሺܽ ܾ݅ሻ and dilates that point from the origin with scale factor ξܽଶ ܾ ଶ .
We can further refine our statement: The transformation ሺ ݔ ݅ݕሻ ՜ ሺܽ ܾ݅ሻሺ ݔ ݅ݕሻ corresponds to a
rotation of the plane about the origin through ሺܽ ܾ݅ሻ and dilation with scale factor ξܽଶ ܾ ଶ .
How does this transformation work on points ሺݔǡ ݕሻ in the plane? Rewrite it to get a transformation in terms
of coordinate points ሺݔǡ ݕሻ.
à We have ሺ ݔ ݅ݕሻ ՜ ሺܽ ܾ݅ሻሺ ݔ ݅ݕሻ and ሺܽ ܾ݅ሻሺ ݔ ݅ݕሻ ൌ ሺܽ ݔെ ܾݕሻ ሺܾ ݔ ܽݕሻ݅, so we can
rewrite this as the transformation ሺݔǡ ݕሻ ՜ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ.
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This is the formula we want for rotation and dilation of points ሺݔǡ ݕሻ in the coordinate plane.
For real numbers ܽ and ܾ, the transformation ܮሺݔǡ ݕሻ ൌ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ corresponds to a
counterclockwise rotation by ሺܽ ܾ݅ሻ about the origin and dilation with scale factor ξܽଶ ܾ ଶ .
We have just written a function in two variables. Let’s practice that. If ܮሺݔǡ ݕሻ ൌ ሺʹݔǡ ݔ ݕሻ, how can we find
ܮሺʹǡ ͵ሻ? Explain this in words.
à We would substitute ʹ in for ݔand ͵ for ݕ.
What is ܮሺʹǡ ͵ሻ?
à ܮሺʹǡ ͵ሻ ൌ ሺͶǡ ͷሻ
How can ܮሺʹǡ ͵ሻ be interpreted?
à When ܮሺʹǡ͵ሻ is multiplied by ܽ ܾ݅, it is transformed to the point ሺͶǡ ͷሻ.
Returning back to our formula, explain how the quantity ܽ ݔെ ܾ ݕwas derived and what it represents in the
formula ܮሺݔǡ ݕሻ.
à When multiplying ሺ ݔ ݅ݕሻ by ሺܽ ܾ݅ሻ, the real component is ܽ ݔെ ܾݕ. This represents the
transformation of the ݔcomponent.
Exercises 1–4 (12 minutes)
These exercises link the geometric interpretation of rotation and dilation to the analytic formulas. Have students work
on these exercises in pairs or small groups. Use these exercises to check for understanding. The exercises can be
modified and/or assigned as instructionally necessary.
Exercises
1.
a. Find values of ࢇ and ࢈ so that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ has the effect of dilation with scale factor
and no rotation.
We need ܚ܉ሺࢇ ࢈ሻ ൌ and ξࢇ ࢈ ൌ . Since ܚ܉ሺࢇ ࢈ሻ ൌ , the point corresponding to ࢇ ࢈ lies
along the positive ࢞-axis, so we know that ࢈ ൌ and ࢇ . Then, we have ξࢇ ࢈ ൌ ξࢇ ൌ ࢇ, so ࢇ ൌ .
Thus, the transformation ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ࢟ǡ ࢞ ࢟ሻ ൌ ሺ࢞ǡ ࢟ሻ has the geometric effect of dilation by
scale factor .
b. Evaluate ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, and identify the resulting transformation.
ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯ ൌ ࡸ ሺ࢞ǡ ࢟ሻ
ൌ ሺ࢞ǡ ࢟ሻ
If we take ࡸ ሺࡸ ሺ࢞ǡ ࢟ሻሻ, we are dilating the point ሺ࢞ǡ ࢟ሻ with scale factor twice. This means that we are
dilating with scale factor ڄ ൌ .
2.
a. Find values of ࢇ and ࢈ so that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ has the effect of rotation about the origin by
ૡι counterclockwise and no dilation.
Since there is no dilation, we have ξࢇ ࢈ ൌ , and ܚ܉ሺࢇ ࢈ሻ ൌ ૡι means that the point ሺࢇǡ ࢈ሻ lies on
the negative ࢞-axis. Then, ࢇ ൏ and ࢈ ൌ , so ξࢇ ࢈ ൌ ξࢇ ൌ ȁࢇȁ ൌ , so ࢇ ൌ െ. Then, the
transformation ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺെ࢞ െ ࢟ǡ ࢞ െ ࢟ሻ ൌ ሺെ࢞ǡ െ࢟ሻ has the geometric effect of rotation by ૡι
without dilation.
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b. Evaluate ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, and identify the resulting transformation.
ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯ ൌ ࡸ ሺെ࢞ǡ െ࢟ሻ
ൌ ൫െሺെ࢞ሻǡ െሺെ࢟ሻ൯
ൌ ሺ࢞ǡ ࢟ሻ
Thus, if we take ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, we are rotating the point ሺ࢞ǡ ࢟ሻ by ૡι twice, which results in a rotation of
ι and has the net effect of doing nothing to the point ሺ࢞ǡ ࢟ሻ. This is the identity transformation.
3.
a. Find values of ࢇ and ࢈ so that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ has the effect of rotation about the origin by
ૢι counterclockwise and no dilation.
Since there is no dilation, we have ξࢇ ࢈ ൌ , and since the rotation is ૢι counterclockwise, we know
that ࢇ ࢈ must lie on the positive imaginary axis. Thus, ࢇ ൌ , and we must have ࢈ ൌ . Then, the
transformation ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ࢟ǡ ࢞ ࢟ሻ ൌ ሺെ࢟ǡ ࢞ሻ has the geometric effect of rotation by ૢι
counterclockwise with no dilation.
b. Evaluate ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, and identify the resulting transformation.
ܮଷ ൫ܮଷ ሺݔǡ ݕሻ൯ ൌ ܮଷ ሺെݕǡ ݔሻ
ൌ ሺെݔǡ െݕሻ
ൌ ܮଶ ሺݔǡ ݕሻ
Thus, if we take ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, we are rotating the point ሺ࢞ǡ ࢟ሻ by ૢι twice, which results in a rotation of
ૡι. This is the transformation ࡸ .
4.
a. Find values of ࢇ and ࢈ so that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ has the effect of rotation about the origin by
ι counterclockwise and no dilation.
Since there is no dilation, we have ξࢇ ࢈ ൌ , and since the rotation is ι counterclockwise, we know
that the point ሺࢇǡ ࢈ሻ lies on the line ࢟ ൌ ࢞, and thus, ࢇ ൌ ࢈. Then, ξࢇ ࢈ ൌ ξࢇ ࢇ ൌ , so ࢇ ൌ ǡ
ඥ ඥ ඥ ඥ ඥ ඥ
and thus ࢇ ൌ , so we also have ࢈ ൌ . Then, the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ൬ ࢞െ ࢟ǡ ࢞ ࢟൰ has
the geometric effect of rotation by ι counterclockwise with no dilation.
(Students may also find the values of ࢇ and ࢈ by ࢈ ൌ ܛܗ܋ሺιሻ ԝܖܑܛሺιሻ.)
b. Evaluate ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, and identify the resulting transformation.
We then have
ξ ξ ξ ξ
ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯ ൌ ࡸ ቆ ࢞ െ ࢟ǡ ࢞ ࢟ቇ
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
ൌቌ ቆ ࢞െ ࢟ቇ െ ቆ ࢞ ࢟ቇ ǡ ቆ ࢞െ ࢟ቇ ቆ ࢞ ࢟ቇቍ
ൌ ቆ൬ ࢞ െ ࢟൰ െ ൬ ࢞ ࢟൰ ǡ ൬ ࢞ െ ࢟൰ ൬ ࢞ ࢟൰ቇ
ൌ ሺെ࢟ǡ ࢞ሻ
ൌ ࡸ ሺ࢞ǡ ࢟ሻ
Thus, if we take ࡸ ൫ࡸ ሺ࢞ǡ ࢟ሻ൯, we are rotating the point ሺ࢞ǡ ࢟ሻ by ι twice, which results in a rotation of ૢι.
This is the transformation ࡸ .
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Exercises 5–6 (10 minutes)
These exercises encourage students to question whether a given analytic formula represents a rotation and/or dilation.
Have students work on these exercises in pairs or small groups.
5. The figure below shows a quadrilateral with vertices ሺǡ ሻ, ሺǡ ሻ, ሺǡ ሻ, and ࡰሺǡ ሻ.
a. Transform each vertex under ࡸ ൌ ሺ࢞ ࢟ǡ ࢟ െ ࢞ሻ, and plot the transformed vertices on the figure.
b. Does ࡸ represent a rotation and dilation? If so, estimate the amount of rotation and the scale factor from
your figure.
The transformed image is roughly three times larger than the original and rotated about ι clockwise.
c. If ࡸ represents a rotation and dilation, calculate the amount of rotation and the scale factor from the
formula for ࡸ . Do your numbers agree with your estimate in part (b)? If not, explain why there are no
values of ࢇ and ࢈ so that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ.
From the formula, we have ࢇ ൌ and ࢈ ൌ െ. The transformation dilates by the scale factor
ȁࢇ ࢈ȁ ൌ ඥ ሺെሻ ൌ ξ ൎ Ǥ and rotates by
࢈
ܚ܉ሺࢇ ࢈ሻ ൌ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ ܖ܉ܜ܋ܚ܉ቀെ ቁ ൎ െૡǤ ι.
ࢇ
6. The figure below shows a figure with vertices ሺǡ ሻ, ሺǡ ሻ, ሺǡ ሻ, and ࡰሺǡ ሻ.
a. Transform each vertex under ࡸ ൌ ሺ࢞ ࢟ǡ ࢞ െ ࢟ሻ, and plot the transformed vertices on the figure.
b. Does ࡸ represent a rotation and dilation? If so, estimate the amount of rotation and the scale factor from
your figure.
The transformed image is dilated and rotated but is also reflected, so transformation ࡸ is not a rotation and
dilation.
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c. If ࡸ represents a rotation and dilation, calculate the amount of rotation and the scale factor from the
formula for ࡸ . Do your numbers agree with your estimate in part (b)? If not, explain why there are no
values of ࢇ and ࢈ so that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ.
Suppose that ሺ࢞ ࢟ǡ ࢞ െ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ. Then, ࢇ ൌ and ࢇ ൌ െ, which is not possible.
This transformation does not fit our formula for rotation and dilation.
Closing (4 minutes)
Ask students to summarize the lesson in writing or orally with a partner. Some key elements are summarized below.
Lesson Summary
For real numbers ࢇ and ࢈, the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ corresponds to a counterclockwise
rotation by ܚ܉ሺࢇ ࢈ሻ about the origin and dilation with scale factor ξࢇ ࢈.
Exit Ticket (5 minutes)
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Name Date
Lesson 20: Exploiting the Connection to Cartesian Coordinates
Exit Ticket
1. Find the scale factor and rotation induced by the transformation ܮሺݔǡ ݕሻ ൌ ሺെ ݔെ ͺݕǡ ͺ ݔെ ݕሻ.
2. Explain how the transformation of complex numbers ܮሺ ݔ ݅ݕሻ ൌ ሺܽ ܾ݅ሻሺ ݔ ݅ݕሻ leads to the transformation of
points in the coordinate plane ܮሺݔǡ ݕሻ ൌ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ.
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Exit Ticket Sample Solutions
1. Find the scale factor and rotation induced by the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺെ࢞ െ ૡ࢟ǡ ૡ࢞ െ ࢟ሻ.
This is a transformation of the form ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ with ࢇ ൌ െ and ࢈ ൌ ૡ. The scale factor is then
ඥሺെሻ ૡ ൌ .
ૡ
The rotation is the ܖ܉ܜ܋ܚ܉ቀ ቁ ൎ െǤ ι.
െ
2. Explain how the transformation of complex numbers ࡸሺ࢞ ࢟ሻ ൌ ሺࢇ ࢈ሻሺ࢞ ࢟ሻ leads to the transformation of
points in the coordinate plane ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ.
First, we associate the complex number ࢞ ࢟ to the point ሺ࢞ǡ ࢟ሻ in the coordinate plane. Then, the point
associated with the complex number ሺࢇ ࢈ሻሺ࢞ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ሻ ሺ࢈࢞ ࢇ࢟ሻ is ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ. Thus, we
can interpret the original transformation of complex numbers as the transformation of points
ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ.
Problem Set Sample Solutions
1. Find real numbers ࢇ and ࢈ so that the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ produces the specified rotation
and dilation.
a. Rotation by ૠι counterclockwise and dilation by scale factor .
We need to find real numbers ࢇ and ࢈ so that ࢇ ࢈ has modulus and argument ૠι. Then, ሺࢇǡ ࢈ሻ lies on
the negative ࢟-axis, so ࢇ ൌ and ࢈ ൏ . We need ൌ ȁࢇ ࢈ȁ ൌ ȁ࢈ȁ ൌ ȁ࢈ȁ, so this means that ࢈ ൌ െ .
Thus, the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ቀ ࢟ǡ െ ࢞ቁ will rotate by ૠι and dilate by a scale factor of .
b. Rotation by ι counterclockwise and dilation by scale factor ξ.
We need to find real numbers ࢇ and ࢈ so that ࢇ ࢈ has modulus ξ and argument ι. Thus, ሺࢇǡ ࢈ሻ lies in
the second quadrant on the diagonal line with equation ࢟ ൌ െ࢞, so we know that ࢇ and ࢈ ൌ െࢇ. Since
ξ ൌ ξࢇ ࢈ and ࢇ ൌ െ࢈, we have ξ ൌ ඥࢇ ሺെࢇሻ so ξ ൌ ξࢇ , and thus ࢇ ൌ . It follows that
࢈ ൌ െ. Then, the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ െ࢞ ࢟ሻ rotates by ι counterclockwise and dilates
by a scale factor of ξ.
c. Rotation by ι clockwise and dilation by scale factor .
We need to find real numbers ࢇ and ࢈ so that ࢇ ࢈ has modulus and argument ι. Thus, ሺࢇǡ ࢈ሻ lies in
the first quadrant on the line with equation ࢟ ൌ ࢞, so we know that ࢇ ൌ ࢈ and ࢇ , ࢈ . Since ൌ
ඥሺࢇ ࢈ ሻ ൌ ξࢇ ࢇ , we know that ࢇ ൌ , and ࢇ ൌ ࢈ ൌ ξ. Thus, the transformation
ࡸሺ࢞ǡ ࢟ሻ ൌ ሺξ࢞ െ ξ࢟ǡ ξ࢞ ξ࢟ሻ rotates by ι counterclockwise and dilates by a scale factor of
.
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d. Rotation by ι counterclockwise and dilation by scale factor .
Rotation by ι counterclockwise has the same effect as rotation by ૡι counterclockwise. Thus, we need
to find real numbers ࢇ and ࢈ so that the argument of ሺࢇ ࢈ሻ is ૡι and ȁࢇ ࢈ȁ ൌ ξࢇ ࢈ ൌ . Since
ܚ܉ሺࢇ ࢈ሻ ൌ ૡι, we know that the point ሺࢇǡ ࢈ሻ lies on the negative ࢞-axis, and we have
ࢇ ൏ and ࢈ ൌ . We then have ࢇ ൌ െ and ࢈ ൌ , so the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺെ࢞ǡ െ࢟ሻ will rotate
by ι counterclockwise and dilate with scale factor .
2. Determine if the following transformations represent a rotation and dilation. If so, identify the scale factor and the
amount of rotation.
a. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ ࢞ ࢟ሻ
If ࡸሺ࢞ǡ ࢟ሻ is of the formሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ, then ࢇ ൌ and ࢈ must be both and Ȃ . Since this is
impossible, this transformation does not consist of rotation and dilation.
b. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺെ࢞ ࢟ǡ െ࢞ െ ࢟ሻ
If we let ࢇ ൌ െ and ࢈ ൌ െ, then ࡸሺ࢞ǡ ࢟ሻ is of the form ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ. Thus, this
transformation does consist of rotation and dilation. The dilation has scale factor ඥሺെሻ ሺെሻ ൌ ,
and the transformation rotates through ܚ܉ሺെ െ ሻ ൌ ܖ܉ܜ܋ܚ܉ቀ ቁ ൎ ૠǤ ૡι.
c. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ െ࢟ ࢞ሻ
If we let ࢇ ൌ and ࢈ ൌ െ, then ࡸሺ࢞ǡ ࢟ሻ is of the form ሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ. Thus, the
transformation does consist of rotation and dilation. The dilation has scale factor ඥሺሻ ሺെሻ ൌ ξ,
and the transformation rotates through ܚ܉ሺ െ ሻ ൌ ι.
3. Grace and Lily have a different point of view about the transformation on cube ࡰ that is shown above. Grace
states that it is a reflection about the imaginary axis and a dilation of factor of . However, Lily argues it should be a
ૢι counterclockwise rotation about the origin with a dilation of a factor of .
a. Who is correct? Justify your answer.
Lily is correct because the vertices of the cube stay the same with respect to each other.
b. Represent the above transformation in the form ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ.
Rotating ૢι with a dilation of a factor of : ࢇ ࢈ ൌ ሺܛܗ܋ሺૢιሻ ή ܖܑܛሺૢιሻሻ ൌ ሺ ሻ ൌ Ǥ
Therefore, ࢇ ൌ ǡ ࢈ ൌ ǡ ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ࢟ǡ ࢞ ࢟ሻ ൌ ሺെ࢟ǡ ࢞ሻǤ
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4. Grace and Lily still have a different point of view on this transformation on triangle shown above. Grace states
that it is reflected about the real axis first, then reflected about the imaginary axis, and then dilated with a factor of
. However, Lily asserts that it is a ૡιcounterclockwise rotation about the origin with a dilation of a factor of .
a. Who is correct? Justify your answer.
Both are correct. Both sequences of transformations result in the same image.
b. Represent the above transformation in the form ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ.
Rotating ૡι with a dilation of a factor of 2:
ࢇ ࢈ ൌ ሺܛܗ܋ሺૡιሻ ή ܖܑܛሺૡιሻሻ ൌ ሺെ ሻ ൌ െ .
Therefore, ࢇ െ ǡ ࢈ ൌ ǡ ࡸሺ࢞ǡ ࢟ሻ ൌ ሺെ࢞ െ ࢟ǡ ࢞ െ ࢟ሻ ൌ ሺെ࢞ǡ െ࢟ሻǤ
5. Given ࢠ ൌ ξ
a. Find the complex number ࢝ that will cause a rotation with the same number of degrees as ࢠ without a
dilation.
ࢠ ൌ ξ ǡ ȁࢠȁ ൌ ǡ ࢝ ൌ ൫ξ ൯
b. Can you come up with a general formula ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ for any complex number
ࢠ ൌ ࢞ ࢟ to represent this condition?
࢝ ൌ ࢞ ࢟ǡ ȁࢠȁ ൌ ඥ࢞ ࢟ ǡ ࢇ ൌ ࢞ǡ ࢈ ൌ ࢟ǡ
ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ή ࢞ െ ࢟ ή ࢟ǡ ࢟ ή ࢞ ࢞ ή ࢟ሻ ൌ ሺ࢞ െ ࢟ ǡ ࢞࢟ሻ
ට࢞ ࢟ ට࢞ ࢟
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Lesson 21: The Hunt for Better Notation
Student Outcomes
Students represent linear transformations of the form ܮሺݔǡ ݕሻ ൌ ሺܽ ݔ ܾݕǡ ܿ ݔ ݀ݕሻ by matrix multiplication
ݔ ܽ ܾ ݔ
ܮቀ ݕቁ ൌ ቀ ቁ ቀ ቁ.
ܿ ݀ ݕ
ݔ ܽ ܾ ݔ
Students recognize when a linear transformation of the form ܮቀݕቁ ൌ ቀ ቁ ቀ ቁ represents rotation and
ܿ ݀ ݕ
dilation in the plane.
ܽ ܾ ݔ
Students multiply matrix products of the formቀ ቁ ቀ ቁ.
ܿ ݀ ݕ
Lesson Notes
This lesson introduces ʹ ൈ ʹ matrices and their use for representing linear transformation through multiplication.
Matrices provide a third method of representing rotation and dilation of the plane, as well as other linear
transformations that students have not yet been exposed to in this module, such as reflection and shearing.
Classwork
Opening Exercise (5 minutes)
Have students work on this exercise in pairs or small groups. Students see how
cumbersome this notation can be. Scaffolding:
Have struggling students
Opening Exercise evaluate ܮଵ ሺͳǡʹሻ and
ܮଶ ሺͳǡ ʹሻ.
Suppose that ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ࢟ǡ ࢞ ࢟ሻ and ࡸ ሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ െ࢟ ࢞ሻ.
Find the result of performing ࡸ and then ࡸ on a point ሺǡ ሻ. That is, find ࡸ ൫ࡸ ሺǡ ሻ൯. Have advanced learners
ࡸ ൫ࡸ ሺǡ ሻ൯ ൌ ࡸ ሺ െ ǡ ሻ
find ܮଶ ൫ܮଵ ሺǡ ݍሻ൯ and
ൌ ൫ሺ െ ሻ ሺ ሻǡ െሺ െ ሻ ሺ ሻ൯
ܮଵ ൫ܮଶ ሺǡ ݍሻ൯ and
ൌ ሺ െ ૢ ૡǡ െૡ ૢ ሻ
determine values of and
ൌ ሺૡ െ ǡ ૡሻ ݍwhere ܮଶ ൫ܮଵ ሺǡ ݍሻ൯ and
ܮଵ ൫ܮଶ ሺǡ ݍሻ൯ are equal.
Discussion (6 minutes)
Use this Discussion to review the answer to the Opening Exercise and to motivate and introduce matrix notation.
What answer did you get to the Opening Exercise?
à ܮଶ ሺܮଵ ሺǡ ݍሻሻ ൌ ሺͳͺ െ ݍǡ ͳͺݍሻ
How do you feel about this notation? Do you find it confusing or cumbersome?
à Answers will vary, but most students will find the composition confusing or cumbersome or both.
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What if I told you there was a simpler way to find the answer? We just have to learn some new mathematics
first.
In the mid-1800s and through the early 1900s, formulas such as ܮሺݔǡ ݕሻ ൌ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ kept popping
up in mathematical situations, and people were struggling to find a simpler way to work with these
expressions. Mathematicians used a representation called a matrix. A matrix is a rectangular array of
ܽ ܽ
numbers that looks like ቀ ቁ or ቂ ቃ. We can represent matrices as soft or hard brackets, but a matrix is a
ܾ ܾ
rectangular array of numbers. These matrices both have ͳ column and ʹ rows. Matrices can be any size.
ܽ ܾ
A square matrix has the same number of rows and columns and could look like ቂ ቃ. We call this a ʹ ൈ ʹ
ܿ ݀
matrix because it has ʹ columns and ʹ rows.
ݔ
A matrix with one column can be used to represent a point ቀݕቁ.
It can also represent a vector from point ܣto point ܤ. If ܣሺܽଵ ǡ ܽଶ ሻ and ܤሺܾଵ ǡ ܾଶ ሻ, then ሬሬሬሬሬԦ
ܤܣcan be represented
ܾଵ െ ܽଵ
as ൨. This translation maps ܣto ܤ.
ܾଶ െ ܽଶ
Explain what we have just said about a matrix and a vector to your neighbor.
Let’s think about what a transformation ܮሺݔǡ ݕሻ ൌ ሺܽ ݔ ܾݕǡ ܿ ݔ ݀ݕሻ does to the components of the point
ݔ
(or vector) ሺݔǡ ݕሻ. It will be helpful to write a point ሺݔǡ ݕሻ as ቀݕቁ. Then, the transformation becomes
ݔ ܽ ݔ ܾݕ
ܮቀ ݕቁ ൌ ൬ ൰Ǥ
ܿ ݔ ݀ݕ
The important parts of this transformation are the four coefficients ܽ, ܾ, ܿ, and ݀. We will record them in a
ܽ ܾ
matrix: ቀ ቁ.
ܿ ݀
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
ܽ ܾ ݔ ܽ ݔ ܾݕ
We can define a new type of multiplication so that ቀ ቁቀ ቁ ൌ ൬ ൰.
ܿ ݀ ݕ ܿ ݔ ݀ݕ
Based on this definition, explain how the entries in the matrix are used in the process of multiplication.
à When we use matrix multiplication, we think of multiplying the first row of the matrix ሺܽ ܾሻ by the
ݔ ݔ
column ቀݕቁ so that ሺܽ ܾሻ ڄቀݕቁ ൌ ܽ ݔ ܾݕ, and we write that result in the first row. (This
ݔ
multiplication ሺܽ ܾሻ ڄቀݕቁ ൌ ܽ ݔ ܾ ݕis called a dot product. The teacher may choose whether or not
to share this terminology with students.) Then, we multiply the second row of the matrix ሺܿ ݀ ሻ by the
ݔ ݔ
column ቀݕቁ so that ሺܿ ݀ ሻ ڄቀݕቁ ൌ ܿ ݔ ݀ݕ, and we write that result in the second row, giving the final
answer.
Example 1 (6 minutes)
Do the following numerical examples to illustrate matrix-vector multiplication. It may be necessary to do more or fewer
examples based on assessment of students’ understanding.
ͳ ʹ ͷ
Evaluate the product ቀ ቁ ቀ ቁ.
͵ Ͷ
ͳ ʹ ͷ ͳڄͷʹڄ ͳ
à ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
͵ Ͷ ͵ڄͷͶڄ ͵ͻ
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ͳ ʹ ʹ
Evaluate the product ቀ ቁ ቀ ቁ.
͵ Ͷ െͷ
ͳ ʹ ʹ ͳ ʹ ڄ ʹ ڄሺെͷሻ െͺ
à ቀ ቁቀ ቁ ൌ ൬ ൰ൌቀ ቁ
͵ Ͷ െͷ ͵ ʹ ڄ Ͷ ڄሺെͷሻ െͳͶ
ͳ ʹ ݔ
Evaluate the product ቀ ቁ ቀ ቁ.
͵ Ͷ ݕ
ͳ ʹ ݔ ݔ ʹݕ
à ቀ ቁቀ ቁ ൌ ൬ ൰
͵ Ͷ ݕ ͵ ݔ Ͷݕ
Exercises 1–2 (6 minutes)
Have students work these exercises in pairs or small groups.
Exercises
1. Calculate each of the following products.
െ
a. ቀ ቁቀ ቁ
െ
െ െૠ
ቀ ቁൌቀ ቁ
െ ૢ
b. ቀ ቁቀ ቁ
െ
െ
ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
െ െ
െ
c. ቀ ቁቀ ቁ
െ െ
െ ૡ
ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
െ െ ૠ
2. Find a value of so that ቀ ቁ ቀ ቁ ൌ ቀ ቁ.
െ
Multiplying this out, we have ቀ ቁቀ ቁ ൌ ቀ ቁ ൌ ቀ ቁǡ so െ ൌ , and thus, ൌ .
െ െ
Example 2 (6 minutes)
Use this example to connect the process of multiplying a matrix by a vector to the geometric transformations of rotation
and dilation in the plane students have been doing in the past few lessons.
We know that a linear transformation ܮሺݔǡ ݕሻ ൌ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ has the geometric effect of a
counterclockwise rotation in the plane by ሺܽ ܾ݅ሻ and dilation with scale factor ȁܽ ܾ݅ȁ. How would we
represent this rotation and dilation using matrix multiplication?
ݔ ܽ െܾ ݔ
à ܮቀݕቁ ൌ ቀ ቁ ቀݕቁ
ܾ ܽ
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ݔ ͳ െʹ ݔ
What is the geometric effect of the transformation ܮቀݕቁ ൌ ቀ ቁ ቀݕቁ?
ʹ ͳ
à This corresponds to the transformation ܮሺݔǡ ݕሻ ൌ ሺܽ ݔെ ܾݕǡ ܾ ݔ ܽݕሻ with ܽ ൌ ͳ and ܾ ൌ ʹ, so the
ʹ
geometric effect of this transformation is a counterclockwise rotation through ቀ ቁ and dilation
ͳ
with scale factor ȁͳ ʹ݅ȁ ൌ ξͷ.
ͳ െʹ ͳ
Evaluate the product ቀ ቁ ቀ ቁ. Scaffolding:
ʹ ͳ Ͳ
Remember from Algebra II
ͳ െʹ ͳ ͳ ͳ ڄ ሺെʹሻ Ͳ ڄ ͳ
à ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ ܾ
ʹ ͳ Ͳ ʹͳڄͳͲڄ ʹ that ߠ ൌ ቀ ቁ
ܽ
ͳ ͳ means students are
The points represented by ቀ ቁ and ቀ ቁ are shown on the axes below. We see finding the angle ߠ such
Ͳ ʹ
ͳ ͳ ܾ
that the point ቀ ቁ is the image of the point ቀ ቁ under rotation by that ሺߠሻ ൌ .
ʹ Ͳ ܽ
ሺͳ ʹ݅ሻ ൌ ሺʹሻ ൎ ͵ǤͶ͵ͷι and dilation byȁͳ ʹ݅ȁ ൌ ξͷ ൎ ʹǤʹͶ. Students know
ߨ
ቀ ቁ ൌ ͳ, so
Ͷ
ߨ
ሺͳሻ ൌ .
Ͷ
Exercises 3–9 (8 minutes)
Have students work in pairs or small group on these exercises.
ࢇ ࢈
3. Find a matrix ቀ ቁ so that we can represent the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ࢟ǡ ࢞ ࢟ሻ by
ࢉ ࢊ
࢞ ࢇ ࢈ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ.
ࢉ ࢊ ࢟
െ
The matrix is ቀ ቁ.
࢞ ࢇ ࢈ ࢞
4. If a transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ has the geometric effect of rotation and dilation, what do you know about
ࢉ ࢊ ࢟
the values ࢇǡ ࢈ǡ ࢉ, and ࢊ?
࢞ ࢇ െ࢈ ࢞
Since the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ െ ࢈࢟ǡ ࢈࢞ ࢇ࢟ሻ has matrix representation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ, we know
࢈ ࢇ ࢟
that ࢇ ൌ ࢊ and ࢉ ൌ െ࢈.
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ࢇ ࢈ ࢞ ࢇ ࢈ ࢞
5. Describe the form of a matrix ቀ ቁ so that the transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ has the geometric effect of
ࢉ ࢊ ࢉ ࢊ ࢟
only dilation by a scale factor ࢘.
The transformation that scales by factor ࢘ has the form ࡸሺ࢞ǡ ࢟ሻ ൌ ࢘ሺ࢞ǡ ࢟ሻ ൌ ሺ࢘࢞ǡ ࢘࢟ሻ ൌ ሺ࢘࢞ െ ࢟ǡ ࢞ ࢘࢟ሻ, so the
࢘
matrix has the form ቀ ቁ.
࢘
ࢇ ࢈ ࢞ ࢇ ࢈ ࢞
6. Describe the form of a matrix ቀ ቁ so that the transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ has the geometric effect of
ࢉ ࢊ ࢉ ࢊ ࢟
only rotation by ࣂ. Describe the matrix in terms of ࣂ.
ࢇ െ࢈
The matrix has the form ቀ ቁ, where ܚ܉ሺࢇ ࢈ሻ ൌ ࣂ. Thus, ࢇ ൌ ܛܗ܋ሺࣂሻ and ࢈ ൌ ܖܑܛሺࣂሻ, so the matrix has
࢈ ࢇ
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ
the form ൬ ൰.
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
ࢇ ࢈ ࢞ ࢇ ࢈ ࢞
7. Describe the form of a matrix ቀ ቁ so that the transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ has the geometric effect of
ࢉ ࢊ ࢉ ࢊ ࢟
rotation by ࣂ and dilation with scale factor ࢘. Describe the matrix in terms of ࣂ and ࢘.
ࢇ െ࢈
The matrix has the form ቀ ቁ, where ܚ܉ሺࢇ ࢈ሻ ൌ ࣂ and ࢘ ൌ ȁࢇ ࢈ȁ. Thus, ࢇ ൌ ࢘᩺ܛܗ܋ሺࣂሻ and ࢈ ൌ ࢘᩺ܖܑܛሺࣂሻ,
࢈ ࢇ
࢘ԝܛܗ܋ሺࣂሻ െ࢘ԝܖܑܛሺࣂሻ
so the matrix has the form ൬ ൰.
࢘ԝܖܑܛሺࣂሻ ࢘ԝܛܗ܋ሺࣂሻ
࢞ ࢞
8. Suppose that we have a transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ.
࢟
a. Does this transformation have the geometric effect of rotation and dilation?
ࢇ െ࢈
No, the matrix is not in the form ቀ ቁ, so this transformation is not a rotation and dilation.
࢈ ࢇ
b. Transform each of the points ൌ ቀ ቁ, ൌ ቀ ቁ, ൌ ቀ ቁ, and ࡰ ൌ ቀ ቁ, and plot the images in the plane
shown.
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࢞ ࢞
9. Describe the geometric effect of the transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ.
࢟
This transformation does nothing to the point ሺ࢞ǡ ࢟ሻ in the plane; it is the identity transformation.
Closing (3 minutes)
Ask students to summarize the lesson in writing or orally with a partner. Some key elements are summarized below.
Lesson Summary
For real numbers ࢇǡ ࢈ǡ ࢉ, and ࢊ, the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺࢇ࢞ ࢈࢟ǡ ࢉ࢞ ࢊ࢟ሻ can be represented using matrix
࢞ ࢇ ࢈ ࢞ ࢇ ࢈ ࢞ ࢇ࢞ ࢈࢟ ࢞
multiplication by ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ, where ቀ ቁቀ ቁ ൌ ൬ ൰ and the ቀ࢟ቁ represents the point ሺ࢞ǡ ࢟ሻ in
ࢉ ࢊ ࢟ ࢉ ࢊ ࢟ ࢉ࢞ ࢊ࢟
the plane.
The transformation is a counterclockwise rotation by ࣂ if and only if the matrix representation is
࢞ ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ ࢞
ࡸ ቀ࢟ቁ ൌ ൬ ൰ ቀ ቁ.
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ ࢟
The transformation is a dilation with scale factor if and only if the matrix representation is
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ.
࢟
The transformation is a counterclockwise rotation by ܚ܉ሺࢇ ࢈ሻ and dilation with scale factor
࢞ ࢇ െ࢈ ࢞
ȁࢇ ࢈ȁ if and only if the matrix representation is ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ࢟ቁ. If we let ࢘ ൌ ȁࢇ ࢈ȁ and
࢈ ࢇ
࢞ ࢘ԝܛܗ܋ሺࣂሻ െ࢘ԝܖܑܛሺࣂሻ ࢞
ࣂ ൌ ܚ܉ሺࢇ ࢈ሻ, then the matrix representation is ࡸ ቀ࢟ቁ ൌ ൬ ൰ ቀ ቁ.
࢘ԝܖܑܛሺࣂሻ ࢘ԝܛܗ܋ሺࣂሻ ࢟
Exit Ticket (5 minutes)
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Name Date
Lesson 21: The Hunt for Better Notation
Exit Ticket
ͳͲ ʹ ͵
1. Evaluate the product ቀ ቁ ቀ ቁ.
െͺ െͷ െʹ
2. Find a matrix representation of the transformation ܮሺݔǡ ݕሻ ൌ ሺ͵ ݔ Ͷݕǡ ݔെ ʹݕሻ.
ݔ ͷ ʹ ݔ
3. Does the transformation ܮቀݕቁ ൌ ቀ ቁ ቀ ቁ represent a rotation and dilation in the plane? Explain how you
െʹ ͷ ݕ
know.
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Exit Ticket Sample Solutions
1. Evaluate the product ቀ ቁ ቀ ቁ.
െૡ െ െ
െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െૡ െ െ െ
ൌቀ ቁ
െ
2. Find a matrix representation of the transformation ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ ࢞ െ ࢟ሻ.
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
െ ࢟
࢞ ࢞
3. Does the transformation ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ represent a rotation and dilation in the plane? Explain how you
െ ࢟
know.
Yes; this transformation can also be represented as ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ሺെሻ࢟ǡ െ࢞ ࢟ሻ, which has the geometric
effect of counterclockwise rotation by ܚ܉ሺ െ ሻ and dilation by ȁ െ ȁ.
Problem Set Sample Solutions
1. Perform the indicated multiplication.
a. ቀ ቁቀ ቁ
ૡ െ
െ
ቀ ቁ
െ
b. ቀ ቁቀ ቁ
െ െ
ቀ ቁ
െૡ
c. ቀ ቁቀ ቁ
െ ૡ
ቀ ቁ
െ
ૠ
d. ቀ ቁቀ ቁ
ૢ
ૠ
ቀ ቁ
ૢ
െ
e. ቀ ቁቀ ቁ
ૠ
െ
ቀ ቁ
െ
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f. ቀ ቁቀ ቁ
ૢ െ
ቀ ቁ
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ ࢞
g. ൬ ൰ቀ ቁ
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ ࢟
࢞ԝܛܗ܋ሺࣂሻ െ ࢟ԝܖܑܛሺࣂሻ
൬ ൰
࢞ԝܖܑܛሺࣂሻ ࢟ԝܛܗ܋ሺࣂሻ
࣊
h. ቀ ቁቀ ቁ
െ࣊ ૠ
࣊ ૠ
ቀ ቁ
െ ૠ࣊
ૠ
2. Find a value of so that ቀ ቁ ቀ ቁ ൌ ቀ ቁ.
We have ቀ ቁቀ ቁ ൌ ቀ ቁ, so ൌ ૠ and ൌ . Thus, ൌ െૡ and ൌ െ, so
ൌ െ.
ૠ
3. Find values of and so that ቀ ቁቀ ቁ ൌ ቀ ቁ.
െ െ
ૠ
We have ቀ ቁቀ ቁ ൌ ቀ ቁǡ so ൌ ૠ and െ ൌ െ. Therefore, ൌ െ and ൌ .
െ െ
4. Find values of and so that ቀ ቁ ቀ ቁ ൌ ቀ ቁ.
െ െૢ
Since ቀ ቁቀ ቁ ൌ ቀ ቁ, we need to find values of and so that ൌ and
െ െ
െ ൌ െૢ. Solving this first equation for gives ൌ െ, and substituting this expression for into the
second equation gives Ȃ ૢ ൌ െሺെሻ ൌ ૢ, so we have ൌ െ. Then, ൌ െ gives ൌ .
Therefore, ൌ and ൌ െ.
5. Write the following transformations using matrix multiplication.
a. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ െ ࢟ǡ ࢞ െ ࢟ሻ
࢞ െ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
െ ࢟
b. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ െ࢞ ࢟ሻ
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
െ ࢟
c. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ ࢟ǡ ૡ࢞ െ ࢟ሻ
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
ૡ െ ࢟
d. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࣊࢞ െ ࢟ǡ െ࢞ ࢟ሻ
࢞ ࣊ െ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ࢟ቁ
െ
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e. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢞ǡ ࢞ሻ
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
࢟
f. ࡸሺ࢞ǡ ࢟ሻ ൌ ሺ࢟ǡ ૠ࢞ሻ
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
ૠ ࢟
6. Identify whether or not the following transformations have the geometric effect of rotation only, dilation only,
rotation and dilation only, or none of these.
࢞ െ ࢞
a. ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
െ ࢟
െ ࢇ െ࢈
The matrix ቀ ቁ cannot be written in the form ቀ ቁ, because ് െ, so this is neither a rotation
െ ࢈ ࢇ
nor a dilation. The transformation ࡸ is not one of the specified types of transformations.
࢞ ࢞
b. ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
࢟
This transformation has the geometric effect of dilation by a scale factor of .
࢞ െ െ ࢞
c. ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
െ ࢟
െ െ ࢇ െ࢈
The matrix ቀ ቁ has the form ቀ ቁ with ࢇ ൌ െ and ࢈ ൌ . Therefore, this transformation has
െ ࢈ ࢇ
the geometric effect of rotation and dilation.
࢞ െ ࢞
d. ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ࢟ቁ
െ
െ ࢇ െ࢈
The matrix ቀ ቁ cannot be written in the form ቀ ቁ, because െ ് െሺെሻ, so this is neither a
െ ࢈ ࢇ
rotation nor a dilation. The transformation ࡸ is not one of the specified types of transformations.
࢞ െૠ ࢞
e. ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
ૠ ࢟
െૠ ࢇ െ࢈
The matrix ቀ ቁ cannot be written in the form ቀ ቁ, because െૠ ് ૠ, so this is neither a rotation
ૠ ࢈ ࢇ
nor a dilation. The transformation ࡸ is not one of the specified types of transformations.
࢞ െ ࢞
f. ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ࢟ቁ
࣊ ࣊
െ ܛܗ܋ቀ ቁ െξԝ ܖܑܛቀ ቁ
We see that ቀ ቁൌቌ ࣊ ࣊ ቍ, so this transformation has the geometric effect of
ξԝ ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
࣊
dilation by ξ and rotation by .
7. Create a matrix representation of a linear transformation that has the specified geometric effect.
a. Dilation by a factor of and no rotation
࢞ ࢞
ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
࢟
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b. Rotation by ૡι and no dilation
࢞ ܛܗ܋ሺૡιሻ െܖܑܛሺૡιሻ ࢞
ࡸ ቀ࢟ቁ ൌ ൬ ൰ ቀ ቁ
ܖܑܛሺૡιሻ ܛܗ܋ሺૡιሻ ࢟
െ ࢞
ൌቀ ቁቀ ቁ
െ ࢟
࣊
c. Rotation by Ȃ ԝ ܌܉ܚand dilation by a scale factor of
࣊ ࣊
࢞ ԝ ܛܗ܋ቀെ ቁ െԝ ܖܑܛቀെ ቁ ࢞
ࡸ ቀ࢟ቁ ൌ ൮
࣊ ࣊ ൲ ቀ࢟ቁ
ԝ ܖܑܛቀെ ቁ ԝ ܛܗ܋ቀെ ቁ
࢞
ൌቀ ቁቀ ቁ
െ ࢟
d. Rotation by ι and dilation by a scale factor of
࢞ ԝܛܗ܋ሺιሻ െԝܖܑܛሺιሻ ࢞
ࡸ ቀ࢟ቁ ൌ ൬ ൰ ቀ ቁ
ԝܖܑܛሺιሻ ԝܛܗ܋ሺιሻ ࢟
࢞
ൌ ൬ξ െ ൰ ቀ࢟ቁ
ξ
8. Identify the geometric effect of the following transformations. Justify your answers.
ඥ ඥ
࢞ െ െ
a. ࡸ ቀ࢟ቁ ൌ ൮ ൲ ቀ࢞ ቁ
ඥ ඥ ࢟
െ
࣊ ඥ ࣊ ඥ
Since ܛܗ܋ቀ ቁ ൌ െ and ܖܑܛቀ ቁ ൌ , this transformation has the form
࣊ ࣊
࢞ ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢞ ࣊
ࡸ ቀ࢟ቁ ൌ ቌ ቍ ቀ࢟ቁ and, thus, represents counterclockwise rotation by with no
࣊ ࣊
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
dilation.
࢞ െ ࢞
b. ࡸ ቀ࢟ቁ ൌ ቀ ቁ ቀ࢟ቁ
࣊ ࣊
࣊ ࣊ ࢞ ԝ ܛܗ܋ቀ ቁ െԝ ܖܑܛቀ ቁ ࢞
Since ܛܗ܋ቀ ቁ ൌ and ܖܑܛቀ ቁ ൌ , this transformation has the form ࡸ ቀ࢟ቁ ൌ ቌ ࣊ ࣊ ቍ ቀ࢟ቁ
ԝ ܖܑܛቀ ቁ ԝ ܛܗ܋ቀ ቁ
࣊
and, thus, represents counterclockwise rotation by and dilation by a scale factor .
࢞ െ ࢞
c. ࡸ ቀ࢟ቁ ൌ ቀ ቁቀ ቁ
െ ࢟
Since ܛܗ܋ሺ࣊ሻ ൌ െ and ܖܑܛሺ࣊ሻ ൌ , this transformation has the form
࢞ ԝܛܗ܋ሺ࣊ሻ െԝܖܑܛሺ࣊ሻ ࢞
ࡸ ቀ࢟ቁ ൌ ൬ ൰ ቀ ቁ and, thus, represents counterclockwise rotation by ࣊ and dilation by
ԝܖܑܛሺ࣊ሻ ԝܛܗ܋ሺ࣊ሻ ࢟
a scale factor .
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࢞ ξ ࢞
d. ࡸ ቀ࢟ቁ ൌ ൬ ൰ ቀ࢟ቁ
െξ
࣊ ࣊ ඥ
Since ܛܗ܋ቀ ቁ ൌ and ܖܑܛቀ ቁ ൌ െ , this transformation has the form
࣊ ࣊
࢞ ԝ ܛܗ܋ቀ ቁ െԝ ܖܑܛቀ ቁ ࢞ ࣊
ࡸ ቀ࢟ቁ ൌ ቌ ቍ ቀ࢟ቁ and, thus, represents counterclockwise rotation by and
࣊ ࣊
ԝ ܖܑܛቀ ቁ ԝ ܛܗ܋ቀ ቁ
dilation with scale factor .
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Lesson 22: Modeling Video Game Motion with Matrices
Student Outcomes
Students use matrix transformations to represent motion along a straight line.
Lesson Notes
This is the first of a two-day lesson where students use their knowledge of ʹ ൈ ʹ matrices and their transformations to
program video game motion. Lesson 22 focuses on straight-line motion. In Lesson 23, students extend that motion
to include rotations. In programming, students multiply matrices and vectors and use matrices to perform
transformations in the plane. This lesson focuses on ŵŽĚĞůŝŶŐĂƐstudentƐ usemathematics (matrices and
transformations) to model a real-world situation (video game programming).
Classwork
Opening Exercise (2 minutes)
Opening Exercise
࢞ ࢞
Let ࡰ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ.
࢟
a. Plot the point ቀ ቁ.
b. Find ࡰ ቀ ቁ and plot it.
࢞ ࢞
c. Describe the geometric effect of performing the transformation ቀ࢟ቁ ՜ ࡰ ቀ࢟ቁ.
Each point in the plane gets dilated by a factor of . In other words, a point ࡼ gets moved to a new location
that is on the line through ࡼ and the origin, but its distance from the origin increases by a factor of .
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Discussion (9 minutes): Motion Along a Line Scaffolding:
ݐ Ͳ ͵ Support students in
Let ݂ሺݐሻ ൌ ቀ ቁ ቀ ቁ.
Ͳ ͳ ݐ understanding the
Find ݂ሺͲሻ, ݂ሺͳሻ, ݂ሺʹሻ, ݂ሺ͵ሻ, and ݂ሺͶሻ. functions defined by
Ͳ Ͳ ͵ Ͳ matrices using simpler
à ݂ሺͲሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ examples.
Ͳ Ͳ ͳ Ͳ
ͳ Ͳ ͵ ͵ Let ݂ሺݐሻ ൌ ͵ݐ. Find
à ݂ሺͳሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
Ͳ ͳ ͳ ͳ ݂ሺͲሻǡ ݂ሺͳሻǡ ݂ሺʹሻǡ ݂ሺ͵ሻǡ ݂ሺͶሻǤ
ʹ Ͳ ͵
à ݂ሺʹሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ Explain how you
Ͳ ʹ ͳ ʹ
͵ Ͳ ͵ ͻ determined your answer.
à ݂ሺ͵ሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
Ͳ ͵ ͳ ͵ Advanced learners can be
Ͷ Ͳ ͵ ͳʹ given the following
à ݂ሺͶሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
Ͳ Ͷ ͳ Ͷ prompt without
Plot each of these points on a graph. What do you notice? supporting questions.
à Each point appears to lie on a straight line through the origin. Suppose you were
designing a computer
game. You want an object
to travel along a line from
the origin to the point
ሺ͵Ͳǡ ͳͲሻ in ʹͲ seconds.
Can you design a function
݄ሺݐሻ that does this?
Let ܲ ൌ ݂ሺݐሻ. How can we be sure that ܲ actually does trace out a straight line as ݐvaries?
à We could check to see if every pair of points forms a segment with the same slope.
What is the slope of the segment that joins ݂ሺͳሻ and ݂ሺͶሻ?
ସିଵ ଷ ଵ
à Since ݂ሺͳሻ ൌ ሺ͵ǡͳሻ and ݂ሺͶሻ ൌ ሺͳʹǡ Ͷሻ, the slope is ൌ ൌ .
ଵଶିଷ ଽ ଷ
ଵ
Now let’s check to see if every pair of points forms a segment with slope also. Let ݐଵ and ݐଶ be two arbitrary
ଷ
times in the domain of ݂. What is the slope of the segment that joins ݂ሺݐଵ ሻ and ݂ሺݐଶ ሻ?
௧మ ି௧భ ௧మ ି௧భ ଵ
à Since ݂ሺݐଵ ሻ ൌ ሺ͵ݐଵ ǡ ݐଵ ሻ and ݂ሺݐଶ ሻ ൌ ሺ͵ݐଶ ǡ ݐଶ ሻ, the slope of the segment is ൌ ൌ .
ଷ௧మ ିଷ௧భ ଷሺ௧మ ି௧భ ሻ ଷ
Since the slope of every segment is constant, we can conclude that the path traced out by ܲ is indeed a
straight line.
Now suppose that ݐrepresents time, measured in seconds, and ݂ሺݐሻ represents the location of an object at
time ݐ. How long would it take the object to travel from the origin to the point ሺ͵Ͳǡ ͳͲሻ?
͵ Ͳ ݐ ͵Ͳ
à We need to find a value of ݐsuch that ቀ ቁ ቀ ቁ ൌ ቀ ቁ. Apparently, ݐൌ ͳͲ works, which means it
Ͳ ͳ ݐ ͳͲ
would take ͳͲ seconds for the object to reach this point.
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ʹݐ Ͳ ͵
Now let ݃ሺݐሻ ൌ ቀ ቁ ቀ ቁ. Do you think the object will reach ሺ͵ͲǡͳͲሻ faster or slower? Go ahead and find
Ͳ ʹͳ ݐ
out.
ͳͲ Ͳ ͵ ͵Ͳ
à If we choose ݐso that ʹ ݐൌ ͳͲ, then we’d have ቀ ቁ ቀ ቁ ൌ ቀ ቁ, so ݐmust be ͷ. Therefore, the
Ͳ ͳͲ ͳ ͳͲ
object reaches the desired location in ͷ seconds.
Suppose you were designing a computer game. You want an object to travel along a line from the origin to the
point ሺ͵Ͳǡ ͳͲሻ in ʹͲ seconds. Can you design a function ݄ሺݐሻ that does this?
݇ ή ʹͲ Ͳ ͵ ͵Ͳ
à We need to find a scale factor ݇ such that ݄ሺʹͲሻ ൌ ቀ ቁ ቀ ቁ ൌ ቀ ቁ. Seeing that we need
Ͳ ݇ ή ʹͲ ͳ ͳͲ
ͳ
ͳ Ͳ ݐ ͵
to have ʹͲ݇ ൌ ͳͲ for this to work, we must have ݇ ൌ . Thus, ݄ሺݐሻ ൌ ቌʹ ͳ
ቍ ቀ ቁ.
ʹ
Ͳ ͳ ݐ
ʹ
Before we move on, try to make sense of the relationship between ݂ሺݐሻ, ݃ሺݐሻǡ and ݄ሺݐሻ. Take about half a
minute to think about this for yourself, then share with a partner, and then we’ll discuss your responses as a
whole class.
ଵ
à ݂ሺݐሻ, ݃ሺݐሻ, and ݄ሺݐሻ use scale factors of ݐ, ʹݐ, and ݐ, respectively.
ଶ
à Since ݃ doubles the ݐ-value, it makes sense that the object is moving twice as fast. For instance, to
make the scale factor equal ͳͲ, we can use ݐൌ ͷ, since ʹሺͷሻ ൌ ͳͲ. So, it takes ͷ seconds instead of ͳͲ
to reach the desired point.
à On the other hand, ݄ cuts the ݐ-value in half, so it would make sense to say that the object should move
only half as fast. In particular, to make the scale factor ͳͲ, we have to use ݐൌ ʹͲ because
ଵ
ሺʹͲሻ ൌ ͳͲ. Thus, it took ʹͲ seconds instead of ͳͲ to reach the desired point, which is twice as much
ଶ
time as it took originally.
Exercises 1–2 (3 minutes)
Give students time to perform the following exercises, and then instruct students to compare their responses with a
partner. Select students to share their responses with the whole class.
Exercises 1–2
࢚
1. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁǡ where ࢚ represents time, measured in seconds. ࡼ ൌ ࢌሺ࢚ሻ represents the position of a
࢚
moving object at time ࢚. If the object starts at the origin, how long would it take to reach ሺǡ ሻ?
࢚
ቀ ቁ ቀ ቁ ൌ ቀ ቁ ǡ ࢚ ൈ ൈ ൌ ǡ ࢚ ൌ ǡ ࢚ ൌ , or
࢚
࢚
ቀ ቁ ቀ ቁ ൌ ቀ ቁ ǡ ൈ ࢚ ൈ ൌ ǡ ࢚ ൌ ǡ ࢚ ൌ
࢚
࢚
2. Let ࢍሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ.
࢚
a. Find the value of that moves an object from the origin to ሺǡ ሻ in just seconds.
࢚ ൌ ǡ ቀ ቁ ቀ ቁ ൌ ቀ ቁ ǡ ൈ ൈ ൌ ǡ ൌ ൌ ǡ or
࢚ ൌ ǡ ቀ ቁ ቀ ቁ ൌ ቀ ቁ ǡ ൈ ൈ ൌ ǡ ൌ ൌ
ૡ
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b. Find the value of that moves an object from the origin to ሺǡ ሻ in seconds.
࢚ ൌ ǡ ቀ ቁ ቀ ቁ ൌ ቀ ቁ ǡ ൈ ൈ ൌ ǡ ൌ ൌ ǡ or
࢚ ൌ ǡ ቀ ቁ ቀ ቁ ൌ ቀ ቁ ǡ ൈ ൈ ൌ ǡ ൌ ൌ
Example 1 (3 minutes)
Ͳ ͵ ݐ
Let’s continue our exploration of the function ݂ሺݐሻ ൌ ቀ ቁ ቀ ቁ. To get some practice with different ways of
ͳ ݐ Ͳ
ݔሺݐሻ
representing transformations, let’s write ݂ሺݐሻ in the form ൬ ൰.
ݕሺݐሻ
͵ Ͳ ݐ ͵ ݐ Ͳ ͵ݐ
à ݂ሺݐሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
Ͳ ͳ ݐ Ͳݐ ݐ
Let’s suppose that an object is moving in a straight line, with the ݔ-coordinate increasing at ͵ units per second
and the ݕ-coordinate increasing at ͳ unit per second, as with ݂ሺݐሻ above. If the object starts at ሺͳʹǡ Ͷሻ, how
long would it take to reach ሺ͵Ͳǡ ͳͲሻ?
à The ݔ-coordinate is increasing at ͵ units per second, so we have ͳʹ ͵ ݐൌ ͵Ͳ, which gives
ݐൌ seconds.
à The ݕ-coordinate is increasing at ͳ unit per second, so we have Ͷ ݐൌ ͳͲ, which also gives
ݐൌ seconds. So, our results corroborate each other.
Can you write a new function ݃ሺݐሻ that gives the position of the object above after ݐseconds?
ͳʹ ͵ݐ
à We have ݔሺݐሻ ൌ ͳʹ ͵ ݐand ݕሺݐሻ ൌ Ͷ ݐ, so that gives ݃ሺݐሻ ൌ ቀ ቁ.
Ͷݐ
Can you find a way to write ݃ሺݐሻ as a matrix transformation?
ͳʹ ͵ݐ ͵ሺͶ ݐሻ
à ݃ሺݐሻ ൌ ቀ ቁൌ൬ ൰ This looks like a dilation of ሺ͵ǡ ͳሻ with scale factor Ͷ ݐ, so the
Ͷݐ ͳሺͶ ݐሻ
Ͷݐ Ͳ ͵
matrix representation of this transformation is ݃ሺݐሻ ൌ ቀ ቁ ቀ ቁ.
Ͳ Ͷͳ ݐ
Exercises 3–4 (3 minutes)
Give students a minute to perform the following exercises; monitor their responses. Then, present the solutions, and
ask students to check their answers.
࢚
3. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ, where ࢚ represents time, measured in seconds, and ࢌሺ࢚ሻ represents the position of a
࢚ ૠ
moving object at time ࢚.
a. Find the position of the object at ࢚ ൌ ǡ ࢚ ൌ ǡ and ࢚ ൌ .
ൈൈૠ
࢚ ൌ ǡ ቀ ቁቀ ቁ ൌ ቀ ቁ ൌ ቀ ቁ or
ૠ ൈൈૠ
ൈൈૠ
࢚ ൌ ǡ ቀ ቁቀ ቁ ൌ ቀ ቁ ൌ ቀ ቁ, or
ૠ ൈൈૠ
ൈൈૠ
࢚ ൌ ǡ ቀ ቁቀ ቁ ൌ ቀ ቁ ൌ ቀ ቁ, or ૡ
ૠ ൈൈૠ ૡ
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࢞ሺ࢚ሻ
b. Write ࢌሺ࢚ሻ in the form ൬ ൰.
࢟ሺ࢚ሻ
࢚ ࢚
ቀ ቁቀ ቁ ൌ ቀ ቁ
࢚ ૠ ૠ࢚
࢚
4. Write the transformation ࢍሺ࢚ሻ ൌ ቀ ቁ as a matrix transformation.
െ െ ࢚
Answers vary based on factoring of factors. However, they start at different points that are all from the line, and
࢚
they all end up having the same result: ࢍሺ࢚ሻ ൌ ቀ ቁ.
െ െ ࢚
࢚ ሺ ࢚ሻ ࢚ ࢚ െሺെ െ ࢚ሻ െ െ ࢚ െ
ቀ ቁൌ൬ ൰ൌቀ ቁ ቀ ቁ or ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
െ െ ࢚ െሺ ࢚ሻ ࢚ െ െ െ ࢚ ሺെ െ ࢚ሻ െ െ ࢚
Example 2 (4 minutes)
ͳെݐ Ͳ ͵
Let ݂ሺݐሻ ൌ ቀ ቁ ቀ ቁ.
Ͳ ͳെͳ ݐ
Graph the path traced out by ܲ ൌ ݂ሺݐሻ with Ͳ ݐ ͵.
à We have ݂ሺͲሻ ൌ ሺ͵ǡͳሻ, ݂ሺͳሻ ൌ ሺͲǡͲሻ, ݂ሺʹሻ ൌ ሺെ͵ǡ െͳሻ, and ݂ሺ͵ሻ ൌ ሺെǡ െʹሻ.
ݔሺݐሻ
Write ݂ሺݐሻ in the form ൬ ൰.
ݕሺݐሻ
ͳെݐ Ͳ ͵ ሺͳ െ ݐሻሺ͵ሻ Ͳ ͵ െ ͵ݐ
à ݂ሺݐሻ ൌ ቀ ቁቀ ቁ ൌ ൬ ൰ൌቀ ቁ
Ͳ ͳെͳ ݐ Ͳ ሺͳ െ ݐሻሺͳሻ ͳെݐ
Now suppose that an object starts at ሺʹͲǡ ͳሻ and moves along a line, reaching the origin in Ͷ seconds.
Write an equation ܲ ൌ ݄ሺݐሻ for the position of the object at time ݐ.
à Looking at the ݔ-coordinates, we see that ʹͲ െ ݇ሺͶሻ ൌ Ͳ, which means that ݇ ൌ ͷ. That is, the
ݔ-coordinate of the point is decreasing at ͷ units per second. Thus, ݔሺݐሻ ൌ ʹͲ െ ͷݐ.
àLooking at the ݕ-coordinates, we see that ͳ െ ݉ሺͶሻ ൌ Ͳ, which means that ݉ ൌ Ͷ. That is, the
ݕ-coordinate of the point is decreasing at Ͷ units per second. Thus, ݕሺݐሻ ൌ ͳ െ Ͷݐ.
ʹͲ െ ͷݐ
à Putting these two results together, we get ݄ሺݐሻ ൌ ቀ ቁ.
ͳ െ Ͷݐ
Write ݄ሺݐሻ as a matrix transformation.
ʹͲ െ ͷݐ ͷሺͶ െ ݐሻ Ͷെݐ Ͳ ͷ
à ݄ሺݐሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
ͳ െ Ͷݐ ͶሺͶ െ ݐሻ Ͳ Ͷെ ݐͶ
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Exercise 5 (2 minutes)
Give students a minute to perform the following exercise, and monitor their responses. Have students compare their
responses with a partner; then, select students to share their responses with the whole class.
5. An object is moving in a straight line from ሺૡǡ ሻ to the origin over a -second period of time. Find a function
࢞ሺ࢚ሻ
ࢌሺ࢚ሻ that gives the position of the object after ࢚ seconds. Write your answer in the form ࢌሺ࢚ሻ ൌ ൬ ൰, and then
࢟ሺ࢚ሻ
express ࢌሺ࢚ሻ as a matrix transformation.
For the ࢞-coordinates, we have ૡ െ ൌ , ൌ . The ࢞-coordinate of the point is decreasing at units per
second. Thus, ࢞ሺ࢚ሻ ൌ ૡ െ ࢚.
For the ࢟-coordinates, we have െ ൌ , ൌ . The ࢟-coordinate of the point is decreasing at units per
second. Thus, ࢞ሺ࢚ሻ ൌ െ ࢚.
ૡ െ ࢚ ሺ െ ࢚ሻ െ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
െ ࢚ ሺ െ ࢚ሻ െ࢚
Discussion (7 minutes): Translations
In a video game, the player controls a character named Steve. When Steve climbs a certain ladder, his vertical
position on the screen increases by ͷ units.
Let ሺݔǡ ݕሻ ՜ ܸሺݔǡ ݕሻ represent the change in Steve’s position when he climbs the ladder. The input represents
his position before climbing the ladder, and the output represents his position after climbing the ladder. Find
the outputs that correspond to each of the following inputs.
ሺ͵ǡͶሻ
à ሺ͵ǡͶሻ ՜ ሺ͵ǡͻሻ
ሺͳͲǡͳʹሻ
à ሺͳͲǡͳʹሻ ՜ ሺͳͲǡͳሻ
ሺǡʹͲሻ
à ሺǡʹͲሻ ՜ ሺǡʹͷሻ
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Let’s look more closely at that last input-output pair. Can you carefully explain the thinking that allowed you to
produce the output here?
à Climbing a ladder does not affect Steve’s horizontal position, so the ݔ-coordinate is still . To get the
new ݕ-coordinate, we add ͷ to ʹͲ, giving ʹͲ ͷ ൌ ʹͷ.
To reveal the underlying structure of this transformation, let’s write ሺǡ ʹͲሻ ՜ ሺǡ ʹͲ ͷሻ. Now let’s
generalize: What is the output that corresponds to a generic input ሺݔǡ ݕሻ?
à ሺݔǡ ݕሻ ՜ ሺݔǡ ݕ ͷሻ
Now let’s write the transformation using the column notation that we have found useful for our work that
ݔ ݔ
involves matrices: ቀݕቁ ՜ ቀ ݕ ͷቁ.
Next, let’s analyze horizontal motion. When the player presses the control pad to the right, Steve moves to the
right ͵ units per second.
Write a function rule that represents a translation that takes each point in the plane ͵ units to the right.
Practice using the column notation.
ݔ ݔ͵
à ቀ ݕቁ ՜ ൬ ൰
ݕ
When Steve jumps while running at super speed, he moves to a new location that is units to the right and
Ͷ units above where he started the jump.
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Write a function rule that represents the change in Steve’s position when he does a jump while running at super
speed. Use column notation.
ݔ ݔ
à ቀ ݕቁ ՜ ൬ ൰
ݕͶ
Do you think there is a way we can represent a translation as a matrix transformation? In particular, can we
ݔ ݔ ݔ ܽ ܾ ݔ
encode the transformation ቀݕቁ ՜ ൬ ൰ as a matrix mapping ቀݕቁ ՜ ቀ ቁ ቀ ቁ?
ݕͶ ܿ ݀ ݕ
Let students consider this question for a few moments.
Here’s a hint: If we take the point ሺͲǡͲሻ as an input, what output is produced in each transformation above?
ݔ ݔ Ͳ Ͳ
à The map ቀݕቁ ՜ ൬ ൰ takes ቀ ቁ ՜ ቀ ቁ ൌ ቀ ቁ.
ݕͶ Ͳ ͲͶ Ͷ
ݔ ܽ ܾ ݔ Ͳ ܽ ܾ Ͳ Ͳ
à On the other hand, the map ቀݕቁ ՜ ቀ ቁ ቀ ቁ takes ቀ ቁ ՜ ቀ ቁ ቀ ቁ ൌ ቀ ቁ.
ܿ ݀ ݕ Ͳ ܿ ݀ Ͳ Ͳ
A matrix transformation always maps the origin to itself, whereas a translation shifts every point in the plane, including
the origin. Thus, there is no way to encode a translation as a matrix transformation.
Exercises 6–9 (2 minutes)
Give students a minute to perform the following exercise, and monitor their responses. Have students compare their
responses with a partner; then, select students to share their responses with the whole class.
6. Write a rule for the function that shifts every point in the plane units to the left.
࢞ ࢞െ ࢞െ
ቀ࢟ቁ ՜ ൬ ൰ ǡ ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰
࢟ ࢟
7. Write a rule for the function that shifts every point in the plane ૢ units upward.
࢞ ࢞ ࢞
ቀ࢟ቁ ՜ ቀ࢟ ૢቁ ǡ ࢌሺ࢞ǡ ࢟ሻ ൌ ቀ࢟ ૢቁ
8. Write a rule for the function that shifts every point in the plane units down and units to the right.
࢞ ࢞ ࢞
ቀ࢟ቁ ՜ ൬ ൰ ǡ ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰
࢟ െ ࢟ െ
࢞ ࢞െૠ
9. Consider the rule ቀ࢟ቁ ՜ ൬ ൰. Describe the effect this transformation has on the plane.
࢟
Every point in the plane is shifted ૠ units to the left and units upward.
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Closing (2 minutes)
Have students take a minute to write a response to the following questions in their notebooks; then, ask them to share
their responses with a partner. Select two students to share their responses with the whole class.
What did you learn today about representing straight-line motion? Give an example of a function that
represents this kind of motion.
ͳͲ െ ʹݐ
à An example would be ݂ሺݐሻ ൌ ቀ ቁǤ
͵ ͷݐ
What did you learn about representing translations? Give an example of a function that represents this kind of
motion.
ݔ ͵ ݔ͵
à An example would be ݂ሺݔǡ ݕሻ ൌ ቀݕቁ ቀ ቁ ൌ ൬ ൰.
Ͷ ݕͶ
Exit Ticket (8 minutes)
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Name Date
Lesson 22: Modeling Video Game Motion with Matrices
Exit Ticket
ݐͷ
1. Consider the function ݄ሺݐሻ ൌ ቀ ቁ. Draw the path that the point ܲ ൌ ݄ሺݐሻ traces out as ݐvaries within the
ݐെ͵
interval Ͳ ݐ Ͷ.
ݐ Ͳ ͷ
2. The position of an object is given by the function ݂ሺݐሻ ൌ ቀ ቁ ቀ ቁ, where ݐis measured in seconds.
Ͳ ʹ ݐ
ݔሺݐሻ
a. Write ݂ሺݐሻ in the form ൬ ൰.
ݕሺݐሻ
b. Find how fast the object is moving in the horizontal direction and in the vertical direction.
3. Write a function ݂ሺݔǡ ݕሻ, which will translate all points in the plane ʹ units to the left and ͷ units downward.
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Exit Ticket Sample Solutions
࢚
1. Consider the function ࢎሺ࢚ሻ ൌ ቀ ቁ. Draw the path that the point ࡼ ൌ ࢎሺ࢚ሻ traces out as ࢚ varies within the
࢚െ
interval ࢚ .
ૠ ૡ ૢ
ࢎሺሻ ൌ ቀ ቁ ǡ ࢎሺሻ ൌ ቀ ቁ ǡ ࢎሺሻ ൌ ቀ ቁ ǡ ࢎሺሻ ൌ ቀ ቁ ǡ ࢎሺሻ ൌ ቀ ቁ
െ െ െ
࢚
2. The position of an object is given by the function ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ, where ࢚ is measured in seconds.
࢚
࢞ሺ࢚ሻ
a. Write ࢌሺ࢚ሻ in the form ൬ ൰.
࢟ሺ࢚ሻ
࢚ ࢚ ൈ ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ ൌ ቀ ቁ Ǥ ࢌሺሻ ൌ ቀ ቁ ǡ ࢌሺሻ ൌ ቀ ቁ, and the slope of the line is
࢚ ൈ࢚ൈ ࢚
ൌ .
a. Find how fast the object is moving in the horizontal direction and in the vertical direction.
The object is moving units upward vertically per second and units to the right horizontally per second.
3. Write a function ࢌሺ࢞ǡ ࢟ሻǡ which will translate all points in the plane units to the left and units downward.
࢞െ
ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰
࢟െ
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Problem Set Sample Solutions
࢞ ࢞
1. Let ࡰ ቀ࢟ቁ ൌ ቀ ቁ ቀ ቁ. Find and plot the following.
࢟
െ െ
a. Plot the point ቀ ቁǡ and find ࡰ ቀ ቁ and plot it.
െ െ െ
ࡰቀ ቁൌቀ ቁቀ ቁ ൌ ቀ ቁ
b. Plot the point ቀ ቁǡ and find ࡰ ቀ ቁ and plot it.
ࡰቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
ૡ
c. Plot the point ቀ ቁǡ and find ࡰ ቀ ቁ and plot it.
ࡰቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
࢚ െ
2. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ. Find ࢌሺሻǡ ࢌሺሻǡ ࢌሺሻǡ ࢌሺሻǡ and plot them on the same graph.
࢚
െ
ࢌሺሻ ൌ ቀ ቁ ቀ ቁ ൌ ቀ ቁǡ
െ െ
ࢌሺሻ ൌ ቀ ቁ ቀ ቁ ൌ ቀ ቁǡ
െ െ
ࢌሺሻ ൌ ቀ ቁ ቀ ቁ ൌ ቀ ቁǡ
െ െ
ࢌሺሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
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࢚
3. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ represent the location of an object at time ࢚ that is measured in seconds.
࢚
a. How long does it take the object to travel from the origin to the point ቀ ቁ?
ૡ
࢚ ൈ ൌ ǡ ࢚ ൌ or ൈ ࢚ ൌ ૡǡ ࢚ ൌ
b. Find the speed of the object in the horizontal direction and in the vertical direction.
࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ The object is moving units upward per second and units to the right per second.
࢚
Ǥ ࢚ ࢚
4. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ ǡ ࢎሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ. Which one will reach the point ቀ ቁ first? The time ࢚ is
Ǥ ࢚ ࢚ ૡ
measured in seconds.
For ࢌሺ࢚ሻǡ Ǥ ࢚ ൈ ൈ ൌ ǡ ࢚ ൌ ൌ seconds.
Ǥ
For ࢎሺ࢚ሻǡ ࢚ ൈ ൈ ൌ ǡ ࢚ ൌ ൌ seconds; therefore, ࢎሺ࢚ሻ will reach the point ቀ ቁ first.
ૡ
࢚ െ
5. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ Ǥ۴ind the value of that moves the object from the origin to ቀ ቁ in seconds.
࢚ െ
െ
ࢌሺሻ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ ǡ ൈ ൈ ൌ െǡ ൌ െ or ൈ ൈ ൌ െǡ ൌ െ
െ
࢞ሺ࢚ሻ
6. Write ࢌሺ࢚ሻ in the form ൬ ൰ if
࢟ሺ࢚ሻ
࢚
a. ࢌሺ࢚ሻ ൌ ቀ ቁቀ ቁ
࢚
࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ
࢚
࢚
b. ࢌሺ࢚ሻ ൌ ቀ ቁቀ ቁ
࢚
࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ
࢚
࢚
െ
c. ࢌሺ࢚ሻ ൌ ቌ ࢚ ቍቀ ቁ
െ െ
࢚ െ
ࢌሺ࢚ሻ ൌ ቀ ቁ
࢚ െ ૡ
࢚
7. Let ࢌሺ࢚ሻ ൌ ቀ ቁ ቀ ቁ represent the location of an object after ࢚ seconds.
࢚
a. If the object starts at ቀ ቁ, how long would it take to reach ቀ ቁ?
ૡ
࢚ ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ Ǣ it starts at ቀ ቁ; therefore, ࢌሺ࢚ሻ ൌ ቀ ቁ.
࢚ ࢚
࢚ ൌ ǡ ࢚ ൌ or ࢚ ൌ ૡǡ ࢚ ൌ
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b. Write the new function ࢌሺ࢚ሻ that gives the position of the object after ࢚ seconds.
࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ
࢚
c. Write ࢌሺ࢚ሻ as a matrix transformation.
࢚ ሺ࢚ ሻ ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
࢚ ሺ࢚ ሻ ࢚
࢚ ሺെ࢚ െ ሻሺെሻ െ࢚ െ െ
or ࢌሺ࢚ሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
࢚ ሺെ࢚ െ ሻሺെሻ െ࢚ െ െ
The answers vary; it depends on how the factoring is applied.
8. Write the following functions as a matrix transformation.
࢚
a. ࢌሺ࢚ሻ ൌ ቀ ቁ
࢚
࢚ ሺ ࢚ሻ ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
࢚ ሺ ࢚ሻ ࢚
െ࢚
b. ࢌሺ࢚ሻ ൌ ቀ ቁ
ૡ࢚ െ
െ࢚ ሺ࢚ െ ሻሺെሻ ࢚ െ െ
ࢌሺ࢚ሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
ૡ࢚ െ ሺ࢚ െ ሻ ࢚ െ
െ࢚ ሺെ࢚ ሻ െ࢚
or ࢌሺ࢚ሻ ൌ ቀ ቁൌ൬ ൰ൌቀ ቁቀ ቁ
ૡ࢚ െ ሺെ࢚ ሻሺെሻ െ࢚ െ
࢞
9. Write a function rule that represents the change in position of the point ቀ࢟ቁ for the following.
a. units to the right and units downward
࢞
ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰
࢟െ
b. units downward and units to the left
࢞െ
ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰
࢟െ
c. units upward, units to the left, and then it dilates by Ǥ
࢞െ ࢞െ
ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰ ǡ ࢌሺ࢞ǡ ࢟ሻ ൌ ቀ ቁ൬ ൰
࢟ ࢟
࣊
d. units upward, units to the left, and then it rotates by counterclockwise.
࣊ ࣊
࢞െ ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢞ െ
ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰ ǡ ࢌሺ࢞ǡ ࢟ሻ ൌ ቌ ࣊ ࣊ ቍ ൬࢟ ൰
࢟ ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
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࢞
10. Annie is designing a video game and wants her main character to be able to move from any given point ቀ࢟ቁ in the
following ways: right unit, jump up unit, and both jump up and move right unit each.
a. What function rules can she use to represent each time the character moves?
࢞ ࢞ ࢞
ࢌሺ࢞ǡ ࢟ሻ ൌ ൬ ൰; ࢍሺ࢞ǡ ࢟ሻ ൌ ቀ࢟ ቁ; ࢎሺ࢞ǡ ࢟ሻ ൌ ൬ ൰
࢟ ࢟
b. Annie is also developing a ski slope stage for her game and wants to model her character’s position using
െ
matrix transformations. Annie wants the player to start at ቀ ቁ and eventually pass through the origin
moving units per second down. How fast does the player need to move to the right in order to pass
through the origin? What matrix transformation can Annie use to describe the movement of the character?
If the far right of the screen is at ࢞ ൌ , how long until the player moves off the screen traveling this path?
If the player is moving units per second down, then she will reach ࢟ ൌ in ࢚ ൌ seconds. Thus, the player
needs to move units per second to the right.
െ ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ
െ ࢚
െሺ െ ࢚ሻ
ൌ൬ ൰
ሺ െ ࢚ሻ
െ࢚ െ
ൌቀ ቁቀ ቁ
െ࢚
The player will leave the screen in seconds.
11. Remy thinks that he has developed matrix transformations to model the movements of Annie’s characters in
࢞
Problem 10 from any given point ቀ࢟ቁ, and he has tested them on the point ቀ ቁ. This is the work Remy did on the
transformations:
ቀ ቁቀ ቁ ൌ ቀ ቁ ቀ ቁቀ ቁ ൌ ቀ ቁ ቀ ቁ ቀ ቁ ൌ ቀ ቁ.
Do these matrix transformations accomplish the movements that Annie wants to program into the game? Explain
why or why not.
These do not accomplish the movements. If we apply the transformations to any other point in the plane, then they
will not produce the same results of moving one unit to the right, one unit up, and one unit up and right.
As a counterexample, any of the three matrix transformations applied to the origin do nothing.
12. Nolan has been working on how to know when the path of a point can be described with matrix transformations
and how to know when it requires translations and cannot be described with matrix transformations. So far, he has
been focusing on the following two functions, which both pass through the point ሺǡ ሻ:
࢚ ࢚
ࢌሺ࢚ሻ ൌ ቀ ቁ and ࢍሺ࢚ሻ ൌ ቀ ቁ.
࢚ ࢚
a. If we simplify these functions algebraically, how does the rule for ࢌ differ from the rule for ࢍ? What does this
say about which function can be expressed with matrix transformations?
ሺ࢚ ሻ
ࢌሺ࢚ሻ ൌ ൬ ൰ Thus, there is a common factor in both the ࢞- and ࢟-coordinate. Because there is a
ሺ࢚ ሻ
common factor, we can pull the factor out as a scalar and rewrite the scalar as a matrix multiplication. ࢍሺ࢚ሻ
does not have a common factor (other than ) between the ࢞- and ࢟-coordinate.
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b. Nolan has noticed that functions that can be expressed with matrix transformations always pass through the
origin; does either ࢌ or ࢍ pass through the origin, and does this support or contradict Nolan’s reasoning?
At ࢚ ൌ െ, the graph of ࢌ passes through the origin. On the other hand, the graph of ࢍ crosses the ࢞-axis at
࢚ ൌ െ and the ࢟-axis at ࢚ ൌ െ, so it does not pass through the origin. This seems to support Nolan’s
reasoning. This agrees with our response to part (a), since the common factor has the same zero and causes
the function to cross the origin.
c. Summarize the results of parts (a) and (b) to describe how we can tell from the equation for a function or
from the graph of a function that it can be expressed with matrix transformations.
If a function has a common factor involving ࢚ that can be pulled out of both the ࢞- and ࢟-coordinates, then the
function can be represented as a matrix transformation. If the graph of the function passes through the
origin, then the function can be represented as a matrix transformation.
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Lesson 23: Modeling Video Game Motion with Matrices
Student Outcomes
Students use matrix transformations to model circular motion.
Students use coordinate transformations to represent a combination of motions.
Lesson Notes
Students have recently learned how to represent rotations as matrix transformations. In this lesson, they apply that
knowledge to represent dynamic motion, as seen in video games. Students analyze circular motion that involves a time
ߨ ߨ
ቀ ή ݐቁ െ ቀ ή ݐቁ
ʹ ʹ
parameter such as ܩሺݐሻ ൌ ቌ ߨ ߨ ቍ. The second part of the lesson involves modeling a combination
ቀ ή ݐቁ ቀ ή ݐቁ
ʹ ʹ
of motions. For instance, students model motion along a circle followed by a translation or motion along a line followed
by a translation.
Classwork
The Opening Exercise allows students to practice matrix transformations and plot the results. This prepares students for
skills needed in this lesson. Work through this as a whole class, asking questions to assess student understanding.
Use this as a way to clear up misconceptions.
Opening Exercise (5 minutes)
Opening Exercise
࣊ ࣊
࢞ ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢞
Let ࡾ ቀ࢟ቁ ൌ ቌ ࣊ ࣊ ቍ ቀ࢟ቁ.
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
࢞ ࢞
a. Describe the geometric effect of performing the transformation ቀ࢟ቁ ՜ ࡾ ቀ࢟ቁ.
࣊
Applying ࡾ rotates each point in the plane about the origin through radians in a counterclockwise direction.
b. Plot the point ቀ ቁ, and then find ࡾ ቀ ቁ and plot it.
࣊
ܛܗ܋ቀ ቁ
ࡾቀ ቁ ൌ ൮ ࣊ ൲
ܖܑܛቀ ቁ
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c. If we want to show that ࡾ has been applied twice to ሺǡ ሻ, we can write ࡾ ቀ ቁ, which represents
ࡾ ቆࡾ ቀ ቁቇ. Find ࡾ ቀ ቁ and plot it. Then, find ࡾ ቀ ቁ ൌ ࡾ ൭ࡾ ቆࡾ ቀ ቁቇ൱, and plot it.
࣊
ܛܗ܋ቀ ቁ ܛܗ܋ሺ࣊ሻ
ࡾ ቀ ቁ ൌ ቌ ቍ; ࡾ ቀ ቁ ൌ ൬ ൰
࣊ ܖܑܛሺ࣊ሻ
ܖܑܛቀ ቁ
࢞ ࢞
d. Describe the matrix transformation ቀ࢟ቁ ՜ ࡾ ቀ࢟ቁ using a single matrix.
࢞ ࣊ ࢞
ࡾ ቀ࢟ቁ is the transformation that rotates points through ή radians, so a formula for ࡾ ቀ࢟ቁ is
࣊ ࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢞
ቌ ࣊ ࣊
ቍ ቀ࢟ቁ.
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
Discussion (3 minutes): Circular Motion over Time
ሺݐሻ െሺݐሻ ͳ
Let ܴሺݐሻ ൌ ൬ ൰ ቀ ቁ.
ሺݐሻ ሺݐሻ Ͳ
Suppose that ݐis measured in degrees. Let’s place several input-output pairs for this function on a graph:
ܴሺ͵Ͳሻ ൌ ሺͲǤͺǡͲǤͷͲሻ ܴሺͶͷሻ ൌ ሺͲǤͳǡͲǤͳሻ ܴሺͲሻ ൌ ሺͲǤͷͲǡͲǤͺሻ
ܴሺͻͲሻ ൌ ሺͲǡͳሻ ܴሺͳͺͲሻ ൌ ሺെͳǡͲሻ ܴሺʹͲሻ ൌ ሺͲǡ െͳሻ ܴሺ͵Ͳሻ ൌ ሺͳǡͲሻ
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What do you notice about the points on the graph?
à The points appear to lie on a circle.
How could we be sure the points are actually on a circle?
à If each point is the same distance from the origin, then the points form a circle.
Now check and see if this is, in fact, the case. Have different students find the distance to the origin from given
points.
à Students check the distance from a given point to the origin and confirm using the distance formula.
For example, ܴሺ͵Ͳሻ ൌ ሺͲǤͺǡ ͲǤͷͲሻ. Its distance from the origin is ඥሺͲǤͺ െ Ͳሻଶ ሺͲǤͷ െ Ͳሻଶ ൌ ͳ.
What is the result of ሺሻ?
ሺݐሻ െሺݐሻ ͳ ሺݐሻ
à ܴሺݐሻ ൌ ൬ ൰ቀ ቁ ൌ ൬ ൰
ሺݐሻ ሺݐሻ Ͳ ሺݐሻ
Does this result ensure points lie on a circle?
à Yes, this would confirm that the points lie on the unit circle because the ݔ-valuecorresponds to cosine
of an angle ݐand the ݕ-value corresponds to sine of the same angle on the unit circle.
Exercise 1 (4 minutes)
This exercise provides students more practice with matrices representing rotations. This time, the angle is different in
each function, allowing them to compare the results. Give students time to work on the following problems
independently; then, call on students to share their responses with the class.
Exercises
࢚ ࢚
ܛܗ܋ሺ࢚ሻ െܖܑܛሺ࢚ሻ ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
1. Let ࢌሺ࢚ሻ ൌ ൬ ൰ ቀ ቁ, and let ࢍሺ࢚ሻ ൌ ቌ ࢚ ࢚ ቍ ቀቁ.
ܖܑܛሺ࢚ሻ ܛܗ܋ሺ࢚ሻ ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
a. Suppose ࢌሺ࢚ሻ represents the position of a moving object that starts at ሺǡ ሻ. How long does it take for this
object to return to its starting point? When the argument of the trigonometric function changes from ࢚ to ࢚,
what effect does this have?
The object will return to ሺǡ ሻ when ࢚ ൌ ࣊. Thus, it will take ࢚ ൌ ࣊ seconds for this to happen. Changing
the argument from ࢚ to ࢚ causes the object to move twice as fast.
b. If the position is given instead by ࢍሺ࢚ሻ,how long would it take the object to return to its starting point?
࢚
When the argument of the trigonometric function changes from ࢚ to , what effect does this have?
࢚
The object will return to ሺǡ ሻ when ൌ ࣊. Thus, it will take ࢚ ൌ ࣊ seconds for this to happen.
࢚
Changing the argument from ࢚ to causes the object to move half as fast.
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Example 1 (4 minutes)
ሺݐሻ െሺݐሻ ͵
Let ܨሺݐሻ ൌ ൬ ൰ ቀ ቁ.
ሺݐሻ ሺݐሻ ʹ
ߨ ͵ߨ
This time, we’ll measure ݐin radians. Find ܨቀ ቁ, ܨሺߨሻ, ܨቀ ቁ, and ܨሺʹߨሻ.
ʹ ʹ
ߨ ߨ
ߨ ቀ ቁ െ ቀ ቁ ͵ Ͳ െͳ ͵ Ͳെʹ െʹ
ʹ ʹ
à ܨቀ ቁ ൌቌ ߨ ߨ ቍ ቀʹቁ ൌ ቀͳ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
ʹ Ͳ ʹ ͵Ͳ ͵
ቀ ቁ ቀ ቁ
ʹ ʹ
ሺɎሻ െ ሺߨሻ ͵ െͳ Ͳ ͵ െ͵ Ͳ െ͵
à ܨሺߨሻ ൌ ൬ ൰ቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
ሺߨሻ ሺߨሻ ʹ Ͳ െͳ ʹ Ͳെʹ െʹ
͵ߨ ͵ߨ
͵ߨ ቀ ቁ െ ቀ ቁ ͵ Ͳ ͳ ͵ Ͳʹ ʹ
ʹ ʹ
à ܨቀ ቁ ൌቌ ͵ߨ ͵ߨ
ቍቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
ʹ ʹ െͳ Ͳ ʹ െ͵ Ͳ െ͵
ቀ ቁ ቀ ቁ
ʹ ʹ
ሺʹɎሻ െ ሺʹߨሻ ͵ ͳ Ͳ ͵ ͵Ͳ ͵
à ܨሺʹߨሻ ൌ ൬ ൰ቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁൌቀ ቁ
ሺʹߨሻ ሺʹߨሻ ʹ Ͳ ͳ ʹ Ͳʹ ʹ
When we plot points, we see once again that they appear to lie on a circle. Make sure this is really true.
ሺݐሻ െሺݐሻ ͵ ͵ԝ ሺݐሻ െ ʹԝሺݐሻ
à ܨሺݐሻ ൌ ൬ ൰ቀ ቁ ൌ ൬ ൰
ሺݐሻ ሺݐሻ ʹ ͵ԝሺݐሻ ʹԝ ሺݐሻ
ሺ͵ԝ ሺݐሻ െ ʹԝሺݐሻሻଶ ሺԝ͵ԝሺݐሻ ʹԝ ሺݐሻሻଶ
ͻԝ ଶሺݐሻ െ ͳʹԝ ሺݐሻ ሺݐሻ Ͷԝଶ ሺݐሻ ͻԝଶ ሺݐሻ ͳʹԝ ሺݐሻ ሺݐሻ Ͷԝ ଶሺݐሻ
ͻሺ ଶሺݐሻ ଶ ሺݐሻሻ Ͷሺଶ ሺݐሻ ଶ ሺݐሻሻ
ͻሺͳሻ Ͷሺͳሻ ൌ ͻ Ͷ ൌ ͳ͵
Thus, each point is ξͳ͵ units from the origin, which confirms that the outputs lie on a circle.
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Discussion (4 minutes): Rotations That Use a Time Parameter
ߨ ߨ
ቀ ή ݐቁ െ ቀ ή ݐቁ ͳ
ʹ ʹ
Let ܨሺݐሻ ൌ ቌ ߨ ߨ ቍ ቀ ቁ.
ቀ ή ݐቁ ቀ ή ݐቁ Ͳ
ʹ ʹ
Draw the path that ܲ ൌ ܨሺݐሻ traces out as ݐvaries within each of the following intervals:
à Ͳݐͳ
à ͳݐʹ
à ʹݐ͵
à ͵ݐͶ
Where will the object be located at ݐൌ ͲǤͷ second?
ߨ ߨ
à ݂ሺͲǤͷሻ ൌ ቀ ቀ ቁ ǡ ቀ ቁቁ ൎ ሺͲǤͳǡͲǤͳሻ
Ͷ Ͷ
How long will it take the object to reach ሺͲǤͳǡ െͲǤͳሻ?
ߨ ߨ ߨ ߨ
à These coordinates represent ቀ ቀ ቁ ǡ ቀ ቁቁ, so ቀ ቀ ή ቁ ǡ ቀ ή ቁቁ. The object reaches this
Ͷ Ͷ ʹ ʹ ʹ ʹ
location at ݐൌ ൌ ͵Ǥͷ seconds.
ʹ
Exercises 2–3 (5 minutes)
Give students time to complete the following exercises; then, ask them to compare their responses with a partner.
Call on students to share their responses with the class.
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࣊ ࣊
ܛܗ܋ቀ ή ࢚ቁ െ ܖܑܛቀ ή ࢚ቁ
2. Let ࡳሺ࢚ሻ ൌ ቌ ࣊ ࣊ ቍ ቀ ቁ.
ܖܑܛቀ ή ࢚ቁ ܛܗ܋ቀ ή ࢚ቁ
a. Draw the path that ࡼ ൌ ࡳሺ࢚ሻ traces out as ࢚ varies within the interval ࢚ .
b. Where will the object be at ࢚ ൌ seconds?
ࡳሺሻ ൌ ሺǡ ሻ
c. How long will it take the object to reach ሺǡ െሻ?
࣊ ࣊
These coordinates represent ሺܛܗ܋ሺ࣊ሻ ǡ ܖܑܛሺ࣊ሻሻ, so ቀ ܛܗ܋ቀ ή ቁ , ܖܑܛቀ ή ቁቁ; the object reaches this location
at ࢚ ൌ seconds. ࡳሺሻ ൌ ሺǡ െሻ, so it will take seconds to reach that location.
࣊ ࣊
ܛܗ܋ቀ ή ࢚ቁ െ ܖܑܛቀ ή ࢚ቁ
3. Let ࡴሺ࢚ሻ ൌ ቌ ࣊ ࣊ ቍ ቀ ቁ.
ܖܑܛቀ ή ࢚ቁ ܛܗ܋ቀ ή ࢚ቁ
a. Draw the path that ࡼ ൌ ࡴሺ࢚ሻ traces out as ࢚ varies within the interval ࢚ .
b. Where will the object be at ࢚ ൌ second?
ࡴሺሻ ൌ ሺെǡ ሻ
c. How long will it take the object to return to its starting point?
ࡴሺሻ ൌ ሺǡ ሻ, so it will take seconds to return to its starting point.
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Example 2 (4 minutes)
ߨ ߨ
ቀ ή ݐቁ െ ቀ ή ݐቁ
Let ݂ሺݐሻ ൌ ቌ ߨ ߨ ቍ ቀ ቁ.
ቀ ή ݐቁ ቀ ή ݐቁ ʹ
Draw the path that ܲ ൌ ݂ሺݐሻ traces out as ݐvaries within each of the following intervals:
à Ͳݐͳ ͳݐʹ ʹݐ͵
à ͵ݐͶ Ͷݐͷ ͷݐ
As an example, can you describe what happens to the object as ݐvaries within the interval Ͳ ݐ ͳ?
à Since ݂ሺͲሻ ൌ ሺǡʹሻ, the object starts its trajectory there. When ݐൌ ͳ, the object will have moved
గ
through radians. So, in the time interval Ͳ ݐ ͳ, the object moves along a circular arc as shown
below.
Exercises 4–5 (3 minutes)
Give students time to complete the following exercises; then ask them to compare their responses with a partner.
Call on students to share their responses with the class, and use this as an opportunity to check for understanding.
4. Suppose you want to write a program that takes the point ሺǡ ሻ and rotates it about the origin to the point
ሺെǡ െሻ over a -second interval. Write a function ࡼ ൌ ࢌሺ࢚ሻ that encodes this rotation.
ܛܗ܋ሺ࣊ ή ࢚ሻ െܖܑܛሺ࣊ ή ࢚ሻ
Let ࢌሺ࢚ሻ ൌ ൬ ൰ ቀ ቁ. We have ࢌሺሻ ൌ ሺǡ ሻ and ࢌሺሻ ൌ ሺെǡ െሻ, as required.
ܖܑܛሺ࣊ ή ࢚ሻ ܛܗ܋ሺ࣊ ή ࢚ሻ
5. If instead you wanted the rotation to take place over a Ǥ -second interval, how would your function change?
࢚ ࢚
ܛܗ܋ቀ࣊ ή ή ቁ െ ܖܑܛቀ࣊ ή ή ቁ
Let ࢌሺ࢚ሻ ൌ ቌ ࢚ ࢚ ቍ ቀ ቁ.We have ࢌሺሻ ൌ ሺǡ ሻ and ࢌሺǤ ሻ ൌ ሺെǡ െሻ, as required.
ܖܑܛቀ࣊ ή ή ቁ ܛܗ܋ቀ࣊ ή ή ቁ
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Example 3 (4 minutes)
ߨ ߨ
͵ ቀ ή ݐቁ െ͵ ቀ ή ݐቁ ͳ
ʹ ʹ
Let’s analyze the transformation ݃ሺݐሻ ൌ ቌ ߨ ߨ ቍ ቀ ቁ. In particular, we will compare ݃ሺͲሻ
͵ ቀ ή ݐቁ ͵ ቀ ή ݐቁ ʹ
ʹ ʹ
and ݃ሺͳሻ.
What is ݃ሺͲሻ? What geometric effect does ݃ሺݐሻ have on ሺͳǡ ʹሻ initially?
à We have ݃ሺͲሻ ൌ ሺ͵ǡ ሻ, which is a dilation of ሺͳǡ ʹሻ using scale factor ͵.
What is ݃ሺͳሻ? Describe what is going on.
à We have ݃ሺͳሻ ൌ ሺെǡ ͵ሻ, which represents a quarter turn of the point ሺ͵ǡ ሻ about the origin in a
counterclockwise direction.
Can you summarize the geometric effect of applying ݃ሺݐሻ to the point ሺͳǡ ʹሻ during the time interval
Ͳ ݐ ͳ?
à This transformation combines a quarter turn about the origin with a scaling by a factor of ͵.
What is ݃ሺʹሻ? Describe what is going on.
à We have ݃ሺʹሻ ൌ ሺെ͵ǡ െሻ, which represents a quarter turn of the point ሺെǡ ͵ሻ about the origin in a
counterclockwise direction.
What is ݃ሺ͵ሻ? Describe what is going on.
à We have ݃ሺ͵ሻ ൌ ሺǡ െ͵ሻ, which represents a quarter turn of the point ሺെ͵ǡ െሻ about the origin in a
counterclockwise direction.
What is ݃ሺͶሻ? Describe what is going on.
à We have ݃ሺͳሻ ൌ ሺ͵ǡ ሻ, which represents a quarter turn of the point ሺǡ െ͵ሻ about the origin in a
counterclockwise direction.
Compare ݃ሺͲሻ and ݃ሺͶሻ. Does this make sense?
à ݃ሺͲሻ ൌ ݃ሺͶሻ This makes sense because 4 quarter turns would be a full rotation, so this would bring
you back to the starting point.
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Closing (4 minutes)
Write one to two sentences in your notebook describing what you learned in today’s lesson; then, share your
response with a partner.
à We learned how to use matrices to describe rotations that happen over a specific time interval. We
also discussed how to model multiple transformations, such as a rotation followed by a translation.
Exit Ticket (5 minutes)
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Name Date
Lesson 23: Modeling Video Game Motion with Matrices
Exit Ticket
Write a function ݂ሺݐሻ that incorporates the following actions. Make a drawing of the path the point follows during the
time interval Ͳ ݐ ͵.
గ
a. During the time interval Ͳ ݐ ͳ, move the point ሺͺǡ ሻ through radians about the origin in a
ସ
counterclockwise direction.
b. During the time interval ͳ ൏ ݐ ͵, move the image along a straight line to ሺǡ െͺሻ.
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Exit Ticket Sample Solutions
Write a function ࢌሺ࢚ሻ that incorporates the following actions. Make a drawing of the path the point follows during the
time interval ࢚ .
࣊
a. During the time interval ࢚ , move the point ሺૡǡ ሻ through radians about the origin in a
counterclockwise direction.
࢚࣊ ࢚࣊
ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰
ૡ
ࢌሺ࢚ሻ ൌ ൮ ࢚࣊ ࢚࣊ ൲ ቀቁ ǡ ࢚
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰
ܛܗ܋ሺሻ െܖܑܛሺሻ ૡ ૡ ૡ
ࢌሺሻ ൌ ൬ ൰ቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ
ܖܑܛሺሻ ܛܗ܋ሺሻ
࣊ ࣊ ξ െξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ۇ
ࢌሺሻ ൌ ൮ ૡ ۊቀૡቁ ൌ ൬ ξ ൰ ൎ ቀǤ ቁ
࣊ ࣊ ൲ ቀቁ ൌ ۈ ۋ ૠξ ૢǤ ૢ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ ξ
ۉ ی
b. During the time interval ൏ ࢚ , move the image along a straight line to ሺǡ െૡሻ.
The image is ൬ ξ ൰ ՜ ቀ ቁ in seconds from ൏ ࢚ .
ૠξ െૡ
ξ െ ࢚ ൌ ξૠ െ ࢚ ൌ െૡ
ξ െ ൌ ૠξ െ ൌ െૡ
ξ െ ૠξ ૡ
ൌ ൌ
൫ξ െ ൯࢚
ۇξ െ ۊ
ࢎሺ࢚ሻ ൌ ۈ
൫ૠξ ૡ൯࢚ۋ
ૠξ െ
ۉ ی
Problem Set Sample Solutions
࣊ ࣊
࢞ ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢞
1. Let ࡾ ቀ࢟ቁ ൌ ቌ ࣊ ࣊
ቍ ቀ࢟ቁ. Find the following.
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
a. ࡾ ൬ξ൰
ξ
࣊ ࣊ ξ ξ
ܛܗ܋ ۇቀ ቁ െ ܖܑܛቀ ቁ ξ ۊ ۇۇ െ ۊ ۊ
ࡾ ൬ξ൰ ൌ ࡾ ൭ࡾ ൬ξ൰൱ ൌ ࡾ ۈ൮ ࣊ ࣊ ൲ ൬ ൰ۋ ൌ ࡾ ۈۈ
ۈ
൬ξ൰
ۋ ۋ
ۋ
ξ ξ ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ ξ ξ ξ
ۉ ی ۉۉ ی ی
࣊ ࣊ ξ ξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ۇ െ ۊ
െξ൰
ൌ ࡾቀ ቁ ൌ ൮ ࣊ ࣊ ൲ ቀቁ ൌ ۈ ۋቀቁ ൌ ൬
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ ξ ξ
ۉ ی
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b. How many transformations do you need to take so that the image returns to where it started?
࣊
It rotates by radians for each transformation; therefore, it takes ૡ times to get to ࣊, which is where it
started.
࢞ ࢞ ࢞
c. Describe the matrix transformation ቀ࢟ቁ ՜ ࡾ ቀ࢟ቁand ࡾ ቀ࢟ቁ using a single matrix.
࢞ ࣊ ࢞
ࡾ ൌ ቀ࢟ቁ is the transformation that rotates the point through ൈ radian, so a formula forࡾ ቀ࢟ቁ is
࣊ ࣊ ࣊ ૈ ࣊ ࣊
ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰ ࢞ ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢞ ࢞ ή ܛܗ܋ቀ ቁ െ ࢟ ή ܖܑܛቀ ቁ
൮ ൲ ቀ࢟ቁ ൌ ൮
࣊ ࣊ ࣊ ࣊ ൲ ቀ࢟ቁ ൌ ൮ ࣊ ࣊ ൲.
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰ ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ࢞ ή ܖܑܛቀ ቁ ࢟ ή ܛܗ܋ቀ ቁ
࣊ ࣊
࢞ ࢞ ή ܛܗ܋ቀ ቁ െ ࢟ ή ܖܑܛቀ ቁ
ࡾ ൌ ቀ࢟ቁ is ൮
࣊ ࣊ ൲ .
࢞ ή ܖܑܛቀ ቁ ࢟ ή ܛܗ܋ቀ ቁ
ܛܗ܋ሺ࢚ሻ െܖܑܛሺ࢚ሻ
2. For ࢌሺ࢚ሻ ൌ ൬ ൰ ቀ ቁ, it takes ࣊ to transform the object at ቀ ቁ back to where it starts. How long
ܖܑܛሺ࢚ሻ ܛܗ܋ሺ࢚ሻ
does it take the following functions to return to their starting point?
ܛܗ܋ሺ࢚ሻ െܖܑܛሺ࢚ሻ
a. ࢌሺ࢚ሻ ൌ ൬ ൰ቀ ቁ
ܖܑܛሺ࢚ሻ ܛܗ܋ሺ࢚ሻ
࣊
࢚ ൌ ࣊ǡ ࢚ൌ
࢚ ࢚
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
b. ࢌሺ࢚ሻ ൌ ቌ ࢚ ࢚
ቍ ቀ ቁ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
࢚
ൌ ࣊ǡ ࢚ ൌ ࣊
࢚ ࢚
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
c. ࢌሺ࢚ሻ ൌ ቌ ቍቀ ቁ
࢚ ࢚
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
࢚
ൌ ࣊ǡ ࢚ ൌ ࣊
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ܛܗ܋ሺ࢚ሻ െܖܑܛሺ࢚ሻ
3. Let ࡲሺ࢚ሻ ൌ ൬ ൰ ቀ ቁ, where ࢚ is measured in radians. Find the following:
ܖܑܛሺ࢚ሻ ܛܗ܋ሺ࢚ሻ
࣊ ૠ࣊
a. ࡲቀ ቁǡࡲቀ ቁǡ and the radius of the path
࣊ ࣊
࣊ ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰
ࡲ൬ ൰ ൌ ൮ ൲ ቀቁ ൌ ቀ ቁ ቀቁ ൌ ቀ ቁ
࣊ ࣊ െ െ
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰
ૠ࣊ ૠ࣊ ξ
ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰ െξ
ૠ࣊ ۇെ ۊ ۊ
ࡲ൬ ൰ ൌ ൮ ൲ቀ ቁ ൌ ۈ ۇ
ૠ࣊ ૠ࣊ ۋቀቁ ൌ ξ
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰ ξ
െ െ െ െ
ۉ ی
ۉ ی
The path of the point from ࢚ ࣊ is a circle with a center at ሺǡ ሻ.
Thus, the radius ൌ ඥ࢞ ࢟ ൌ ඥሺሻ ሺሻ ൌ ξ or
ඥ
the radius ൌ ඥ࢞ ࢟ ൌ ඨቀെξ ቁ ൬െ െ ൰ ൌ ξ.
ܛܗ܋ሺ࢚ሻ െܖܑܛሺ࢚ሻ ࢞
b. Show that the radius is always ඥ࢞ ࢟ for the path of this transformation ሺ࢚ሻ ൌ ൬ ൰ ቀ ቁ.
ܖܑܛሺ࢚ሻ ܛܗ܋ሺ࢚ሻ ࢟
ܛܗ܋ሺ࢚ሻ െܖܑܛሺ࢚ሻ ࢞ ࢞ԝܛܗ܋ሺ࢚ሻ െ ࢟ԝܖܑܛሺ࢚ሻ
ࡲሺ࢚ሻ ൌ ൬ ൰ቀ ቁ ൌ ൬ ൰
ܖܑܛሺ࢚ሻ ܛܗ܋ሺ࢚ሻ ࢟ ࢞ԝܖܑܛሺ࢚ሻ ࢟ԝܛܗ܋ሺ࢚ሻ
The radius ൌ ටሺ࢞ԝܛܗ܋ሺ࢚ሻ െ ࢟ԝܖܑܛሺ࢚ሻሻ ൫࢞ԝܖܑܛሺ࢚ሻ ࢟ԝܛܗ܋ሺ࢚ሻ൯
ൌ ඥ࢞ ܛܗ܋ ሺ࢚ሻ െ ࢞࢟ԝܛܗ܋ሺ࢚ሻܖܑܛሺ࢚ሻ ࢟ ܖܑܛ ሺ࢚ሻ ࢞ ܖܑܛ ሺ࢚ሻ ࢞࢟ԝܖܑܛሺ࢚ሻܛܗ܋ሺ࢚ሻ ࢟ ܛܗ܋ ሺ࢚ሻ
ൌ ඥ࢞ ܛܗ܋ ሺ࢚ሻ ࢞ ܖܑܛ ሺ࢚ሻ ࢟ ܖܑܛ ሺ࢚ሻ ࢟ ܛܗ܋ ሺ࢚ሻ
ൌ ට࢞ ൫ ܛܗ܋ ሺ࢚ሻ ܖܑܛ ሺ࢚ሻ൯ ࢟ ൫ܛܗ܋ ሺ࢚ሻ ܖܑܛ ሺ࢚ሻ൯
ൌ ඥ࢞ ࢟ Ǥ
࢚࣊ ࢚࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
4. Let ࡲሺ࢚ሻ ൌ ቌ ࢚࣊ ࢚࣊
ቍ ቀ ቁ, where ࢚is a real number.
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
a. Draw the path that ࡼ ൌ ࡲሺ࢚ሻ traces out as ࢚ varies within each of the following intervals:
i. ࢚
ii. ࢚
iii. ࢚
iv. ࢚
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b. Where will the object be located at ࢚ ൌ Ǥ seconds?
࣊ ࣊ െξ ξ
ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰
ࡲሺǤ ሻ ൌ ൮ ൲ ቀቁ ൌ ۇ ۊ
࣊ ࣊ ۈെξ െξ ۋቀቁ ൌ ൬െξ൰
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰
ۉ ی
െૡξ
c. How long does it take the object to reach ൬ ൰?
ૡξ
െૡξ ࢚࣊ ૡξ ࣊ ࢚࣊ ࣊
The point ൬ ൰ is in Quadrant II; the reference angle is ൌ ܖ܉ܜ܋ܚ܉ቀ ቁ ൌ ǡ ൌ ǡ࢚ ൌ
ૡξ ૡξ
second.
ૠ
It takes Ǥ seconds to rotate the point to ࣊; therefore, Ǥ െ ൌ second.
࢚࣊ ࢚࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ െ
5. Let ࡲሺ࢚ሻ ൌ ቌ ࢚࣊ ࢚࣊
ቍ൬ ൰.
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ െξ
a. Draw the path that ࡼ ൌ ࡲሺ࢚ሻ traces out as ࢚ varies within the interval ࢚ .
b. How long does it take the object to reach ൫ξǡ ൯?
The point ൫ξǡ ൯ lies on the ࢞-axis. Therefore, ࢚ ൌ seconds to rotate to the point ൫െǡ െξ൯.
c. How long does it take the object to return to its starting point?
It takes seconds.
6. Find the function that will rotate the point ሺǡ ሻ about the origin to the point ሺെǡ െሻ over the following time
intervals.
a. Over a -second interval
ܛܗ܋ሺ࢚࣊ሻ െܖܑܛሺ࢚࣊ሻ
ࢌሺ࢚ሻ ൌ ൬ ൰ቀ ቁ
ܖܑܛሺ࢚࣊ሻ ܛܗ܋ሺ࢚࣊ሻ
b. Over a -second interval
࢚࣊ ࢚࣊
ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰
ࢌሺ࢚ሻ ൌ ൮ ࢚࣊ ࢚࣊ ൲ ቀቁ
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰
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c. Over a -second interval
ܛܗ܋ሺ࢚࣊ሻ െܖܑܛሺ࢚࣊ሻ
ࢌሺ࢚ሻ ൌ ൬ ൰ቀ ቁ
ܖܑܛሺ࢚࣊ሻ ܛܗ܋ሺ࢚࣊ሻ
d. How about rotating it back to where it starts over a -second interval?
࢚࣊ ࢚࣊
ܛܗ܋൬ ൰ െ ܖܑܛ൬ ൰
ࢌሺ࢚ሻ ൌ ൮ ൲ ቀቁ
࢚࣊ ࢚࣊
ܖܑܛ൬ ൰ ܛܗ܋൬ ൰
7. Summarize the geometric effect of the following function at the given point and the time interval.
࢚࣊ ࢚࣊
ԝ ܛܗ܋ቀ ቁ െԝ ܖܑܛቀ ቁ
a. ࡲሺ࢚ሻ ൌ ቌ ࢚࣊ ࢚࣊
ቍ ቀ ቁ ǡ ࢚
ԝ ܖܑܛቀ ቁ ԝ ܛܗ܋ቀ ቁ
At ࢚ ൌ , the point ቀ ቁ is dilated by a factor of to ቀ ቁǤ
࣊
At ࢚ ൌ , the image ቀ ቁ then is rotated by radians counterclockwise about the origin.
࢚࣊ ࢚࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
b. ࡲሺ࢚ሻ ൌ ቌ
ቍቀ ቁǡ ࢚
࢚࣊ ࢚࣊
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
At ࢚ ൌ , the point ቀ ቁ is dilated by a factor of to ቀ ቁǤ
࣊
At ࢚ ൌ , the image ቀ ቁ then is rotated by radians counterclockwise about the origin.
8. In programming a computer video game, Grace coded the changing location of a rocket as follows:
࢞
At the time ࢚ second between ࢚ ൌ seconds and ࢚ ൌ seconds, the location ቀ࢟ቁ of the rocket is given by
࣊ ࣊
ܛܗ܋ቀ ࢚ቁ െ ܖܑܛቀ ࢚ቁ
൮ ξ
࣊ ࣊ ൲൬ ൰.
ܖܑܛቀ ࢚ቁ ܛܗ܋ቀ ࢚ቁ ξ
At a time of ࢚ seconds between ࢚ ൌ and ࢚ ൌ ૡ seconds, the location of the rocket is given by
ξ
ۇെξ ሺ࢚ െ ሻۊ
ۈ ۋ.
ξ
െξ ሺ࢚ െ ሻ
ۉ ی
a. What is the location of the rocket at time ࢚ ൌ ? What is its location at time ࢚ ൌ ૡ?
ܛܗ܋ሺሻ െܖܑܛሺሻ ξ ξ
At ࢚ ൌ ǡ ൬ ൰൬ ൰ ൌ ቀ ቁ ൬ ൰ ൌ ൬ξ൰.
ܖܑܛሺሻ ܛܗ܋ሺሻ ξ ξ ξ
ඥ
െξ ሺૡ െ ሻ െξ ξ
At ࢚ ൌ ǡ ൮ ൲ൌ൬ ൰ ൌ ൬ξ൰.
ඥ െξ ξ ξ
െξ ሺૡ െ ሻ
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b. Mason is worried that Grace may have made a mistake and the location of the rocket is unclear at time
࢚ ൌ seconds. Explain why there is no inconsistency in the location of the rocket at this time.
ܛܗ܋ሺ࣊ሻ െܖܑܛሺ࣊ሻ െ െξ
At ࢚ ൌ ǡ ൬ ൰ቀ ቁ ൌ ቀ ቁ ൬ξ൰ ൌ ൬ ൰.
ܖܑܛሺ࣊ሻ ܛܗ܋ሺ࣊ሻ െ ξ െξ
ඥ
െξ ሺ െ ሻ െξ
At ࢚ ൌ ǡ ൮ ൲ൌ൬ ൰.
ඥ െξ
െξ ሺ െ ሻ
These are consistent.
c. What is the area of the region enclosed by the path of the rocket from time ࢚ ൌ to ࢚ ൌ ૡ?
࣊࢘ ࣊
The path traversed is a semicircle with a radius of ; the area enclosed is ൌ ൌ ൌ ࣊ square units.
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Lesson 24: Matrix Notation Encompasses New
Transformations!
Student Outcomes
Students work with ʹ ൈ ʹ matrices as transformations of the plane.
Students understand the role of the multiplicative identity matrix.
Lesson Notes
In the preceding lessons, students learned how to represent rotations and dilations in matrix notation. They saw that
not all transformations can be represented this way (translations, for example). In Lessons 24 and 25, they continue to
explore the power of this new notation for finding new transformations, some that they may not have even conceived
of. This lesson begins as students study the multiplicative identity matrix and discover it is similar to multiplying
by ͳ within the real number system. Students experiment with various matrices to discover the transformation each
produces. The use of a transparency and a dry erase marker assists students in conducting their exploration.
Scaffolding:
Classwork These more specific matrices
Opening (5 minutes) can be used to help students
visualize the results of matrix
Review the types of matrices that, when used to transform a point or vector, produce multiplication.
rotations, dilations, or both. ሺͻͲιሻ െሺͻͲιሻ
൬ ൰ is
Show each matrix, and ask students to describe the effect of the matrix in words. ሺͻͲιሻ ሺͻͲιሻ
a ͻͲι rotation
ሺߠሻ െሺߠሻ counterclockwise.
൬ ൰
ሺߠሻ ሺߠሻ Ͷ Ͳ
ቀ ቁ is a dilation with a
à Produces a counterclockwise rotation of ߠι Ͳ Ͷ
݇ Ͳ scale factor of Ͷ.
ቀ ቁ Ͷ െ͵
Ͳ ݇ ቀ ቁ is a dilation and a
àProduces a dilation with a scale factor of ݇ ͵ Ͷ
rotation.
ܽ െܾ
ቀ ቁ For advanced students, instead
ܾ ܽ
of Example 1, pose the
à Produces a counterclockwise rotation and a dilation
following: Multiply several
ͳ Ͳ ͳ Ͳ
ቀ ቁ matrices by the matrix ቀ ቁ.
Ͳ ͳ Ͳ ͳ
à No effect; this is the multiplicative identity matrix. What effect does this matrix
have on other matrices? Can
you predict the name of this
matrix and what number it is
similar to in the real number
system?
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Example 1 (8 minutes)
Students should work on this example individually; after they have finished, pull the class together to debrief.
Example 1
Determine the following:
a. ቀ ቁቀ ቁ ቀ ቁ
െ െ
െૠ െૠ
b. ቀ ቁቀ ቁ ቀ ቁ
c. ቀ ቁቀ ቁ ቀ ቁ
െ െ
െ െ െ െ
d. ቀ ቁቀ ቁ ቀ ቁ
െૠ െૠ
ૢ ૢ
e. ቀ ቁቀ ቁ ቀ ቁ
െ െ െ െ
ࢇ ࢈ ࢇ ࢈
f. ቀ ቁቀ ቁ ቀ ቁ
ࢉ ࢊ ࢉ ࢊ
࢞ ࢟ ࢞ ࢟
g. ቀ ቁቀ ቁ ቀ ቁ
ࢠ ࢝ ࢠ ࢝
What did you notice about the result of each matrix multiplication problem?
à The result was always one of the matrices.
Which matrix?
ͳ Ͳ
à The one that was not ቀ ቁ
Ͳ ͳ
What is this similar to in the real number system?
à When we multiply by the number ͳ
What do we call the number ͳ when multiplying real numbers? Explain.
à The multiplicative identity because you always get the number that was multiplied by ͳ as the product.
ͳ Ͳ
Can you predict what we call the matrix ቀ ቁ?
Ͳ ͳ
à The multiplicative identity matrix
Explain to your neighbor what the multiplicative identity matrix is and why it is called the multiplicative
identity.
à Students explain.
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Example 2 (8 minutes)
Give students time to think about the question individually and then pair up to discuss with a partner. After each pair
has come up with an answer, discuss as a class.
Example 2
Can the reflection about the real axis ࡸሺࢠሻ ൌ ࢠത be expressed in matrix notation?
࢞ ࢞ ࢞
Yes, using the matrix ቀ ቁቀ ቁ ՜ ቀ ቁ ቀ ቁ ൌ ቀെ࢟ቁ
െ ࢟ െ ࢟
ܽ െܾ
Is the matrix you found in the same form as those studied in the previous lessons, ቀ ቁ?
ܾ ܽ
à No
How did you verify that the matrix you found did in fact produce a reflection about the real axis?
à
I multiplied it by a sample point to see if it reflected the point about the real axis. Teacher note: Allow
students to share various responses, even incorrect ones, until the class comes to a consensus on the
correct answer.
ܽଵଵ ܽଵଶ
Do you think all matrices in the form ቀܽ ቁ correspond to a transformation of some kind?
ଶଵ ܽଶଶ
ܽଵଵ ܽଵଶ ݔ
à Yes, multiplying ቀܽ ܽ ቁ times some point ቀݕቁ will affect the point in some way, thereby producing
ଶଵ ଶଶ
a transformation of some sort.
Exercises 1–3 (8 minutes) Scaffolding:
If students are struggling,
Allow students time to work on Exercises 1–3 either individually or in pairs, and then have them work with a
debrief. sample point before
generalizing.
Exercises
1. Express a reflection about the vertical axis in matrix notation. Prove that it produces the
desired reflection by using matrix multiplication.
െ
ቀ ቁ
࢞ െ ࢞ െ࢞
ቀ࢟ቁ ՜ ቀ ቁቀ ቁ ൌ ቀ ࢟ ቁ
࢟
What are the coordinates
2. Express a reflection about both the horizontal and vertical axes in matrix notation. Prove
of ሺʹǡ ͵ሻ after a reflection
that it produces the desired reflection by using matrix multiplication. in the ݔ-axis?
െ What are the coordinates
ቀ ቁ
െ of ሺʹǡ ͵ሻafter a reflection
࢞ െ ࢞ െ࢞ in the ݕ-axis?
ቀ࢟ቁ ՜ ቀ ቁ ቀ ቁ ൌ ቀെ࢟ቁ
െ ࢟
What happens to the
coordinates of a point
after a reflection in the
ݔ-axis? ݕ-axis?
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3. Express a reflection about the vertical axis and a dilation with a scale factor of in matrix notation. Prove that it
produces the desired reflection by using matrix multiplication.
െ
ቀ ቁ
࢞ െ ࢞ െ࢞
ቀ࢟ቁ ՜ ቀ ቁቀ ቁ ൌ ൬ ൰
࢟ ࢟
For Exercise 2, could this transformation also be viewed as a rotation?
à Yes. Reflecting about both axes is the same as a ͳͺͲιrotation.
Is it possible to have a dilation where the ݔ-coordinate is increased by a different scale factor than the
ݕ-coordinate?
ʹ Ͳ
à Yes. For example, ቀ ቁ would dilate the ݔby a factor of ʹ and the ݕby a factor of ͵.
Ͳ ͵
Exercises 4–8 (15 minutes)
Provide each student with a transparency and a dry erase marker. Instruct them to use the graph provided to
experiment with the effect of each matrix. While circulating around the room providing assistance as needed, allow
students time to work either individually or in pairs.
Explore the transformation given by each matrix below. Use the graph of the rectangle provided to assist in the
exploration. Describe the effect on the graph of the rectangle, and then show the general effect of the transformation by
using matrix multiplication.
Matrix Transformation of the Rectangle General Effect of the Matrix
࢞ ࢞ ࢞
4. ቀ ቁ None; the points do not change. ቀ࢟ቁ ՜ ቀ ቁ ቀ ቁ ൌ ቀ࢟ቁ
࢟
The ࢞- and ࢟-coordinates switched. The ࢞ ࢞ ࢟
5. ቀ ቁ ቀ࢟ቁ ՜ ቀ ቁቀ ቁ ൌ ቀ ቁ
rectangle flipped sideways. ࢟ ࢞
࢞ ࢞
6. ቀ ቁ It takes all the points to the origin. ቀ࢟ቁ ՜ ቀ ቁቀ ቁ ൌ ቀ ቁ
࢟
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The points on the ࢞-axis remained fixed, but
࢞ ࢞ ࢞࢟
7. ቀ ቁ all other points shifted horizontally by ࢟ units. ቀ࢟ቁ ՜ ቀ ቁቀ ቁ ൌ ቀ ࢟ ቁ
࢟
The rectangle became a parallelogram.
The points on the ࢟-axis remained fixed, but
࢞ ࢞ ࢞
8. ቀ ቁ all other points shifted vertically by ࢞ units. ቀ࢟ቁ ՜ ቀ ቁ ቀ ቁ ൌ ቀ࢟ ࢞ ቁ
࢟
The rectangle became a parallelogram.
ͳ Ͳ
What effect did the matrix ቀ ቁ have on the graph?
Ͳ ͳ
à None
This is called an identity matrix because it is equivalent to multiplying by ͳ.
Ͳ ͳ
Is the matrix ቀ ቁ also an identity matrix?
ͳ Ͳ
à No. It did change the coordinates. Teacher note: The teacher may demonstrate or lead students to the
fact that the graph was reflected about the line ݕൌ ݔ, but it is not the focal point of this lesson.
Ͳ Ͳ
Can we view the zero matrix ቀ ቁ as a dilation?
Ͳ Ͳ
à Yes, with a scale factor of Ͳ
How would you classify the last two transformations?
à It was sort of like a translation because some of the points moved to the right, but some of them
remained fixed. It does not exactly fit into any of our transformations. Teacher note: These are both
examples of shears, but the main point is that all of these matrices produce some sort of
transformation but not necessarily ones that we know about at this point.
Closing (4 minutes)
Ask students to summarize the types of transformation matrices seen in this lesson. Add these to the list started at the
beginning of class.
Lesson Summary
ࢇ ࢇ
All matrices in the form ቀࢇ ࢇ ቁ correspond to a transformation of some kind.
The matrix ቀ ቁ reflects all coordinates about the horizontal axis.
െ
െ
The matrix ቀ ቁ reflects all coordinates about the vertical axis.
The matrix ቀ ቁ is the identity matrix and corresponds to a transformation that leaves points alone.
The matrix ቀ ቁ is the zero matrix and corresponds to a dilation of scale factor .
Exit Ticket (5 minutes)
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Name Date
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Exit Ticket
What type of transformation is shown in the following examples? What is the resulting matrix?
Ͳ െͳ ͵
1. ቀ ቁቀ ቁ
ͳ Ͳ ʹ
͵ Ͳ ͵
2. ቀ ቁቀ ቁ
Ͳ ͵ ʹ
െͳ Ͳ ͵
3. ቀ ቁቀ ቁ
Ͳ ͳ ʹ
ͳ Ͳ ͵
4. ቀ ቁቀ ቁ
Ͳ െͳ ʹ
5. What is the multiplicative identity matrix? What is it similar to in the set of real numbers? Explain your answer.
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Exit Ticket Sample Solutions
What type of transformation is shown in the following examples? What is the resulting matrix?
െ
1. ቀ ቁቀ ቁ
࣊ െ
It is a pure rotation. The point ቀ ቁ is rotated radians, and the image is ቀ ቁ.
2. ቀ ቁቀ ቁ
ૢ
It is dilation with a factor of . The point ቀ ቁ is dilated by a factor of , and the image isቀ ቁ.
െ
3. ቀ ቁቀ ቁ
െ
It is a reflection about the ࢟-axis. The point ቀ ቁ is reflected about the ࢟-axis, and the image is ቀ ቁ.
4. ቀ ቁቀ ቁ
െ
It is a reflection about the ࢞-axis. The point ቀ ቁ is reflected about the ࢞-axis, and the image is ቀ ቁ.
െ
5. What is the multiplicative identity matrix? What is it similar to in the set of real numbers? Explain your answer.
The multiplicative identity matrix is ቂ ቃ. It is similar to the number in the real number system because any
matrix multiplied by ቂ ቃ produces the original matrix.
Problem Set Sample Solutions
1. What matrix do you need to use to reflect the following points about the ࢟-axis? What is the resulting matrix when
this is done? Show all work and sketch it.
a. ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
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b. ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ
c. ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ
d. ቀ ቁ
െ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
e. ቀ ቁ
െ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
f. ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
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2. What matrix do you need to use to reflect the following points about the ࢞-axis? What is the resulting matrix when
this is done? Show all work and sketch it.
a. ቀ ቁ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
b. ቀ ቁ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
െ
c. ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
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െ
d. ቀ ቁ
െ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
e. ቀ ቁ
െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
െ
f. ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
3. What matrix do you need to use to dilate the following points by a given factor? What is the resulting matrix when
this is done? Show all work and sketch it.
a. ቀ ቁ, a factor of
ቀ ቁቀ ቁ ൌ ቀ ቁ
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b. ቀ ቁ, a factor of
ቀ ቁቀ ቁ ൌ ቀ ቁ
c. ቀ ቁ, a factor of
െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
െ
d. ቀ ቁ, a factor of
െ
െ െ
൮ ൲ቀ ቁ ൌ ቀ ቁ
െ െ
ૢ
e. ቀ ቁ, a factor of
ૢ
൮ ൲ቀ ቁ ൌ ቀ ቁ
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f. ൬ ξ ൰, a factor ofξ
ξ
൬ξ ൰ ൬ ξ ൰ ൌ ൬ ξ ൰
ξ ξ ξ
4. What matrix will rotate the given point by the given radian measures? What is the resulting matrix when this is
done? Show all work and sketch it.
࣊
a. ቀ ቁ, radians
࣊ ࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
െ
൮ ࣊ ࣊ ൲ ቀቁ ൌ ቀ ቁ ቀቁ ൌ ቀቁ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
࣊
b. ቀ ቁ, radians
࣊ ࣊ ξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ۇ െ ۊ
൮ ࣊ ࣊ ൲ ቀቁ ൌ ۈ ቀ ቁൌۊ ۇ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ ۋ ξ
ۉی
ۉ ی
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࣊
c. ቀ ቁ, radians
࣊ ࣊ ξ ξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ െ ۊ
൮ ൲ ቀቁ ൌ ۇ ቀቁ ൌ ۇ ۊ
࣊ ࣊ ۈ ۋ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ
ۉی
ۉ ی
࣊
d. ቀ ቁ, radians
࣊ ࣊ ξ ξ ξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ െ ۊ
൮ ൲ ቀቁ ൌ ۇ ቀቁ ൌ ۇ ۊ
࣊ ࣊ ۈ ۋ ۋ ۈ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ ξ ξ
ۉ ی ۉی
ඥ
࣊
e. ቌ ቍ, radians
࣊ ࣊ ξ ξ ξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ۇ െ ۊ
ۇ ۊ ۇ ۊ ۇ ۊ
൮ ࣊ ࣊ ൲ ൌۈ ۋ ൌ ξ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ
ۉی ۉی ۉ ی ی
ۉ
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ඥ
࣊
f. ൮ ൲, radians
ඥ
࣊ ࣊ ξ ξ ξ ξ
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ۇ ۊ ۇ െ ۊ ۇۊ
൮ ࣊ ࣊ ൲ ۋ ۈൌ ۈ ۋ ۈ ۋൌ ቀቁ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ξ ξ ξ ξ
ۉۉ ی ۉی ی
ඥ
g. ቌ ቍ, ࣊ radians
ξ ξ ξ
ܛܗ܋ሺ࣊ሻ െܖܑܛሺ࣊ሻ ۇ ۊ െ ۇ ۇ ۊെ ۊ
൬ ൰ ൌቀ ቁ ൌ
ܖܑܛሺ࣊ሻ ܛܗ܋ሺ࣊ሻ െ
െ
ۉی ۉ ۉ یی
࣊
h. ቀ ቁ, െ radians
࣊ ࣊ ξ ξ
ܛܗ܋ቀെ ቁ െ ܖܑܛቀെ ቁ ۇ ۊ
ۇ ۊ
൮ ࣊ ࣊ ൲ ቀቁ ൌ ۈ ۋቀ ቁ ൌ
ܖܑܛቀെ ቁ ܛܗ܋ቀെ ቁ ξ
െ െ
ۉی
ۉ ی
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5. For the transformation shown below, find the matrix that will transform point to Ԣ, and verify your answer.
ൌ ሺǡ ሻǡ ࡻ ൌ ඥሺሻ ሺሻ ൌ ξ
ᇱ ൌ ሺെǡ ሻǡ ࡻ ൌ ඥሺെሻ ሺሻ ൌ ξ ൌ ξ; therefore, it has a
dilation of .
࣊ ࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
െ െ
൮ ࣊ ࣊ ൲ ൌ ቀ ቁ ൌ ቀ ቁ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
6. In this lesson, we learned ቂ ቃ will produce a reflection about the line ࢟ ൌ ࢞. What matrix will produce a
reflection about the line ࢟ ൌ െ࢞? Verify your answers by testing the given point ቀ ቁ and graphing them on the
coordinate plane.
െ
ቀ ቁ
െ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
7. Describe the transformation and the translations in the diagram below. Write the matrices that will perform the
tasks. What is the area that these transformations and translations have enclosed?
First, is rotated a ࣊ radian counterclockwise to ᇱ ; next, it is
translated to ԢԢ; finally, it is translated back to .
ܛܗ܋ሺ࣊ሻ െܖܑܛሺ࣊ሻ െ
From to AԢ : ൬ ൰ൌቀ ቁ
ܖܑܛሺ࣊ሻ ܛܗ܋ሺ࣊ሻ െ
࢞ െ ࢞
From ᇱ to ԢԢ: ቀ࢟ െ ࢟ ቁ
࢞ ࢞
From ᇱᇱ to : ቀ࢟ െ ࢟ ቁ
࣊ȁ࢞ି࢞ ȁ ȁ࢞ି࢞ ȁήȁ࢟ ȁ
Area: Semicircletriangle, which is
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8. Given the kite figure ࡰ below, answer the following questions.
a. Explain how you would create the star figure above using only rotations.
࣊
By rotating the kite figure ࡰ radians counterclockwise or clockwise four times
b. Explain how to create the star figure above using reflections and rotation.
࣊
Answers will vary. One explanation is to rotate the kite figure ࡰ radians counterclockwise to get to
െ
ԢԢԢࡰԢ and then reflect both figures about the line ࢟ ൌ െ࢞, which is ቀ ቁ.
െ
c. Explain how to create the star figure above using only reflections. Explain your answer.
First, reflect the kite figure ࡰ about the line ࢟ ൌ ࢞, which is ቀ ቁ; then, reflect ԢԢԢࡰԢ about the
െ
line ࢟ ൌ െ࢞, which is ቀ ቁ; next, reflect ԢԢԢԢԢԢࡰԢԢ about the line ࢟ ൌ ࢞, which is ቀ ቁ; finally,
െ
െ
reflect ԢԢԢԢԢԢԢԢԢࡰԢԢԢ about the line ࢟ ൌ െ࢞, which is ቀ ቁ.
െ
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9. Given the rectangle ࡰ below, answer the following questions.
a. Can you transform the rectangle ԢԢԢࡰԢ above using only rotations? Explain your answer.
No, because no matter how it is rotated, the vertices of the rectangle will not stay the same with respect to
each other.
b. Describe a way to create the rectangleԢԢԢࡰԢ.
െ
Answers will vary. One way is to reflect the rectangle ࡰ about the ࢟-axis first, which isቀ ቁ, and
࣊ െ
then rotate radians counterclockwise, which is ቀ ቁ.
c. Can you make the rectangle ԢԢԢࡰԢ above using only reflections? Explain your answer.
െ
Yes, reflect the rectangle ࡰ about the line ࢟ ൌ െ࢞ǡ which is ቀ ቁ.
െ
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Lesson 25: Matrix Multiplication and Addition
Student Outcomes
Students work with ʹ ൈ ʹ matrices as transformations of the plane.
Students combine matrices using matrix multiplication and addition.
Students understand the role of the zero matrix in matrix addition.
Lesson Notes
In Lesson 24, students continued to explore matrices and their connection to transformations. In this lesson, students
work with the zero matrix and discover that it is the additive identity matrix with a role similar to Ͳ in the real number
system. Students focus on the result of performing one transformation followed by another and discover:
ܽ ܿ ݎ ݔ
If ܮis given by ቀ ቁ and ܯis given by ቀݍ ݏቁ, then ܮܯቀݕቁ is the same as applying the matrix
ܾ ݀
ܽ ܿ ܾݎ ݀ݎ ݔ
൬ ൰ to ቀݕቁ.
ܽݍ ܿݍ ܾݏ ݀ݏ
This motivates the definition of matrix multiplication.
Classwork
Opening Exercise (8 minutes)
Allow students time to complete the Opening Exercise independently. Encourage students to think/write independently,
chat with a partner, and then share as a class.
Opening Exercise
Consider the point ቀ ቁ that undergoes a series of two transformations: a dilation of scale factor followed by a
reflection about the horizontal axis.
a. What matrix produces the dilation of scale factor ? What is the coordinate of the point after the dilation?
The dilation matrix is ቀ ቁǤ
ቀ ቁቀ ቁ ൌ ቀ ቁ
The coordinate is now ቀ ቁ.
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b. What matrix produces the reflection about the horizontal axis? What is the coordinate of the point after the
reflection?
The reflection matrix is ቀ ቁǤ
െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
The coordinate is now ቀ ቁ.
െ
c. Could we have produced both the dilation and the reflection using a single matrix? If so, what matrix would
both dilate by a scale factor of and produce a reflection about the horizontal axis? Show that the matrix
you came up with combines these two matrices.
Yes, by using the matrix ቀ ቁ
െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
The dilation matrix was ቀ ቁ. The rotation matrix was ቀ ቁ. The product of these matrices gives the
െ
matrix that produces a dilation and then a rotation. ቀ ቁቀ ቁൌቀ ቁ
െ െ
How did you come up with the dilation matrix?
݇ Ͳ
à I know that a dilation matrix is in the form ቀ ቁ where ݇ the scale factor is.
Ͳ ݇
How did you come up with the reflection matrix?
ͳ Ͳ
à I know that the matrix ቀ ቁ reflects coordinates about the horizontal axis.
Ͳ െͳ
How did you come up with a matrix that was both a dilation and a reflection?
à I knew that I wanted to multiply both the ݔand ݕby a factor of Ͷ and that I also wanted to multiply the
Ͷ Ͳ
ݕby െͳ; from that, I combined the two matrices to get ቀ ቁǤ
Ͳ െͶ
In what sense did we combine the two matrices?
à We multiplied Ͷ ൈ െͳ.
We know that transformations are produced through matrix multiplication. What if we have more than one
transformation? Could we multiply the two transformation matrices together first instead of completing the
transformations in two separate steps?
Write this on the board:
Ͷ Ͳ ͳ Ͳ
ቀ ቁቀ ቁൌ
Ͳ Ͷ Ͳ െͳ
This should be equivalent to applying the dilation and then the reflection. So, what should the product equal?
Ͷ Ͳ
à ቀ ቁ
Ͳ െͶ
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Ask students to think about how we multiply a ʹ ൈ ʹ matrix and a ʹ ൈ ͳ matrix. Based on the fact that this
Ͷ Ͳ
product should be ቀ ቁ and what you know about multiplying a matrix by a vector, develop an
Ͳ െͶ
explanation for how to multiply these two matrices together.
à ቀ
Ͷ Ͳ ͳ Ͳ
ቁቀ ቁൌቀ
Ͷ ൈ ͳ Ͳ ൈ Ͳ Ͷ ൈ Ͳ Ͳ ൈ െͳ
ቁൌቀ
Ͷ Ͳ
ቁ
Ͳ Ͷ Ͳ െͳ Ͳ ൈ ͳ Ͳ ൈ Ͳ Ͷ ൈ Ͳ Ͷ ൈ െͳ Ͳ െͶ
Does this technique align with our earlier definition of multiplying a ʹ ൈ ʹ times a ʹ ൈ ͳ?
à Yes, we follow the same process to get the numbers in column 2 that we did to get the numbers in
column 1—multiplying each row by the numbers in the column and then adding.
͵ ͳ െͳ
Can matrices of any size be multiplied together? For example, can you multiply ቀ ቁ ቀ ቁ? Why or why
ʹ Ͷ ͷ
not?
à No, the number of rows and columns do not match up.
What must be true about the dimensions of matrices in order for them to be able to be multiplied?
à The number of columns of the first matrix must equal the number of rows of the second matrix.
Example 1 (7 minutes): Is Matrix Multiplication Commutative?
Conduct each part of the example as a think-pair-share. Allow students time to think of Scaffolding:
the answer independently, pair with a partner to discuss, and then share as a class.
For students who are
struggling, start with a
Example 1: Is Matrix Multiplication Commutative?
review of the commutative
࣊
a. Take the point ቀ ቁ through the following transformations: a rotation of and a property of multiplication.
reflection across the ࢟-axis. If ܽ and ܾ are real
ቀ ቁ
numbers, then
ܽ ൈ ܾ ൌ ܾ ൈ ܽ.
Have them highlight or
b. Will the resulting point be the same if the order of the transformations is reversed? circle each row and
No, if the reflection is applied first, followed by the rotation, the resulting point column as they multiply.
െ
is ቀ ቁ.
െ What does it mean for
multiplication of real
c. Are transformations commutative?
numbers to be
commutative? Explain
Not necessarily; the order in which the transformations are applied can affect the
with an example.
results in some cases.
Can you think of an
operation that is not
െ െ
d. Let ൌ ቀ ቁ and ൌ ቀ ቁ. Find and then . commutative? Explain
െ െ െ with an example.
ൌ ቀ ቁቀ ቁൌቀ ቁ
െ
െ െ
ൌ ቀ ቁቀ ቁൌቀ ቁ
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e. Is matrix multiplication commutative?
No, ്
f. If we apply matrix to the point ቀ ቁ, in what order are the transformations applied?
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
The reflection is applied first, followed by the dilation.
g. If we apply matrix to the point ቀ ቁ, in what order are the transformations applied?
ቀ ቁቀ ቁ ൌ ቀ ቁ
The dilation is applied first, followed by the reflection.
h. Can we apply ቀ ቁ to matrix ?
ቀ ቁቀ ቁ
No, ቀ ቁ has two rows and one column. It would have to have two columns to be able to multiply by the ൈ
matrix .
Exercises 1–3 (10 minutes)
Allow students time to work on the exercises either independently or in groups. Circulate around the room providing
assistance as needed, particularly watching for students who are struggling with matrix multiplication.
Exercises 1– 3
െ
1. Let ࡵ ൌ ቀ ቁ and ࡹ ൌ ቀ ቁ.
െ
a. Findࡵࡹ.
െ
ቀ ቁ
െ
b. Find ࡹࡵ.
െ
ቀ ቁ
െ
c. Do these results make sense based on what you know about the matrix ቀ ቁ?
Yes, the matrix ቀ ቁ is the identity matrix. It corresponds to a transformation that leaves points alone.
Therefore, geometrically we must have ࡵࡹ ൌ ࡹࡵ ൌ ࡹ.
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2. Calculate and then . Is matrix multiplication commutative?
a. ൌቀ ቁǡ ൌ ቀ ቁ
െ
Scaffolding:
ൈൈ ൈൈ
ൌ ቀ ቁൌቀ ቁ For students who like a
െ ൈ ൈ െ ൈ ൈ െ െૠ
ൈ ൈ െ ൈ ൈ െૡ challenge, pose one of the
ൌ ቀ ቁൌቀ ቁ
ൈ ൈ െ ൈ ൈ െ following:
് ; matrix multiplication is not commutative. Multiply two ͵ ൈ ͵
matrices.
b. ൌቀ
െ
ቁǡ ൌ ቀ
െ
ቁ Could we use our
ૠ െ
definition for multiplying
െ ൈ െ ൈ െ ൈ ൈ െ െ
ൌ ቀ
ൈ െ ૠ ൈ ൈ ૠ ൈ െ
ቁൌቀ
ૢ െ
ቁ matrices to multiply
a ʹ ൈ ͳ times a ʹ ൈ ʹ?
െ ൈ െ ൈ െ ൈ ൈ ૠ
ൌ ቀ ቁൌቀ ቁ
ൈ െ െ ൈ ൈ െ ൈ ૠ െ െ
് ; matrix multiplication is not commutative.
3. Write a matrix that would perform the following transformations in this order: a rotation of ૡι, a dilation by a
scale factor of , and a reflection across the horizontal axis. Use the point ቀ ቁ to illustrate that your matrix is
correct.
െ െ
ቀ ቁቀ ቁൌቀ ቁ
െ െ
െ െૡ
ቀ ቁቀ ቁ ൌ ቀ ቁ
Example 2 (5 minutes): More Operations on Matrices
Discuss this question as a class before completing the example.
We know that there is matrix multiplication. Does it seem logical that there would be matrix addition?
ʹ Ͳ ͳ ͷ ͵ ͷ
If ቀ ቁቀ ቁൌቀ ቁ, explain how to add matrices.
െʹ ͵ Ͳ ͳ െʹ Ͷ
à Add the numbers that are in the same position in the corresponding matrices.
Each number within a matrix is called an element. In order to add matrices, the elements in the same row and
same column are added. We refer to elements in the same row and same column as corresponding elements.
So, to recap, in order to add matrices, we add corresponding elements.
How would you subtract matrices?
àSubtract corresponding elements.
ʹ Ͳ ͳ ͷ
Find ቀ ቁെቀ ቁǤ
െʹ ͵ Ͳ ͳ
ͳ െͷ
à ቀ ቁ
െʹ ʹ
͵ ͷ ͳ
Can you add the following matrices: ቀ ቁ ቀ ቁ? Explain your answer.
െʹ Ͷ െ͵
à No, the matrix is not the same size, so there are not corresponding elements.
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When can matrices be added and subtracted?
à When they are the same size
In Lesson 24, we studied the multiplicative identity matrix. In what ways is the identity matrix similar to the
number ͳ within the set of real numbers? Why?
à The identity matrix is similar to the number ͳ in the real number system because any number times ͳ is
itself, and any matrix times the identity matrix is itself.
What is the additive identity in the real number system? Why?
à Ͳ, because any number plus Ͳ is itself.
What matrix would you hypothesize has the same effect in matrix addition? Write the matrix.
Ͳ Ͳ
à ቀ ቁ is the additive identity matrix because any matrix plus this matrix is itself.
Ͳ Ͳ
We call this the zero matrix. What would be the impact of multiplying a matrix by the zero matrix? How does
the impact of multiplying a matrix by the zero matrix compare to multiplying by zero in the real number
system?
à Any number multiplied by Ͳ is Ͳ, and any matrix multiplied by the zero matrix has all terms of Ͳ.
Note that it is difficult to know what matrix addition means geometrically in terms of transformations, but we
will see a natural interpretation of matrix addition in Module 3.
Example 2: More Operations on Matrices
Find the sum. ቀ ቁቀ ቁ
െ
ቀ ቁቀ ቁൌቀ ቁൌቀ ቁ
െ െ െ
Find the difference. ቀ ቁെቀ ቁ
െ
െ െ െ
ቀ ቁെቀ ቁൌቀ ቁൌቀ ቁ
െ െ െ െ െ
Find the sum. ቀ ቁቀ ቁ
െ
ቀ ቁቀ ቁൌቀ ቁൌቀ ቁ
െ െ െ
Exercises 4–5 (5 minutes)
Allow students time to work on the exercises either independently or in groups. Circulate around the room providing
assistance as needed.
Exercises 4–5
4. Express each of the following as a single matrix.
െ െ ૡ
a. ቀ ቁቀ ቁ
െ െ
ቀ ቁ
െ
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െ ૠ
b. ቀ ቁቀ ቁ ቀ ቁ
െ
ቀ ቁ
ૡ െ
c. ቀ ቁെቀ ቁ
െ ૡ
ቀ ቁ
െ
5. In arithmetic, the additive identity says that for some number ࢇ, ࢇ ൌ ࢇ ൌ . What would be an additive
identity in matrix arithmetic?
ࢇ ࢉ ࢇ ࢉ
We would use the zero matrix. ቀ ቁቀ ቁൌቀ ቁ
࢈ ࢊ ࢈ ࢊ
Closing (5 minutes)
Ask students to summarize what they have learned about matrix multiplication and addition either in writing or orally.
When we multiply two matrices, what is the geometric interpretation?
à It is a series of transformations.
Can all matrices be multiplied? Why or why not?
à Matrices can be multiplied if the number of columns of the first matrix is equal to the number of rows of
the second matrix.
Can two matrices be combined through addition? If so, explain how.
à Yes, if the matrices are the same size, they can be added by adding corresponding elements.
Lesson Summary
ࢇ ࢉ ࢘ ࢞
If ࡸ is given by ቀ ቁ and ࡹ is given by ቀ ࢙ቁ, then ࡹࡸ ቀ࢟ቁ is the same as applying the matrix
࢈ ࢊ
ࢇ ࢘࢈ ࢉ ࢘ࢊ ࢞
൬ ൰ to ቀ࢟ቁ.
ࢇ ࢙࢈ ࢉ ࢙ࢊ
ࢇ ࢉ
If ࡸis given by ቀ ቁ and ࡵ is given by ቀ ቁ, then ࡵ acts as a multiplicative identity,
࢈ ࢊ
and ࡵࡸ ൌ ࡸࡵ ൌ ࡸ.
ࢇ ࢉ
If ࡸis given by ቀ ቁ and ࡻ is given by ቀ ቁ, then ࡻ acts as an additive identity,
࢈ ࢊ
and ࡻ ࡸ ൌ ࡸ ࡻ ൌ ࡸ.
Exit Ticket (5 minutes)
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Name Date
Lesson 25: Matrix Multiplication and Addition
Exit Ticket
1. Carmine has never seen matrices before but must quickly understand how to add, subtract, and multiply matrices.
Explain the following problems to Carmine.
ʹ ͵ Ͳ ͳ ʹ Ͷ
a. ቀ ቁቀ ቁൌቀ ቁ
െͳ ͷ Ͷ െ͵ ͵ ʹ
ʹ ͵ Ͳ ͳ ʹ ʹ
b. ቀ ቁെቀ ቁൌቀ ቁ
െͳ ͷ Ͷ െ͵ െͷ ͺ
ʹ ͵ Ͳ ͳ ͳʹ െ
c. ቀ ቁቀ ቁൌቀ ቁ
െͳ ͷ Ͷ െ͵ ʹͲ െͳ
2. Explain to Carmine the significance of the zero matrix and the multiplicative identity matrix.
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Exit Ticket Sample Solutions
1. Carmine has never seen matrices before but must quickly understand how to add, subtract, and multiply matrices.
Explain the following problems to Carmine.
a. ቀ ቁቀ ቁൌቀ ቁ
െ െ
To add matrices, add the corresponding elements. So, add the and the because they are both in the first
row, first column.
b. ቀ ቁെቀ ቁൌቀ ቁ
െ െ െ ૡ
To subtract matrices, subtract the corresponding elements. So, subtract the from the because they are
both in the first row, first column.
െૠ
c. ቀ ቁቀ ቁൌቀ ቁ
െ െ െ
To multiply matrices, multiply the elements in the first row by the elements in the first column, and then add
the products together.
ൈൈ ൈ ൈ െ െૠ
ቀ ቁൌቀ ቁ
െ ൈ ൈ െ ൈ ൈ െ െ
2. Explain to Carmine the significance of the zero matrix and the multiplicative identity matrix.
The zero matrix is ቀ ቁ and is similar to in the real number system. Any matrix added to the zero matrix is
itself, and any matrix multiplied by the zero matrix has all terms of .
The multiplicative identity matrix is ቀ ቁ and is similar to in the real number system. Any matrix times the
multiplicative identity matrix has a product of itself.
Problem Set Sample Solutions
1. What type of transformation is shown in the following examples? What is the resulting matrix?
ܛܗ܋ሺ࣊ሻ െܖܑܛሺ࣊ሻ
a. ൬ ൰ቀ ቁ
ܖܑܛሺ࣊ሻ ܛܗ܋ሺ࣊ሻ
െ
It is a pure rotation. The point ቀ ቁ is rotated ࣊ radians, and the image is ቀ ቁ.
െ
െ
b. ቀ ቁቀ ቁ
࣊ െ
It is a pure rotation. The point ቀ ቁ is rotated radians, and the image is ቀ ቁ.
c. ቀ ቁቀ ቁ
ૢ
It is dilation with a factor of . The point ቀ ቁ is dilated by a factor of , and the image isቀ ቁ.
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െ
d. ቀ ቁቀ ቁ
െ
It is a reflection about the ࢟-axis. The point ቀ ቁ is reflected about the ࢟-axis, and the image is ቀ ቁ.
e. ቀ ቁቀ ቁ
െ
It is a reflection about the ࢞-axis. The point ቀ ቁ is reflected about the ࢞-axis, and the image is ቀ ቁ.
െ
ܛܗ܋ሺ࣊ሻ െܖܑܛሺ࣊ሻ
f. ൬ ൰ቀ ቁ
ܖܑܛሺ࣊ሻ ܛܗ܋ሺ࣊ሻ
It is a pure rotation. The point ቀ ቁ is rotated ࣊ radians, and the image is ቀ ቁ.
ξ ξ
െ
g. ቌ
ቍቀ ቁ
ξ ξ
ξ
࣊
It is a pure rotation. The point ቀ ቁ is rotated radians, and the image is ቌ ቍ.
ξ
ඥ
െ
h. ൮ ൲ ቀቁ
ඥ
ඥ
࣊ െ
It is a pure rotation. The point ቀ ቁ is rotated radians, and the image is ቌ ቍ.
െ ξ
2. Calculate each of the following products.
a. ቀ ቁቀ ቁ
ቀ ቁ
b. ቀ ቁቀ ቁ
ቀ ቁ
ૠ ૡ
െ െ െ
c. ቀ ቁቀ ቁ
െ െ െ
ቀ ቁ
െ െ െ െ
d. ቀ ቁቀ ቁ
െ െ െ
ቀ ቁ
ૡ
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e. ቀ ቁቀ ቁ
െ
ቀ ቁ
െ
f. ቀ ቁቀ ቁ
െ െ െ
ቀ ቁ
ૡ
3. Calculate each sum or difference.
a. ቀ ቁቀ ቁ
ቀ ቁ
െ െ െ
b. ቀ ቁቀ ቁ
െ െૠ െ
െ െ
ቀ ቁ
െૠ െ
െ
c. ቀ ቁቀ ቁ
െ
ቀ ቁ
ૠ
d. ቀ ቁെቀ ቁ
െ ૢ
െ
ቀ ቁ
െ
െ െ െ
e. ቀ ቁെቀ ቁ
െ െૠ െ
െ െૡ
ቀ ቁ
െ െ
4. In video game programming, Fahad translates a car, whose coordinate is ቀ ቁ, units up and units to the right,
࣊ ࣊
rotates it radians counterclockwise, reflects it about the ࢞-axis, reflects it about the ࢟-axis, rotates it radians
counterclockwise, and finally translates it units down and units to the left. What point represents the final
location of the car?
ቀ ቁ՜ቀ ቁൌቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
െ െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ െ
െ
ቀ ቁቀ ቁ ൌ ቀ ቁ
െ
െ
ቀ ቁ՜ቀ ቁൌቀ ቁ
െ
The final location of the car is its initial starting point.
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Lesson 26: Getting a Handle on New Transformations
Student Outcomes
Students understand that the absolute value of the determinant of a ʹ ൈ ʹ matrix is the area of the image of
the unit square.
Lesson Notes
This is day one of a two-day lesson on transformations using matrix notation. Students begin with the unit square and
look at the geometric results of simple transformations on the unit square. Students then calculate the area of the
transformed figure and understand that it is the absolute value of the determinant of the ʹ ൈ ʹ matrix representing the
transformation.
Classwork
Have students work on the Opening Exercise individually, and then check solutions as a class. This exercise allows
students to practice the matrix operations of addition and subtraction and prepares them for concepts they need in
Lessons 26 and 27.
In the next few exercises, matrices are represented with square brackets. Discuss with students that matrices can be
ͳ ʹ ͳ ʹ
represented with soft or square brackets. ቂ ቃ and ቀ ቁ represent the same matrix.
͵ Ͷ ͵ Ͷ
Opening Exercise (8 minutes)
Opening Exercise
Perform the following matrix operations:
Scaffolding:
െ െ
a. ቂ
ቃቂ ቃ
To aid students in matrix operations,
െૢ ask questions, such as “How do we
ቂ ቃ
multiply matrices?” or “How do we add
matrices?” The images below can be
െ displayed in the classroom as guides.
b. ቂ ቃቂ ቃ
To multiply matrices:
െ
ቂ ቃ ܽ ܾ ݔ ܽ ݔ ܾݕ
ቂ ቃቂ ቃ ൌ ൨
ܿ ݀ ݕ ܿ ݔ ݀ݕ
ܽ ܾ ݕ ݔ ܽ ݔ ܾ ݕܽ ݖ ܾݓ
c. ቂ
െ െ
ቃቂ ቃ ቂ ቃቂ ቃൌ ൨
ܿ ݀ ݓ ݖ ܿ ݔ ݀ ݕܿ ݖ ݀ݓ
െ െૠ To add matrices:
ቂ ቃ
ૠ ݕ ݔ ܾܽ ݔݕ
ܽ ܾ
ቀ ቁቀ ቁൌቀ ቁ
ܿ ݀ ݓ ݖ ܿ݀ ݖݓ
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െ ࢇ ࢈
d. ቂ ቃቂ ቃ
ࢉ ࢊ
ࢇ െ ࢉ ࢈ െ ࢊ
ቂ ቃ
ࢇ ࢉ ࢈ ࢊ
െ
e. ቂ ቃቂ ቃ
െ
ቂ ቃ
െ െ
f. ቂ ቃቂ ቃ
െ
ቂ ቃ
ૢ
െ ࢇ ࢈
g. ቂ ቃቂ ቃ
ࢉ ࢊ
ࢇ െ ࢈
ቂ ቃ
ࢉ ࢊ
h. Can you add the two matrices in part (a)? Why or why not?
No, the matrices do not have the same dimensions, so they cannot be added.
Exploratory Challenge (20 minutes)
In this Exploratory Challenge, students discover what matrix transformations do to a unit Scaffolding:
square geometrically. Students calculate the area of the image of the unit square and
Some student pairs may
discover that the area is the absolute value of the determinant of the resulting ʹ ൈ ʹ
need targeted one-to-one
matrix. In this challenge, let students work in pairs, but lead the class together from step
guidance on this challenge.
to step. Students should have graph paper and a ruler.
For advanced students,
ܽ ܾ give them the challenge
We have seen that every matrix ቂ ቃ corresponds to some kind of
ܿ ݀ without guiding questions,
transformation of the plane, but it can be hard to see what the transformation
and allow them to work in
ͳͲͻ ͵
actually does. For example, what does the matrix ቂ ቃ do to points, pairs on their own,
ͳ െʹ
shapes, and lines in the plane? checking their steps
periodically.
à Allow students to share ideas. We answer this question later after
Provide unlabeled graphs
looking at more basic matrices.
for students who have
Let’s draw the unit square in the coordinate plane with each side ͳ inch long. difficulties with eye-hand
à Students draw the unit square. (Check to make sure the squares coordination or fine motor
are ͳ inch ൈ ͳ inch.) skills.
Now label the vertices of the square.
à Students label the vertices as shown.
Write a set of matrices that represents the vertices of the unit square.
Ͳ ͳ Ͳ ͳ
à ቂ ቃ ǡ ቂ ቃ ǡ ቂ ቃ, and ቂ ቃ
Ͳ Ͳ ͳ ͳ
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ܽ െܾ
As we learned in previous lessons, any matrix of the form ቂ ቃ is a
ܾ ܽ
rotation and a dilation. Perform this transformation on the vertices of
the unit square if ܽ Ͳ and ܾ Ͳ. Show your work.
ܽ െܾ Ͳ Ͳ ܽ െܾ ͳ ܽ ܽ െܾ Ͳ െܾ
à ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ,
ܾ ܽ Ͳ Ͳ ܾ ܽ Ͳ ܾ ܾ ܽ ͳ ܽ
ܽ െܾ ͳ ܽെܾ
and ቂ ቃቂ ቃ ൌ ቂ ቃ
ܾ ܽ ͳ ܾܽ
What are the coordinates of the image of ሺͳǡ Ͳሻ and ሺͲǡ ͳሻ?
à ሺͳǡ Ͳሻ ՜ ሺܽǡ ܾሻ
à ሺͲǡ ͳሻ ՜ ሺെܾǡ ܽሻ
Graph the image on the same graph as the original
unit square in a different color.
à See the diagram to the right.
Label the coordinates of the vertices.
ሺͲǡ Ͳሻǡ ሺܽǡ ܾሻǡ ሺെܾǡ ܽሻ, and
ሺܽ െ ܾǡ ܾ ܽሻ
This picture allows you to see the rotation and
dilation that took place on the unit square.
Let’s try another transformation.
Draw another unit square with side lengths
of ͳ inch.
à Students draw a second unit square.
ܽ ܿ
Perform the general transformation ቂ ቃ on the
ܾ ݀
vertices of the unit square.
ܽ ܿ Ͳ Ͳ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ܾ ݀ Ͳ Ͳ
ܽ ܿ ͳ ܽ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ܾ ݀ Ͳ ܾ
ܽ ܿ Ͳ ܿ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ܾ ݀ ͳ ݀
ܽ ܿ ͳ ܽܿ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ܾ ݀ ͳ ܾ݀
What are the coordinates of the image of ሺͳǡ Ͳሻ
and ሺͳǡ ͳሻ?
à ሺͳǡ Ͳሻ ՜ ሺܽǡ ܾሻ
à ሺͳǡ ͳሻ ՜ ሺܽ ܿǡ ܾ ݀ሻ
Graph the image on the same graph as the second
unit square in a different color, and label the
vertices.
à See the diagram to the right.
The vertices are ሺͲǡ Ͳሻǡ ሺܽǡ ܾሻǡ ሺܿǡ ݀ሻ, and
ሺܽ ܿǡ ܾ ݀ሻ.
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Look at the two diagrams that we have created. The original unit
square had four straight sides. After the transformations, were the
straight segments mapped to straight segments? Was the square
mapped to a square? Explain.
à Straight segments were mapped to straight segments.
à The square was mapped to a parallelogram.
Does this transformation change the area of the unit square?
à It seems to—yes.
Let’s find the area of the image from the general transformation. Allow students to work in pairs to find the
area of the image by enclosing the parallelogram in a rectangle and subtracting the areas of the right triangles
and rectangles surrounding the parallelogram. The area of the first image may be slightly easier to find.
ͳ ͳ
ൌ ሺܽ ܿሻሺܾ ݀ሻ െ ʹ ൬ ܾܽ ܿ݀ ܾܿ൰ Scaffolding:
ʹ ʹ
ൌ ܾܽ ܽ݀ ܾܿ ܿ݀ െ ܾܽ െ ܿ݀ െ ʹܾܿ If students are struggling
to understand the need
ൌ ܽ݀ െ ܾܿ
for absolute value with
When we drew the image, we kept the orientation of the vertices; in other variables, have them
words, we mapped ሺͳǡ Ͳሻ to ሺܽǡ ܾሻ and ሺͳǡ ͳሻ to ሺܽ ܿǡ ܾ ݀ሻ. We could have perform this activity using
switched the order of vertices ሺܽǡ ܾሻ and ሺܿǡ ݀ሻ. Redraw the picture and the following matrices.
calculate the area of the parallelogram image. Do you get the same area? ͵ ͳ
ቂ ቃ has a determinant
Explain. ʹ Ͷ
of ͳͲ.
à The area is the opposite of what we calculated before.
ͳ ͵
The area is ܾܿ െ ܽ݀. ቂ ቃ has a determinant
Ͷ ʹ
What could we do to ensure this formula always works for the area regardless of of െͳͲ.
the orientation of the vertices? Both give the same
à Take the absolute value. transformed figure, so to
get the area, we must take
Write the general formula for the area of the parallelogram that is the image of
the absolute value of the
the transformation of the unit square.
determinant.
à ൌ ȁܽ݀ െ ܾܿȁ
ܽ ܿ
The determinant of a ʹ ൈ ʹ matrix ቂ ቃ is ȁܽ݀ െ ܾܿȁ. Explain this geometrically to your neighbor.
ܾ ݀
à The determinant of a ʹ ൈ ʹ matrix is the area of the image of the unit square that has undergone the
ܽ ܿ
transformation ቂ ቃ.
ܾ ݀
DETERMINANT: The area of the image of the unit square under the linear transformation represented by
a ʹ ൈ ʹ matrix is called the determinant of that matrix.
Exercises 1–3 (10 minutes)
In Exercises 1 and 2, two problems discussed in the Exploratory Challenge are revisited, and questions are answered
using what students have discovered. Exercise 3 revisits the pure dilation and rotation matrices to see their effect on
area. Students should work on these exercises in pairs and then debrief as a class. Any problems not completed can be
assigned for homework.
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Exercises
ૢ
1. Perform the transformation ቂ ቃ on the unit square.
െ
a. Sketch the image. What is the shape of
the image?
The image is a parallelogram.
b. What are the coordinates of the vertices of the image?
ሺǡ ሻǡ ሺǡ െሻǡ ሺૢǡ ሻ, and ሺǡ െሻ
c. What is the area of the image? Show your work.
܉܍ܚۯൌ ȁሺૢሻሺെሻ െ ሺሻሺሻȁ ൌ ȁെȁ ൌ
ࢇ െ࢈
2. In the Exploratory Challenge, we drew the image of a general rotation/dilation of the unit square ቂ ቃ.
࢈ ࢇ
a. Calculate the area of the image by enclosing the image in a rectangle
and subtracting the area of the surrounding right triangles. Show
your work.
܉܍ܚۯൌ ሺࢇ ࢈ሻ െ ൬ ࢇ࢈൰
܉܍ܚۯൌ ࢇ ࢇ࢈ ࢈ െ ࢇ࢈
܉܍ܚۯൌ ࢇ ࢈
b. Confirm the area using the determinant of the resulting matrix.
ࢇ െ࢈
ቂ ቃ
࢈ ࢇ
܉܍ܚۯൌ ȁሺࢇሻሺࢇሻ െ ሺ࢈ሻሺെ࢈ሻȁ ൌ ࢇ ࢈
3. We have looked at several general matrix transformations in Module 1. Answer the questions below about these
familiar matrices, and explain your answers.
a. What effect does the identity transformation have on the unit square? What is the area of the image?
Confirm your answer using the determinant.
The identity transformation does nothing to the unit square. The area is , as is the determinant of the unit
matrix.
b. How does a dilation with a scale factor of change the area of the unit square? Calculate the determinant of
a matrix representing a pure dilation of .
The dilation changes all areas by . The pure dilation matrix is ቂ ቃ, which has a determinant of .
c. Does a rotation with no dilation change the area of the unit square? Confirm your answer by calculating the
determinant of a pure rotation matrix, and explain it.
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ
A pure rotation does not change the area. The pure rotation matrix is ൨. Its determinant is
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
ሺܛܗ܋ሺࣂሻሻ െ ൫െܖܑܛሺࣂሻ൯ ൌ ܛܗ܋ ሺࣂሻ ܖܑܛ ሺࣂሻ ൌ , which confirms that the area does not change.
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Closing (2 minutes)
Have students do a 30-second quick write on the following question, and then debrief as a class.
ܽ ܿ
What effect does the general transformation ቂ ቃ have on the unit square?
ܾ ݀
à The image of this transformation is a parallelogram with vertices
ሺͲǡ Ͳሻǡ ሺܽǡ ܾሻǡ ሺܿǡ ݀ሻ, and ሺܽ ܿǡ ܾ ݀ሻ.
What is the easiest way to calculate the area of the image of this transformation?
à Calculate the determinant of the resulting matrix ȁܽ݀ െ ܾܿȁ.
Lesson Summary
Definition
The area of the image of the unit square under the linear transformation represented by a ൈ matrix
is called the determinant of that matrix.
Exit Ticket (5 minutes)
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Name Date
Lesson 26: Getting a Handle on New Transformations
Exit Ticket
െʹ ͷ
Perform the transformation ቂ ቃ on the unit square.
Ͷ െͳ
a. Draw the unit square and the image after this transformation.
b. Label the vertices. Explain the effect of this transformation on the unit square.
c. Calculate the area of the image. Show your work.
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Exit Ticket Sample Solutions
െ
Perform the transformation ቂ ቃ on the unit square.
െ
a. Draw the unit square and the image after this
transformation.
b. Label the vertices. Explain the effect of this
transformation on the unit square.
ሺǡ ሻ ՜ Ԣሺെǡ ሻ
ࡰሺǡ ሻ ՜ ࡰԢሺǡ െሻ
ሺǡ ሻ ՜ Ԣሺǡ ሻ
ሺǡ ሻ ՜ ሺǡ ሻ
c. Calculate the area of the image. Show your work.
ȁሺെሻሺെሻ െ ሺሻሺሻȁ ൌ ȁ െ ȁ ൌ ȁെૡȁ ൌ ૡ
Problem Set Sample Solutions
1. Perform the following transformation on the unit square: Sketch and state the area of the image.
െ
a. ቂ ቃ
െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡ ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ࢇ ࢈ ൌ ൌ
The area is units .
െ
b. ቂ ቃ
െ െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ࢇ ࢈ ൌ ൌ
The area is units .
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െ
c. ቂ ቃ
െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ሾሿǡ ቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ࢇ ࢈ ൌ ൌ
The area is units .
െ
d. ቂ ቃ
െ െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ࢇ ࢈ ൌ ൌ
The area is units .
2. Perform the following transformation on the unit square: Sketch the image, find the determinant of the given
matrix, and find the area of the image.
a. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ െ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area is units .
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b. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
ૠ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ െ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area is units .
c. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area is units .
d. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ െ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area isunits .
e. The determinants in parts (a), (b), (c), and (d) have positive or negative values. What is the value of the
determinants if the vertices (b, c) and (c, d) are switched?
The value is negative.
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3. Perform the following transformation on the unit square: Sketch the image, find the determinant of the given
matrix, and find the area of the image.
െ െ
a. ቂ ቃ
െ െ
െ െ െ െ െ െ െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
െ െ െ െ െ െ െ െૠ െ െ െ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ െ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area is units .
െ െ
b. ቂ ቃ
െ െ െ െ െ െ െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁെ ȁ ൌ
The area is units .
c. ቂ ቃ
െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
െ െ െ െ െ െ െ െ െ െ െ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁെ ȁ ൌ
The area is units .
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െ
d. ቂ ቃ
െ
െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
െ െ െ െ െ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁെ ȁ ൌ
The area is units .
െ
e. ቂ ቃ
െ
െ െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
െ െ െ െ െ െ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ െ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁെ ȁ ൌ
The area is units .
െ
f. ቂ ቃ
െ
െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
െ െ െ െ െ െ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ െ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area is units .
െ
g. ቂ ቃ
െ
െ െ െ െ െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
െ െ െ െ െ
Determinant: ࢇࢊ െ ࢈ࢉ ൌ െ ൌ ܉܍ܚۯൌ ȁࢇࢊ െ ࢈ࢉȁ ൌ ȁ െ ȁ ൌ
The area is units .
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Lesson 27: Getting a Handle on New Transformations
Student Outcomes
Students understand that ʹ ൈ ʹ matrix transformations are linear transformations taking straight lines to
straight lines.
Students understand that the absolute value of the determinant of a ʹ ൈ ʹ matrix is the area of the image of
the unit square.
Lesson Notes
This is day two of a two-day lesson on transformations using matrix notation. In Lesson 26, students looked at general
matrix transformations on the unit square and discovered that the area of the image was the determinant of the
resulting matrix. In Lesson 27, students get more practice with this concept and connect it to the study of linearity.
Classwork
This Opening Exercise reminds students of general matrices studied in prior lessons and their geometric effect. Show
one matrix at a time to the class, and discuss the geometric significance of each matrix.
Opening Exercise (8 minutes)
Opening Exercise
Explain the geometric effect of each matrix.
a. ቂ
ࢇ െ࢈
ቃ
Scaffolding:
࢈ ࢇ
Have students create an example of each
࢈
A rotation of ܖ܉ܜ܋ܚ܉ቀ ቁ and a dilation with scale factor transformation and show it graphically in a
ࢇ
ξࢇ ࢈ graphic organizer (see sample below).
Ask advanced learners to create a matrix
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ that produces a dilation of ͵ and a rotation
b. ൨
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ of ͵Ͳι counterclockwise.
A pure rotation of ࣂ Matrix Transformation Picture
݇ Ͳ Pure dilation of
ቂ ቃ
c. ቂ
ቃ Ͳ ݇ scale factor ݇
A pure dilation of scale factor
d. ቂ ቃ
The multiplicative identify matrix has no geometric effect.
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e. ቂ ቃ
The additive identify matrix maps all points to the origin.
ࢇ ࢉ
f. ቂ ቃ
࢈ ࢊ
Transforms the unit square to a parallelogram with vertices ሺǡ ሻ, ሺࢇǡ ࢈ሻ, ሺࢉǡ ࢊሻ, and ሺࢇ ࢉǡ ࢈ ࢊሻ with area
of ȁࢇࢊ െ ࢈ࢉȁ.
Example 1 (10 minutes)
In Example 1, students perform a transformation on the unit square, calculate and confirm the area of the image, and
then solve a system of equations that would map the transformation to a given point. Students should complete this
example in groups with guiding questions from the teacher as needed.
Example 1
Scaffolding:
Given the transformation ቂ ቃ with :
Give advanced students a
a. Perform this transformation on the vertices of the unit square. Sketch the image, and
single task: “Write a
label the vertices.
formula for the application
of this transformation ݊
ቂ ቃቂ ቃ ൌ ቂ ቃ
times.” Ask them to
ቂ ቃቂ ቃ ൌ ቂ ቃ develop an answer
without the questions
ቂ ቃቂ ቃ ൌ ቂ ቃ shown.
ቂ
ቃቂ ቃ ൌ ቂ
ቃ
Provide labeled graphs for
students who have
difficulties with eye-hand
coordination or fine motor
skills.
b. Calculate the area of the image using the dimensions of the image parallelogram.
The parallelogram is unit high, and the perpendicular distance between parallel bases is units wide, so the
area is ή ൌ square units.
c. Confirm the area of the image using the determinant.
The area of the unit square is ǡ and the determinant of the transformation matrix is ȁሺሻሺሻ െ ሺሻሺሻȁ ൌ .
The area of the parallelogram is ή ൌ square units. The area is confirmed.
࢞
d. Perform the transformation on ቂ࢟ቃ.
࢞ ࢞
ቂ ቃቂ ቃ ൌ ൨
࢟ ࢞ ࢟
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e. In order for two matrices to be equivalent, each of the corresponding elements must be equivalent. Given
࢞
that, if the image of this transformation is ቂ ቃ, find ቂ࢟ቃ.
࢞
൨ൌቂ ቃ
࢞ ࢟
࢞ ൌ ࢞ ࢟ ൌ
࢞ൌ ൬ ൰ ࢟ ൌ
࢟ൌ
࢟ ൌ െ
࢞
ቂ࢟ቃ ൌ ቈ
െ
f. Perform the transformation on ቂ ቃ. Write the image matrix.
ቂ ቃቂ ቃ ൌ ቂ ቃ
g. Perform the transformation on the image again, and then repeat until the transformation has been
performed four times on the image of the preceding matrix.
ቂ ቃቂ ቃൌ ൨
ቂ
ቃ ൨ൌ ൨
ቂ
ቃ ൨ൌ ൨
What are the vertices of the image?
à ሺͲǡ Ͳሻ, ሺ݇ǡ ݇ሻ, ሺͲǡ ͳሻ, and ሺ݇ǡ ݇ ͳሻ
What is the formula for the area of a parallelogram?
à The area of a parallelogram is ൈ .
What is the base and height of the parallelogram that is the image of this transformation? How do you know?
à The base is the length of one of the parallel sides, which is ͳ unit. The height is the perpendicular
distance between parallel sides, and that is ݇ units.
Using the formula, calculate the area of the parallelogram.
à The area is ݇ square units.
Now find the area using the determinant. Is the area confirmed?
à ൌ ȁሺ݇ሻሺͳሻ െ ሺͲሻ݇ȁ ൌ ݇ square units. This is the same area.
Now perform the transformation on a point ሺݔǡ ݕሻ. What is the matrix that results?
݇ݔ
à ൨
݇ ݔ ݕ
If we want the image of this transformation on ሺݔǡ ݕሻ to map to ሺͷǡ Ͷሻ, how could we find ሺݔǡ ݕሻ?
à We could write a system of equations and solve for ሺݔǡ ݕሻ.
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Write the system of equations.
à ݇ ݔൌ ͷ
à ݇ ݔ ݕൌ Ͷ
Solve for ݔand ݕin terms of ݇. Which variable is easiest to solve for? Explain and solve for it.
ͷ
à It is easiest to solve for ݔbecause the first equation only has ݔ, not ݕ. ݔൌ
݇
Now solve for the other variable.
ͷ
à ݕൌ Ͷ െ ݇ ቀ ቁ ൌ Ͷ െ ͷ ൌ െͳ
݇
ݔ
So, ቂݕቃ is equal to what?
ͷ
à ቈ݇
െͳ
Exercise 1 (8 minutes)
ݔ
This exercise should be completed in pairs and gives students practice solving for ݕ൨ and writing a general formula to
represent ݊ transformations. Some groups may need the leading questions presented in Example 1 to help them.
Exercise 1
1. Perform the transformation ቂ ቃ with on the vertices of the unit square.
a. What are the vertices of the image?
ሺǡ ሻ, ሺǡ ሻ, ሺǡ ሻ, and ሺǡ ሻ
b. Calculate the area of the image.
࢞ െ ࢞
c. If the image of the transformation on ቂ࢟ቃ is ቂ ቃ, find ቂ࢟ቃ in terms of .
െ
࢞ ࢞ െ
ቂ ቃቂ ቃ ൌ ൨ൌቂ ቃ
࢟ ࢞ ࢟ െ
࢞ ൌ െ ࢞ ࢟ ൌ െ
െ െ
࢞ൌ ࢟ ൌ െ
࢟ ൌ െ
െ
࢟ൌ
െ
࢞
ቂ࢟ቃ ൌ ൦ ൪
െ
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Example 2 (10 minutes)
In Lesson 26, the claim was made that a matrix transformation takes straight lines to straight lines. This example
explores that claim, because students discover that matrix transformations are indeed linear, and sets the groundwork
for the work of Lessons 27–30. Students should work in pairs with the teacher leading the discussion.
Example 2
Consider the matrix ࡸ ൌ ቂ ቃ. For each real number ࢚ , consider the point ሺ ࢚ǡ ࢚ሻ.
െ
a. Find point when ࢚ ൌ .
ሺǡ ሻ
b. Find point when ࢚ ൌ .
ሺǡ ሻ
c. Show that for ࢚ ൌ ,ሺ ࢚ǡ ࢚ሻ is the midpoint of തതതത
.
തതതത is ቀ ૠ
When ࢚ ൌ , point ࡹ is ቆ ǡ ቀ ቁቇ, or ሺǤ ǡ ሻ. The midpoint of ǡ ቁ, or ቀ ǡ ቁ.
The midpoint is at ࢚ ൌ .
d. Show that for each value of ࢚ǡ ሺ ࢚ǡ ࢚ሻ is a point on the line through and .
െ
The equation of the line through A and B is ࢟ െ ൌ ሺ࢞ െ ሻ, or ࢟ െ ൌ ሺ࢞ െ ሻ, or ࢟ ൌ ࢞ .
െ
If we substitute ሺ ࢚ǡ ࢚ሻ into the equation, we get ࢚ ൌ ሺ ࢚ሻ or ࢚ ൌ ࢚ ,
which is a statement that is true for all real values of ࢚. Therefore, the point ሺ ࢚ǡ ࢚ሻ lies on the line
through and for all values of ࢚.
e. Find ࡸ and ࡸ.
ࡸ ൌ ቂ ቃቂ ቃ ൌ ቂ ቃ
െ ૠ
ૡ
ࡸ ൌ ቂ ቃቂ ቃ ൌ ቂ ቃ
െ
f. What is the equation of the line through ࡸ and ࡸ?
െૠ
The line through ࡸ and ࡸ is ࢟ െ ૠ ൌ ሺ࢞ െ ሻ or ࢟ െ ૠ ൌ ሺ࢞ െ ሻ.
ૡെ
࢚
g. Show that ࡸ ቂ ቃ lies on the line through ࡸ and ࡸ.
࢚
࢚ ࢚ ࢚ ࢚ ࢚
ࡸቂ ቃൌቂ ቃቂ ቃൌቂ ቃൌቂ ቃ
࢚ െ ࢚ െ െ ࢚ ࢚ ࢚ ૠ
ሺ࢚ ૠሻ െ ૠ ൌ ൫ሺ࢚ ሻ െ ൯
࢚
࢚ ൌ ࢚, which is true for all real values of ࢚, so ࡸ ቂ ቃ and lies on the line through ࡸ and ࡸ.
࢚
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ͳ
Will the midpoint always occur at ݐൌ ? Explain.
ʹ
ଵ
à It will always occur at the ሺ ͳݐ ʹݐሻ. Since ݐଵ ݐଶ ൌ ͳ in this problem, the midpoint occurred
ଶ
ͳ
at ݐൌ .
ʹ
Write an equation for the line through ܣand ܤ. Explain your work.
ͳʹെͳͲ
à ܣሺ͵ǡ ͳͲሻ and ܤሺͶǡ ͳʹሻ, so the slope is ݉ ൌ ൌ ʹ. In point slope form, the equation is
Ͷെ͵
ݕെ ͳͲ ൌ ʹሺ ݔെ ͵ሻ or ݕെ ͳʹ ൌ ʹሺ ݔെ Ͷሻ. In slope-intercept form, the equation is ݕൌ ʹ ݔ Ͷ.
Substitute ݔൌ ͵ ݐand ݕൌ ͳͲ ʹ ݐinto this equation. What do you discover?
à ͳͲ ʹ ݐൌ ʹሺ͵ ݐሻ Ͷ
ͳͲ ʹ ݐൌ ʹ ݐ Ͷ
ͳͲ ʹ ݐൌ ͳͲ ʹݐ
We get a statement that is true for all real values of ݐ.
What does this mean?
à The point ሺ͵ ݐǡ ͳͲ ʹݐሻ lies on the line through ܣand ܤfor all values of ݐ.
Write an equation for the line through ܣܮand ܤܮ.
ͷ ͷ
à ݕെ ʹ ൌ ሺ ݔെ ͷሻ or ݕെ ͵ʹ ൌ ሺ ݔെ ͺሻ
ͳʹ ͳʹ
͵ݐ
Does every point on ܮቂ ቃ lie on the line through ܣܮand ?ܤܮExplain.
ͳͲ ʹݐ
ͷ
à Yes ሺͷ ݐ ʹሻ െ ʹ ൌ ൫ሺͳʹ ݐ ͷሻ െ ͷ൯
ͳʹ
͵ݐ
à ͷ ݐൌ ͷݐǡ which is true for all real values of ;ݐtherefore, ܮቂ ቃ lies on the line through ܣܮand ܤܮ.
ͳͲ ʹݐ
Closing (4 minutes)
Have students explain to a neighbor everything that they learned about matrix transformations in Lessons 26 and 27;
then, pull the class together to debrief.
Explain to your neighbor everything that you learned about matrix transformations in Lessons 26 and 27.
à The image of this transformation is a parallelogram with vertices.
à ሺͲǡ Ͳሻ, ሺܽǡ ܾሻ,ሺܿǡ ݀ሻ, and ሺܽ ܿǡ ܾ ݀ሻ
à The determinant of the ʹ ൈ ʹ transformation matrix is the area of the image of the unit square after
the transformation.
à A ʹ ൈ ʹ transformation can rotate, dilate, and/or change the shape of the unit square.
à A ʹ ൈ ʹ transformation takes straight lines and maps them to straight lines.
Exit Ticket (5 minutes)
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Name Date
Lesson 27: Getting a Handle on New Transformations
Exit Ticket
Ͳ ݇
Given the transformation ቂ ቃ with ݇ Ͳ:
ͳ ݇
a. Find the area of the image of the transformation performed on the unit matrix.
ݔ ͳ ݔ
b. The image of the transformation on ቂݕቃ is ቂ ቃ; find ቂݕቃ in terms of ݇. Show your work.
ͷ
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Exit Ticket Sample Solutions
Given the transformation ቂ ቃ with :
a. Find the area of the image of the transformation performed on the unit matrix.
ȁሺሻሺሻ െ ሺሻሺሻȁ ൌ ȁെȁ ൌ
࢞ ࢞
b. The image of the transformation on ቂ࢟ቃ is ቂ ቃ; find ቂ࢟ቃ in terms of . Show your work.
࢞ ࢟
ቂ ቃቂ ቃ ൌ ൨ൌቂ ቃ
࢟ ࢞ ࢟
࢟ ൌ ࢞ ࢟ ൌ
࢟ൌ ࢞ ൬ ൰ ൌ
࢞ ൌ
࢞ൌ
࢞
ቂ࢟ቃ ൌ ൩
Problem Set Sample Solutions
1. Perform the following transformation on the vertices of the unit square. Sketch the image, label the vertices, and
find the area of the image parallelogram.
a. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units
b. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
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c. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
d. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
e. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ૢ
The area is ૢunits .
f. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
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g. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
h. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
i. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
ૡ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ
The area is units .
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࢞ ࢞ ࢞
2. Given ቂ ቃቂ ቃ ൌ ൨. Find ቂ࢟ቃ if the image of the transformation is the following:
࢟ ࢞ ࢟
a. ቂ ቃ
࢞
൨ൌቂ ቃ
࢞ ࢟
࢞ ൌ ࢞ ࢟ ൌ
࢞ൌ ࢟ൌ
࢟ൌ
࢟ൌ
࢞
ቂ࢟ቃ ൌ
െ
b. ቂ ቃ
࢞ െ
൨ൌቂ ቃ
࢞ ࢟
࢞ ൌ െ ࢞ ࢟ ൌ
െ
࢞ൌെ ࢟ ൌ
െ ࢟ ൌ
࢟ൌ
࢞ െ
ቂ࢟ቃ ൌ
c. ቂ ቃ
െ
࢞
൨ൌቂ ቃ
࢞ ࢟ െ
࢞ ൌ ࢞ ࢟ ൌ െ
࢞ൌ ࢟ ൌ െ
࢟ ൌ െ
࢟ ൌ െ
࢞
ቂ࢟ቃ ൌ
െ
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࢞ ࢞
3. Given ቂ ቃቂ ቃ ൌ ൨. Find the value of so that:
࢟ ࢞ ࢟
࢞
a. ቂ࢟ቃ ൌ ቂ ቃ and the image is ቂ ቃ.
െ
ቂ ቃቂ ቃ ൌ ቂ ቃ
െ
ൌ OR െ ൌ
ൌૡ ൌૡ
࢞ ૠ ૡ
b. ቂ࢟ቃ ൌ ቂ ቃ and the image is ቂ ቃ.
ૠ ૡ
ቂ ቃቂ ቃ ൌ ቂ ቃ
ૠ ൌ ૡ OR ૠ ൌ
ૡ ૡ
ൌ ൌ ൌ ൌ
ૠ ૠ
࢞ ࢞
4. Given ቂ ቃቂ ቃ ൌ ൨. Find the value of so that:
࢟ ࢞ ࢟
࢞ െ െ
a. ቂ࢟ቃ ൌ ቂ ቃ and the image is ቂ ቃ.
െ െ
ቂ ቃቂ ቃ ൌ ቂ ቃ
െ ൌ െ OR െ ൌ
ൌ ൌ
െ
࢞
b. ቂ࢟ቃ ൌ ൦ ൪ and the image is .
െ
ૢ
ې ۍ െ
ێۑ
ቂ ቃ ۑ ێൌ
ێۑ െ
ےૢۏ
OR
ൌ െ ൌെ
ૢ
ൌ െૢ ൌ െૢ
5. Perform the following transformation on the vertices of the unit square. Sketch the image, label the vertices, and
find the area of the image parallelogram.
a. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ൌ
The area is units .
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b. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ൌ ૢ
The area is ૢunits .
c. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ൌ
The area is units .
d. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ൌ
The area is units .
e. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ൌ
The area is units .
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f. ቂ ቃ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃ ቂ ቃ ൌ ቂ ቃ, and ቂ ቃቂ ቃ ൌ ቂ ቃ
ૡ
܉܍ܚۯൌ ȁ ൈ െ ൈ ȁ ൌ ൌ ૢ
The area is ૢunits .
6. Consider the matrix ࡸ ൌ ቂ ቃ. For each real number ࢚ , consider the point ሺ ࢚ǡ ࢚ሻ.
a. Find the point when ࢚ ൌ .
ሺǡ ሻ
b. Find the point when ࢚ ൌ .
ሺǡ ሻ
c. Show that for ࢚ ൌ ǡ ሺ ࢚ǡ ࢚ሻ is the midpoint of തതതത
.
When ࢚ ൌ , the point ࡹ is ሺ ǡ ሻ, or ሺǡ ሻ.
And the midpoint of തതതത
is ቀ ǡ ቁ, or ሺǡ ሻ. Thus, the midpoint is at ࢚ ൌ .
d. Show that for each value of ࢚, ሺ ࢚ǡ ࢚ሻ is a point on the line through and .
തതതത:
The equation of the line through
െ
࢟ െ ൌ ሺ࢞ െ ሻ
െ
࢟ ൌ࢞ૢ
If we substitute ሺ ࢚ǡ ࢚ሻ into the equation, we get ࢚ ൌ ࢚ ૢ, or ࢚ ൌ ࢚,
which is a statement that is true for all real values of ࢚. Therefore, the point ሺ ࢚ǡ ࢚ሻ lies on the line
through and for all values of ࢚.
e. Find ࡸ and ࡸ.
ૢ
ࡸ ൌ ቂ ቃቂ ቃ ൌ ቂ ቃ
ૠ
ࡸ ൌ ቂ ቃቂ ቃ ൌ ቂ ቃ
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f. What is the equation of the line through ࡸ and ࡸ?
െ
࢟ െ ൌ ሺ࢞ െ ૢሻ
ૠ െ ૢ
࢟ െ ൌ ሺ࢞ െ ૢሻ
࢚
g. Show that ࡸ ቂ ቃ lies on the line through ࡸ and ࡸ.
࢚
࢚ ࢚ ࢚ ࢚ ૢ ૡ࢚
ࡸቂ ቃൌቂ ቃቂ ቃൌቂ ቃൌቂ ቃ
࢚ ࢚ ࢚ ૡ ૡ࢚ ࢚
We substitute it into the equation in part (f):
࢚ െ ൌ ሺૢ ૡ࢚ െ ૢሻ
࢚ ൌ ሺૡ࢚ሻ
࢚ ൌ ࢚,
࢚
which is true for all real values of ࢚, so ࡸ ቂ ቃ lies on the line through ࡸ and ࡸ.
࢚
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Lesson 28: When Can We Reverse a Transformation?
Student Outcomes
Students determine inverse matrices using linear systems.
Lesson Notes
In the final three lessons of this module, students discover how to reverse a transformation by discovering the inverse
matrix. In Lesson 28, students are introduced to inverse matrices and find inverses of matrices with a determinant of ͳ
by solving a system of equations. Lesson 29 expands this idea to include inverses of matrices with a determinant other
than ͳ and finding a general formula for an inverse matrix. In Lesson 30, students discover that matrices with
determinants of zero do not have an inverse.
Classwork
The Opening Exercise can be done individually or in pairs. It allows students to practice a ʹ ൈ ʹ matrix transformation on
a unit square. Students need graph paper.
Opening Exercise (8 minutes)
Opening Exercise
െ
Perform the operation ቂ ቃ on the unit square.
a. State the vertices of the transformation.
ሺǡ ሻ, ሺǡ ሻ, ሺെǡ ሻ, andሺǡ ሻ
b. Explain the transformation in words.
ሺǡ ሻ stays at the origin, the vertex ሺǡ ሻ moves to ሺǡ ሻ, ሺǡ ሻ moves to ሺെǡ ሻ, and ሺǡ ሻ moves to
ሺǡ ሻ.
c. Find the area of the transformed figure.
ȁሺሻሺሻ െ ሺെሻሺሻȁ ൌ
The area is units2.
d. If the original square was ൈ instead of a unit square, how would the transformation change?
The coordinates of the vertices of the image would all double. The vertices would be ሺǡ ሻ, ሺǡ ሻ, ሺെǡ ሻ,
and ሺǡ ሻ.
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e. What is the area of the image? Explain how you know.
The area of the image is square units. The area of the original square was square units; multiply that by
the determinant, which is , and the area of the new figure is ൈ ൌ square units.
Discussion (10 minutes)
This Discussion is a whole-class discussion that wraps up the Opening Scaffolding:
Exercise and gets students to think about reversing transformations.
Advanced learners can do the
What are some differences between the unit square and a ʹ ൈ ʹ Discussion in small homogenous groups
square? and do Example 1 with no guiding
à The coordinates of the vertices of the ʹ ൈ ʹ square were questions.
double the coordinates of the vertices of the unit square. Remind students how to multiply ʹ ൈ ʹ
matrices by displaying this graphic:
à The area of the ʹ ൈ ʹ square is four times the area of the
ܽ ܾ ݏ ݎ ܾܽ ܾ ݏܽ ݐ ܾݑ
unit square. ቂ ቃቂ ቃൌቂ ቃ.
ܿ ݀ ݑ ݐ ܿ ݎ ݀ ݏܿ ݐ ݀ݑ
Was the same thing true when the matrix transformation was Remind students of the trigonometry
applied? Pythagorean identities by displaying the
à Yes following:
How can the determinant of the transformation matrix be used to ଶ ሺߠሻ ଶ ሺߠሻ ൌ ͳ.
find the area of a transformed image if the original image was not a
unit square?
à Find the area of the original square, and then multiply that area by the value of the determinant.
What matrix have we studied that produces only a counterclockwise rotation through an angle ߠ about the
origin? Call it ܴఏ .
ሺߠሻ െ ሺߠሻ
à ܴఏ ൌ ൨
ሺߠሻ ሺߠሻ
What transformation would “undo” this transformation? Describe it in words or symbols.
à We can undo this transformation by rotating in the opposite direction or through an angle of െߠ.
à ሺെߠሻ ൌ െ ሺߠሻ because it is an odd function and is symmetric about the origin. ݂ሺെݔሻ ൌ െ݂ሺݔሻ
à ሺെߠሻ ൌ ሺߠሻ because it is an even function and is symmetric about the ݕ-axis. ݂ሺെݔሻ ൌ ݂ሺݔሻ
Write the matrix that represents the rotation through െߠǤ Call it ܴିఏ .
ሺെߠሻ െ ሺെߠሻ ሺߠሻ ሺߠሻ
à ܴିఏ ൌ ൨ൌ ൨
ሺെߠሻ ሺെߠሻ െሺߠሻ ሺߠሻ
What do you think will happen if we apply ܴఏ and then ܴିఏ ?
à We should end up with what we started with.
Let’s confirm this. What matrix do you expect to see when you compute the product ܴିఏ ܴఏ ?
ͳ Ͳ
à The identity matrix ቂ ቃ
Ͳ ͳ
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Perform this operation. Were you correct?
ሺߠሻ ሺߠሻ ሺߠሻ െ ሺߠሻ
à ൨ ൨ൌ
െሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ
ଶ ሺߠሻ ଶ ሺߠሻ െ ሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ ͳ Ͳ
ଶ ଶ ൨ൌቂ ቃ
െሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ Ͳ ͳ
à Yes, we got the identity matrix.
What about ܴఏ ܴିఏ ?
ሺߠሻ െሺߠሻ ሺߠሻ ሺߠሻ
à ൨ ൨ൌ
ሺߠሻ ሺߠሻ െሺߠሻ ሺߠሻ
ଶ ሺߠሻ ଶ ሺߠሻ െሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ ͳ Ͳ
൨ൌቂ ቃ
െ ሺߠሻ ሺߠሻ ሺߠሻ ሺߠሻ ଶ ሺߠሻ ଶሺߠሻ Ͳ ͳ
à Yes, we get the identify matrix again.
Explain to your neighbor what we have just discovered.
ͳ Ͳ
à ܴିఏ ܴఏ ൌ ܴఏ ܴିఏ ൌ ቂ ቃ, the identify matrix
Ͳ ͳ
à If the transformation ܴఏ is performed, it can be reversed by performing the transformation ܴିఏ .
Example 1 (10 minutes)
In this example, students solve a system of equations to find the transformation that reverses the pure dilation matrix
with a scale factor of ݇. This example concludes with students writing their own definition of an inverse matrix and then
comparing it to the formal definition. Students should work in small homogenous groups or pairs. Some groups can
work through without guided questions while others may need targeted teacher support.
Example
What transformation reverses a pure dilation from the origin with a scale factor of ?
ࢇ ࢉ
a. Write the pure dilation matrix, and multiply it by ቂ ቃ.
࢈ ࢊ
ࢇ ࢉ ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ ࢈ ࢊ
b. What values of ࢇǡ ࢈ǡ ࢉ, and ࢊ would produce the identity matrix? (Hint: Write and solve a system of
equations.)
ࢇ ࢉ
ቂ ቃൌቂ ቃ
࢈ ࢊ
ࢇ ൌ ǡ ࢉ ൌ ǡ ࢈ ൌ ǡ ࢊ ൌ
ࢇ ൌ ǡ ࢉ ൌ ǡ ࢈ ൌ ǡ ࢊ ൌ
c. Write the matrix, and confirm that it reverses the pure dilation with a scale factor of .
൦ ൪ቂ ቃൌቂ ቃ
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What is the pure dilation matrix with a scale factor of ݇?
݇ Ͳ
à ቂ ቃ
Ͳ ݇
ܽ ܿ
Multiply by a general matrix ቂ ቃ. What is the resulting matrix?
ܾ ݀
ܽ݇ ܿ݇
à ቂ ቃ
ܾ݇ ݀݇
What matrix would this have to be equal to if the transformation had been reversed? Write that matrix.
ͳ Ͳ
à The identity matrix, ቂ ቃ
Ͳ ͳ
Equate the two matrices, and write a system of equations that would have to be true for the matrices to be
equal.
ܽ݇ ܿ݇ ͳ Ͳ
à ቂ ቃൌቂ ቃ
ܾ݇ ݀݇ Ͳ ͳ
à ܽ݇ ൌ ͳ, ܾ݇ ൌ Ͳ, ܿ݇ ൌ Ͳ, and݀݇ ൌ ͳ
Solve this system for ܽ, ܾ, ܿ, and ݀ in terms of ݇.
ͳ ͳ
à ܽ ൌ , ܾ ൌ Ͳ, ܿ ൌ Ͳ, and ݀ ൌ
݇ ݇
Write the matrix that reverses the pure dilation transformation with a scale factor of ݇.
ͳ
Ͳ
à ݇ ͳ
Ͳ
݇
Confirm that this is the matrix that reverses the transformation. Explain.
ͳ
Ͳ ݇ Ͳ ͳ Ͳ
݇
à ቂ
ͳ Ͳ
ቃൌቂ ቃ
Ͳ ݇ Ͳ ͳ
݇
à When the two matrices are multiplied, you get the identity matrix, which means that the
transformation has been reversed.
If the transformations were done in the reverse order, would they still “undo” each other? Show your work.
ͳ
݇ Ͳ ݇ Ͳ ͳ Ͳ
à Yes ቂ ቃ ͳ
ൌቂ ቃ
Ͳ ݇ Ͳ Ͳ ͳ
݇
Let’s call original matrix ܣ, the matrix that reverses the transformation ܤ, and the identity matrix ܫ. Write a
statement that is true that relates the three matrices.
à ܤܣൌ ܫ, ܣܤൌ ܫ
We call matrix ܤan inverse matrix of matrix ܣ. Write a definition of the inverse matrix.
à Matrix ܤis an inverse matrix to matrix ܣif ܤܣൌ ܫand ܣܤൌ ܫ.
Exercises 1–3 (10 minutes)
Students find the inverse matrices of each matrix given. Exercises 1 and 2 require students to solve a system of four
equations and four variables. This is not as difficult as it may seem, since two of the equations are equal to zero. In
Lesson 29, a general formula is developed for the inverse of any matrix; in this exercise, students start seeing patterns
relating the inverse matrix and the original matrix.
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These problems were all chosen because their determinant is zero, so students can focus on the movement of terms and
changing of signs. All students should do Exercises 1 and 2. Early finishers can also do Exercise 3. The results of this
exercise are used in the Opening Exercise of Lesson 29, asking if students see a pattern.
Exercises
Find the inverse matrix and verify.
1. ቂ ቃ
ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ
ࢇࢉ ࢉ
ቂ ቃൌቂ ቃ
࢈ࢊ ࢊ
ࢇ ࢉ
ቂ ቃൌቂ ቃ
࢈ ࢊ െ
2. ቂ ቃ
ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ
ࢇ ࢉ ࢇ ࢉ
ቂ ቃൌቂ ቃ
࢈ ࢊ ࢈ ࢊ
ࢇ ࢉ െ
ቂ ቃൌቂ ቃ
࢈ ࢊ െ
െ െ
3. ቂ ቃ
ࢇ ࢉ െ െ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ
െࢇ ࢉ െࢇ ࢉ
ቂ ቃൌቂ ቃ
െ࢈ ࢊ െ࢈ ࢊ
ࢇ ࢉ
ቂ ቃൌቂ ቃ
࢈ ࢊ െ െ
Closing (2 minutes)
Students should do a 30-second quick write and then share with the class the answer to the following:
What is an inverse matrix?
à An inverse matrix is a matrix that when multiplied by a given matrix, the product is the identity matrix.
à An inverse matrix “undoes” a transformation.
Explain how to find an inverse matrix.
ܽ ܿ
à Multiply a general matrix ቂ ቃ by a given matrix, and set it equal to the identify matrix. Solve the
ܾ ݀
system of equations for ܽ, ܾ, ܿ, and ݀Ǥ
Exit Ticket (5 minutes)
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Name Date
Lesson 28: When Can We Reverse a Transformation?
Exit Ticket
Ͷ െʹ ͵ ʹ
ܣൌቂ ቃ ܤൌ ቂ ቃ
െͳ ͵ ͳ Ͷ
1. Is matrix ܣthe inverse of matrix ?ܤShow your work, and explain your answer.
2. What is the determinant of matrix ?ܤOf matrix ?ܣ
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Exit Ticket Sample Solutions
െ
ൌቂ ቃ ൌ ቂ ቃ
െ
1. Is matrix the inverse of matrix ? Show your work, and explain your answer.
No, the product of the two matrices is not the identity matrix.
െ
ቂ ቃቂ ቃൌቂ ቃ
െ
2. What is the determinant of matrix ? Of matrix ?
The determinant of matrix ൌ ڿሺሻሺሻ െ ሺെሻሺെሻ ۀൌ .
The determinant of matrix ൌ ȁሺሻሺሻ െ ሺሻሺሻȁ ൌ .
Problem Set Sample Solutions
1. In this lesson, we learned ࡾࣂ ࡾିࣂ ൌ ቂ ቃ. Chad was saying that he found an easy way to find the inverse matrix,
൨
which is ࡾିࣂ ൌ . His argument is that if we have ࢞ ൌ , then ࢞ ൌ .
ࡾࣂ
a. Is Chad correct? Explain your reason.
Chad is not correct. Matrices cannot be divided.
b. If Chad is not correct, what is the correct way to find the inverse matrix?
To find the inverse of ࡾିࣂ , calculate the determinant, switch the terms on the forward diagonal and change
the signs on the back diagonal, and then divide all terms by the absolute value of the determinant.
2. Find the inverse matrix and verify it.
a. ቂ ቃ
ૠ
ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
ૠ ࢈ ࢊ
ࢇ ࢈ ࢉ ࢊ
ቂ ቃൌቂ ቃ ǡࢇ ࢈ ൌ ǡࢉ ࢊ ൌ ǡૠࢇ ࢈ ൌ ǡૠࢉ ࢊ ൌ
ૠࢇ ࢈ ૠࢉ ࢊ
ࢇ ࢉ െ
Solve ࢇǡ ࢈ǡ ࢉǡ ࢊǣ ቂ ቃൌቂ ቃ
࢈ ࢊ െૠ
െ Ȃ ൨ ൌ ቂ ቃ
Verify: ቂ ቃቂ ቃ ൌ െ
ૠ െૠ െ െ
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െ െ
b. ቂ ቃ
െ െ ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ
െࢇ െ ࢈ െࢉ െ ࢊ
ቂ ቃൌቂ ቃǡെ ࢇ െ ࢈ ൌ ǡ െ ࢉ െ ࢊ ൌ ǡࢇ ࢈ ൌ ǡࢉ ࢊ ൌ
ࢇ ࢈ ࢉ ࢊ
ࢇ ࢉ
Solve ࢇ, ࢈, ࢉ, ࢊ: ቂ ቃൌቂ ቃ
࢈ ࢊ െ െ
െ െ െ െ
Verify: ቂ ቃቂ ቃൌቂ ቃൌቂ ቃ
െ െ െ െ
െ
c. ቂ ቃ
െ
The determinant is ; therefore, there is no inverse matrix.
d. ቂ ቃ
െ
ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
െ ࢈ ࢊ
࢈ ࢊ
ቂ ቃൌቂ ቃ ǡ࢈ ൌ ǡࢊ ൌ ǡ െ ࢇ ࢈ ൌ ǡ െ ࢉ ࢊ ൌ
െࢇ ࢈ െࢉ ࢊ
ࢇ ࢉ െ
Solve ࢇ, ࢈, ࢉ, ࢊ: ቂ ቃൌቂ ቃ
࢈ ࢊ
െ
Verify: ቂ ቃቂ ቃൌቂ ቃൌቂ ቃ
െ െ
e. ቂ ቃ
ࢇ ࢉ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ
ࢇ ࢈ ࢉ ࢊ ࢇ ࢈ ࢉ ࢊ
ቂ ቃൌቂ ቃ ǡࢇ ࢈ ൌ ǡ ൌ ǡࢇ ࢈ ൌ ǡࢉ ࢊ ൌ
ࢇ ࢈ ࢉ ࢊ ࢇ ࢈ ࢉ ࢊ
ࢇ ࢉ
Solve ࢇ, ࢈, ࢉ, ࢊ: ቂ ቃ ൌ ቈ െ
࢈ ࢊ
െ
െ െ Ȃ ൨ ൌ ቂ ቃ
Verify: ቂ ቃ ቈ ൌ
െ െ െ
࢞
3. Find the starting point ቂ࢟ቃ if:
a. The point ቂ ቃ is the image of a pure dilation with a factor of .
ې ۍ
࢞
ቂ࢟ቃ ൌ ۑۑ ێൌ ቂ ቃ
ێ
ێۑ
ۏے
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b. The point ቂ ቃ is the image of a pure dilation with a factor of .
ۍ ې
ێ ۑ
࢞
ቂ࢟ቃ ൌ ێ ۑൌ ቂૡቃ
ێ ۑ
ێ ۑ
ۏ ے
െ
c. The point ቂ ቃ is the image of a pure dilation with a factor of .
െ
ۍ ې
࢞ ێ ۑ െ
ቂ࢟ቃ ൌ ێ ۑൌቂ ૠ ቃ
ێ ۑ
ۏ ے
d. The point ૢ is the image of a pure dilation with a factor of .
ې ૢ ۍ
ێ ۑ
࢞ ێ ۍ ۑې
ቂ࢟ቃ ൌ ێ ۑൌۑ ێ
ێ ێێ ۑૡۑۑ
ێ ۏ ۑૠے
ێ ۑ
ۏ ے
4. Find the complex number if:
a. is the image of a reflection about the real axis.
ࢠത ൌ െ
b. is the image of a reflection about the imaginary axis.
തതതതതതതത
െࢠത ൌ െሺ ଙሻ ൌ െሺ െ ሻ ൌ െ
c. is the image of a reflection about the real axis and then the imaginary axis.
ധധധധധധധധ
െࢠധ ൌ െ൫ തതതതതതതത
ଙ൯ ൌ െሺ െ ଙሻ ൌ െሺ ሻ ൌ െ െ
d. െ െ is the image of a ࣊ radians counterclockwise rotation.
࢞
ቂ࢟ቃ ൌ ൌ
ή
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ
5. Let’s call the pure counterclockwise rotation of the matrix ൨ as ࡾࣂ , and the “undo” of the pure
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
ܛܗ܋ሺെࣂሻ െܖܑܛሺെࣂሻ
rotation is ൨ as ࡾିࣂ .
ܖܑܛሺെࣂሻ ܛܗ܋ሺെࣂሻ
ܛܗ܋ሺെࣂሻ െܖܑܛሺെࣂሻ
a. Simplify ൨.
ܖܑܛሺെࣂሻ ܛܗ܋ሺെࣂሻ
ܛܗ܋ሺെࣂሻ െܖܑܛሺെࣂሻ ܛܗ܋ሺࣂሻ ܖܑܛሺࣂሻ
൨ൌ ൨
ܖܑܛሺെࣂሻ ܛܗ܋ሺെࣂሻ െܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
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b. What would you get if you multiply ࡾࣂ to ࡾିࣂ ?
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ ܖܑܛሺࣂሻ
ࡾࣂ ൈ ࡾିࣂ ൌ ൨ ൨
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
࢙ܗ܋ ሺࣂሻ ܖܑܛ ሺࣂሻ ܛܗ܋ሺࣂሻ ή ܖܑܛሺࣂሻ െ ܖܑܛሺࣂሻ ή ܛܗ܋ሺࣂሻ
ൌ ൨
ܖܑܛሺࣂሻ ή ܛܗ܋ሺࣂሻ െ ܛܗ܋ሺࣂሻ ή ܖܑܛሺࣂሻ ܖܑܛ ሺࣂሻ ܛܗ܋ ሺࣂሻ
ൌቂ ቃ
࣊
c. Write the matrix if you want to rotate radians counterclockwise.
࣊ ࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
െ
൦ ࣊ ࣊ ൪ ൌ ቂ ቃ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ
࣊
d. Write the matrix if you want to rotate radians clockwise.
࣊ ࣊
ܛܗ܋ቀെ ቁ െ ܖܑܛቀെ ቁ
൦ ࣊ ࣊ ൪ ൌ ቂെ ቃ
ܖܑܛቀെ ቁ ܛܗ܋ቀെ ቁ
࣊
e. Write the matrix if you want to rotate radians counterclockwise.
࣊ ࣊ ۍξ െ ې
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
൦ ێ ۑ
࣊ ࣊ ൪ ൌ ێ ξ ۑ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ێ ۑ
ۏ ے
࣊
f. Write the matrix if you want to rotate radians counterclockwise.
࣊ ࣊ ۍξ െ ξې
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
൦ ێ ۑ
࣊ ࣊ ൪ ൌ ێξ ξ ۑ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ێ ۑ
ۏ ے
࣊
g. If the point ቂ ቃ is the image of a point that has been rotated radians counterclockwise,
࢞
find the coordinates of the original point ቂ࢟ቃ.
࣊ ࣊ ۍξ ξې
࢞ ܛܗ܋ቀെ ቁ െ ܖܑܛቀെ ቁ
ێ ۑ ξ
ቂ࢟ቃ ൌ ൦ ࣊ ࣊ ൪ ቂ ቃ ൌ ێξ ξ ۑቂቃ ൌ ൨
ܖܑܛቀെ ቁ ܛܗ܋ቀെ ቁ ێ ۑ ξ
ۏെ ے
࣊
h. If the point ඥ is the image of a point that has been rotated radians counterclockwise, find the
࢞
coordinates of the original point ቂ࢟ቃ.
࣊ ࣊ ξ ې
ܛܗ܋ቀെ ቁ െ ܖܑܛቀെ ቁ ۍ ۍ ې ۍ ۍ ېξې
࢞ ێ ۑ ۑൌ ێ ۑ
ቂ࢟ቃ ൌ ൦ ൪ ێ ۑൌ ێێ ۑ
࣊ ࣊
ܖܑܛቀെ ቁ ܛܗ܋ቀെ ቁ ێξێ ۑെ ξێ ۑξ ێ ۑ ۑ
ۏ ۏ ے ۏ ے ۏ ے ے
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Lesson 29: When Can We Reverse a Transformation?
Student Outcomes
Students understand that an inverse transformation, when represented by a ʹ ൈ ʹ matrix, exists precisely
when the determinant of that matrix is nonzero.
Lesson Notes
Lesson 29 is the second of a three-day lesson sequence. In Lesson 28, students were introduced to inverse matrices and
asked to find inverses of matrices with a determinant of ͳ by solving a system of equations. Lesson 29 has students
finding the inverse of any matrix and understanding when a matrix does not have an inverse.
Classwork
The Opening Exercise can be done individually or in pairs. Students use the skills learned in Lesson 28 to find an inverse
matrix and then compare that inverse to inverses of other matrices determined in Lesson 28. Students see a pattern.
Then, they see that that pattern only works if the determinant is ͳ. This leads to a general formula for any matrix
followed by the question, “Do all matrices have inverses?”
Opening Exercise (5 minutes)
Opening Exercise
െૠ െ
Find the inverse of ቂ
ቃ. Show your work. Confirm that the matrices are inverses.
ࢇ ࢉ െૠ െ
ቂ ቃቂ ቃൌቂ ቃ
࢈ ࢊ
െૠࢇ ࢉ ൌ ǡ െࢇ ࢉ ൌ ǡ െૠ࢈ ࢊ ൌ ǡ െ࢈ ࢊ ൌ
ࢇ ൌ ǡ ࢈ ൌ െǡ ࢉ ൌ ǡandࢊ ൌ െૠ
െૠ െ
ቂ ቃቂ ቃൌቂ ቃ
െ െૠ
Exploratory Challenge (10 minutes)
In this Exploratory Challenge, students look at the patterns of matrices and their inverses that they found in
Exercises 1–3 of Lesson 28 and the Opening Exercise of Lesson 29. This leads to the discovery of the general formula for
the inverse of any matrix. Students should work in small groups.
Do you think all matrices have inverses? Explain why or why not.
Answers will vary. Allow students to state their opinions and explain. Do not add to the discussion; students discover
the correct answer in this Exploratory Challenge.
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Post or project the following:
Matrix Inverse
ͳ Ͳ ͳ Ͳ
ቂ ቃ ቂ ቃ
ͳ ͳ െͳ ͳ
͵ ͳ ʹ െͳ
ቂ ቃ ቂ ቃ
ͷ ʹ െͷ ͵
െʹ െͷ ʹ ͷ
ቂ ቃ ቂ ቃ
ͳ ʹ െͳ െʹ
െ െʹ ͳ ʹ
ቂ ቃ ቂ ቃ
Ͷ ͳ െͶ െ
In Lesson 28, you found the inverses of the first three matrices, and in the
Scaffolding:
Opening Exercise, you found the inverse of the last matrix. Do you see any
patterns between the original matrix and its inverse? Some student pairs may
need targeted one-to-one
à The numbers in the top left and bottom right corners seem to change guidance on this challenge.
places. Consider pairing groups
à The numbers in the top right and bottom left corners change signs. and having a larger group
Do you think this is true for the inverse of all matrices? that is teacher led.
Give advanced students a
Answers will vary, but most students will think that yes, this is true. single task: “Write a
Let’s see if we are right. Find the inverse of the matrix in Exercise 1 using the formula for an inverse
pattern we discovered, and confirm that it is indeed the inverse. matrix after studying the
patterns, and verify your
formula.” Ask them to
Exercise 1 (3 minutes) develop an answer
without the questions
shown.
Exercises
1. Find the inverse of ቂ ቃ. Confirm your answer.
െ
ቂ ቃቂ ቃൌቂ ቃ
െ
Was the matrix that you found using the pattern the inverse? What was missing?
à No. Where we needed ͳ’s, we had ͳͶ’s.
Let’s look at this a little further. Look at the matrices in the table. Find the determinant of the matrices.
(Assign different groups/pairs different matrices from above.)
à All of the determinants were ͳ.
Do you think that makes a difference? What was the determinant of the matrix in Exercise 1?
à The determinant was ͳͶ.
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How does that compare to the matrix that resulted from multiplying the matrices in Exercise 1?
à That was the number that was in the position that should have been a ͳ.
How do you think this ties into the way we find an inverse matrix?
à We can still use our pattern, but we need to divide each term by the determinant. (Answers may vary,
but let students try out their hypotheses to come up with the right answer.)
Try it on the inverse matrix in Exercise 1. Write the inverse matrix.
Ͷ ͵
െ
ͳͶ ͳͶ
à ʹ ͷ
െ
ͳͶ ͳͶ
Verify that is the inverse. Were you correct?
Ͷ ͵
െ ͳ Ͳ
ͳͶ ͳͶ ͷ ͵
à ʹ ͷ
ቂ ቃൌቂ ቃ
െ ʹ Ͷ Ͳ ͳ
ͳͶ ͳͶ
à Yes. We get the identify matrix again.
Explain to your neighbor how to find the inverse of a matrix.
à Switch the numbers in the top left and bottom right. Change the signs of the numbers in the top right
and bottom left. Divide all of the terms by the determinant of the original matrix.
Exercises 2–4 (10 minutes)
In Exercises 2 and 3, students practice finding an inverse matrix and confirm their results. In Exercise 4, students find the
inverse matrix of a general matrix. Choose exercises based on the needs of students; Exercises 2 and 3 are simpler while
Exercise 4 is more complicated. Students should complete this exercise in small groups and then present their findings
to the class.
Find the inverse matrix and verify.
െ
2. ቂ ቃ
Determinant ൌ ሺሻሺሻ െ ሺെሻሺሻ ൌ ൌ
െ
൦ ൪ቂ ቃൌቂ ቃ
െ
െ
3. ቂ ቃ
െ
Determinant ൌ ሺሻሺെሻ െ ሺെሻሺሻ ൌ െ ૡ ൌ െૠ
െ
൦ૠ ૠ൪ ቂ െቃ ൌ ቂ ቃ
െ
െ
ૠ ૠ
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ࢇ ࢉ
4. ቂ ቃ
࢈ ࢊ
Determinant ൌ ሺࢇሻሺࢊሻ െ ሺࢉሻሺ࢈ሻ ൌ ࢇࢊ െ ࢉ࢈
ࢊ ࢉ
െ
൦ ࢇࢊ െ ࢉ࢈ ࢇࢊ െ ࢉ࢈൪ ቂࢇ ࢉ ቃ ൌ ቂ ቃ
࢈ ࢇ ࢈ ࢊ
െ
ࢇࢊ െ ࢉ࢈ ࢇࢊ െ ࢉ࢈
Example 1 (10 minutes)
In this example, students calculate the determinant of a matrix and find that it is Ͳ; they then try to find the inverse of
the matrix. They discover that there is no inverse and then explore what that means about the resulting image.
Students conclude that matrices with a determinant of Ͳdo not have inverses. Students need graph paper.
Example 1
Find the determinant of ቂ ቃ.
The determinant is .
ͳ ʹ
Now that we have calculated the determinant and found it to be Ͳ, let’s examine the inverse of ቂ ቃ.
ʹ Ͷ
Students may struggle, but they should see that they cannot divide by Ͳ, so there will be an issue finding the inverse.
Let’s try to solve for the inverse with a system of equations.
ܽ ܿ ͳ ʹ ܽ ʹܿ ʹܽ Ͷܿ ͳ Ͳ
à ቂ ቃቂ ቃൌቂ ቃൌቂ ቃ
ܾ ݀ ʹ Ͷ ܾ ʹ݀ ʹܾ Ͷ݀ Ͳ ͳ
à ܽ ʹܿ ൌ ͳ, ʹܽ Ͷܿ ൌ Ͳ, ܾ ʹ݀ ൌ Ͳ, ʹܾ Ͷ݀ ൌ ͳ
What did you discover?
à We get a system of equations with no solutions.
What do you think this means about the inverse of this matrix?
à This matrix does not have an inverse.
Let’s explore this further using what we know about matrix
transformations of the unit square.
ͳ ʹ
Perform the operation ቂ ቃ on the unit square. What are the
ʹ Ͷ
coordinates of the vertices of the unit square on the image?
ͳ ʹ Ͳ Ͳ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ʹ Ͷ Ͳ Ͳ
ͳ ʹ ͳ ͳ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ʹ Ͷ Ͳ ʹ
ͳ ʹ Ͳ ʹ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ʹ Ͷ ͳ Ͷ
ͳ ʹ ͳ ͵
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ʹ Ͷ ͳ
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Plot the unit square and the transformation. What do you notice?
à The image is a line.
What is the area of the image?
à The image is a line, not a parallelogram, so the area is Ͳ.
What does the determinant of the transformation represent?
à It represents the area of the image of the unit square after the transformation.
Is the area confirmed?
à Yes. The determinant is Ͳ, so the area of the transformation is Ͳ.
ͳ
The points ሺͳǡ Ͳሻ and ቀͲǡ ቁ are both on the unit square. Perform this transformation on each of these points.
ʹ
ͳ ʹ ͳ ͳ
à ቂ ቃቂ ቃ ൌ ቂ ቃ
ʹ Ͷ Ͳ ʹ
ͳ ʹ Ͳ ͳ
à ቂ ቃ ቈ ͳ ൌ ቂ ቃ
ʹ Ͷ ʹ ʹ
What does this mean?
à Both points map to the same location.
When the unit square “collapses” to a straight line under a transformation, we will always have more than one
point mapping to the same location. This means that we cannot “undo” this transformation, because there is
ͳ
no clear way to reverse the transformation. Would ሺͳǡ ʹሻ map back to ሺͳǡ Ͳሻor ቀͲǡ ቁ? We are not sure.
ʹ
When does a matrix not have an inverse?
à When the image of the unit square “collapses” to a figure of Ͳ area, we have distinct points mapping to
the same location, so there is no inverse.
à When the determinant of the matrix is Ͳ
Closing (2 minutes)
Students should do a 30-second quick write and then share with the class the answer to the following:
What is an inverse matrix?
à An inverse matrix is a matrix that when multiplied by a given matrix, the product is the identity matrix.
à An inverse matrix “undoes” a transformation.
Explain how to find an inverse matrix.
ܽ ܿ
à Multiply a general matrix ቂ ቃ by a given matrix, and set it equal to the identify matrix.
ܾ ݀
Solve the system of equations for ܽ, ܾ, ܿ, and ݀Ǥ
Exit Ticket (5 minutes)
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Name Date
Lesson 29: When Can We Reverse a Transformation?
Exit Ticket
Ͷ െʹ
ܣൌቂ ቃ
െͳ ͵
1. Find the inverse of ܣ. Show your work, and confirm your answer.
͵
2. Explain why the matrix ቂ ቃ has no inverse.
Ͷ ʹ
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Exit Ticket Sample Solutions
െ
ൌቂ ቃ
െ
1. Find the inverse of . Show your work, and confirm your answer.
Determinant ൌ ሺሻሺሻ െ ሺെሻሺെሻ ൌ െ ൌ
െ
൦ ൪ቂ ቃൌቂ ቃ
െ
2. Explain why the matrix ቂ ቃ has no inverse.
Determinant ൌ ሺሻሺሻ െ ሺሻሺሻ ൌ
This means the area of the image is because the image of the unit square maps to a straight line, which has no
area. This also means that distinct points map to the same location, so the transformation cannot be reversed.
Problem Set Sample Solutions
Find the inverse matrix of the following.
a. ቂ ቃ
Determinantൌ െ ൌ Inverse matrix: ቂ ቃ
Verify:
ቂ ቃቂ ቃൌቂ ቃ
b. ቂ ቃ
Determinant ൌ െ ൌ െ Inverse matrix: ቂ ቃ
Verify:
ቂ ቃቂ ቃൌቂ ቃ
c. ቂ ቃ
Determinant ൌ െ ൌ No inverse matrix
d. ቂ ቃ
Determinant ൌ No inverse matrix
e. ቂ ቃ
Determinant ൌ No inverse matrix
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െ
f. ቂ ቃ
െ
െ
Determinant ൌ െૡ ൌ Inverse matrix: ቈ
െ
Verify:
െ
െ
ቂ ቃ ൩ൌቂ ቃ
െ െ
g. ቂ ቃ
ૡ
െ
Determinant ൌ െ ൌ Inverse matrix: ቈ
Verify:
െ
ቂ ቃ ൩ൌቂ ቃ
ૡ
െૢ
h. ቂ ቃ
െૠ
ૠ
െ
Determinant ൌ െ ൌ Inverse matrix:
െ
Verify:
ૠ
െ
െૢ
ቂ ቃ൦ ൪ൌቂ ቃ
െૠ
െ
i. ቈ െ
െ
െ െ
Determinant ൌ െ ൌ െ Inverse matrix:
െ
Verify:
െ െ
െ ൪ ൌ ቂ ቃ
൩ ൦
െ െ
Ǥ ૡ Ǥ
j. ቂ ቃ
െǤ ૠ െǤ
െ
Determinant ൌ െǤ Ǥ ൌ െǤ Inverse matrix: ቂ ቃ
െૠǤ െૡ
Verify:
Ǥ ૡ Ǥ െ
ቂ ቃቂ ቃൌቂ ቃ
െǤ ૠ െǤ െૠǤ െૡ
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Student Outcomes
Students understand that an inverse transformation, when represented by a ʹ ൈ ʹ matrix, exists precisely
when the determinant of that matrix is nonzero.
Lesson Notes
Lesson 30 is the last of a three-day lesson sequence and the last lesson of Module 1. In Lessons 28 and 29, students
studied inverse matrices and found that some matrices do not have inverses. Lesson 30 allows students to practice
these concepts while revisiting rotations and dilations.
Classwork
The Opening Exercise serves as a review of concepts studied in the second half of Module 1. Conduct this exercise as a
Rapid White Board Exchange, using it as a way to informally assess students. This allows teachers to assign
homogeneous groups for the lesson.
Teachers should show one problem at a time either by projecting them or writing them on a personal white board.
Allow students time to write answers on their personal white boards, and then signal students when to show their
answers. Simple mistakes can be explained immediately. Students struggling can be assigned to groups that get more
teacher attention during the lesson.
Opening Exercise (13 minutes)
Give students five minutes to complete the Opening Exercise individually, and then group students to compare answers.
Have groups present their answers to each question and discuss.
Opening Exercise
a. What is the geometric effect of the following matrices?
i. ቂ ቃ
A pure dilation with a scale factor of
ࢇ െ࢈
ቂ ቃ
ii.
࢈ ࢇ
A pure rotation
ܛܗ܋ሺࣂሻ െ ܖܑܛሺࣂሻ
iii. ൨
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
A pure rotation ࣂι counterclockwise
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b. Jadavis says that the identity matrix is ቂ ቃ. Sophie disagrees and states that the identity matrix is ቂ ቃ.
i. Their teacher, Mr. Kuzy, says they are both correct and asks them to explain their thinking about
matrices to each other but to also use a similar example in the real number system. Can you state each
of their arguments?
Jadavis says that any matrix added to the matrix ቂ ቃ does not change. This matrix is similar to in
the real number system, so it is the additive identity matrix.
Sophie explains that when a matrix is multiplied by ቂ ቃ, the matrix does not change, just like the
number in the real number system; so, this matrix is the multiplicative identity.
ii. Mr. Kuzy then asks each of them to explain the geometric effect that their matrix would have on the
unit square.
ቂ ቃ would collapse the entire square to ሺǡ ሻ.
Scaffolding:
ቂ ቃ would have no effect on the unit square.
Ask advanced learners, in
place of Example 1, to
c. Given the matrices below, answer the following: determine and explain the
meaning of the matrix that
ൌቂ ቃ ൌቂ ቃ
is the inverse of
i. Which matrix does not have an inverse? Explain algebraically and
geometrically how you know. ͳ ඥ͵
െ
Matrix does not have an inverse. The determinant is , which means it would
൦ʹ ʹ൪
ඥ͵ ͳ
transform the unit square to a straight line with no area. ʹ ʹ
After assessing students in
ii. If a matrix has an inverse, find it. the Opening Exercise,
choose a small group for
െ
ି
ൌ൦ ൪ targeted instruction. This
െ is an opportunity to
solidify the concepts of
this module with all
students.
Example 1 (5 minutes)
Example 1 has students determine the transformation performed on the unit circle, determine the area of the image,
determine the exact transformation, and then find the inverse matrix. This problem should be modeled with the entire
class. Students need graph paper.
Example
ඥ
െ
Given൦ ൪
ඥ
a. Perform this transformation on the unit square, and sketch the
results on graph paper. Label the vertices.
See diagram at right.
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b. Explain the transformation that occurred to the unit square.
The transformation is a ι rotation counterclockwise about the origin.
c. Find the area of the image.
ξ ξ
ቤ൬ ൰ ൬ ൰ െ ቆെ ቇ ቆ ቇቤ ൌ ൌ
The area is units .
d. Find the inverse of this transformation.
ۍ ξې
ێ ۑ
ێ ۑۑ
ێെ ξ
ۏ ے
e. Explain the meaning of the inverse transformation on the unit square.
The inverse is a െι rotation counterclockwise about the origin.
What are the vertices of the transformed image?
ͳ ඥ͵ ඥ͵ ͳ ͳ ඥ͵ ඥ͵ ͳ
à ሺͲǡ Ͳሻ, ൬ ǡ ൰, ൬െ ǡ ൰, and൬ െ ǡ ൰
ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ
ͳ ඥ͵
Look at the special triangle formed connecting the origin, the vertex of the image ൬ ǡ ൰, and the ݔ-axis.
ʹ ʹ
What type of transformation occurred? Explain.
à This is a ͵Ͳι-Ͳι-ͻͲι triangle. The unit square was rotated Ͳι counterclockwise about the origin.
What is the area of the image? Explain.
à Only a rotation occurred, so the area did not change. The area is ͳ square unit. This is confirmed
because the determinant is ͳ.
What is the inverse matrix?
ͳ ඥ͵
à ൦ ʹ ʹ൪
ඥ͵ ͳ
െ
ʹ ʹ
Explain this transformation.
à If the original transformation was a Ͳι counterclockwise rotation about the origin, the inverse is a
െͲι rotation, or a Ͳι clockwise rotation about the origin.
Exercises 1–8 (20 minutes)
Students should work on these exercises in pairs or small groups. All students should complete Exercises 1 and 2. The
other exercises can be assigned to specific groups or all groups. If problems are assigned to specific groups, have groups
work their assigned problems and then post solutions and either present them to the class or have a gallery walk.
Exercises 3 and 4 are similar, but Exercise 4 is more challenging. Exercises 5, 6, and 7 are also similar, with 6 and 7 being
more challenging. Exercise 8 can be assigned to stronger students. Students need graph paper.
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Exercises
1. Given
a. Perform this transformation on the unit square, and sketch the
results on graph paper. Label the vertices.
b. Explain the transformation that occurred to the unit square.
The transformation is a dilation with a scale factor of .
c. Find the area of the image.
ฬ൬ ൰ ൬ ൰ െ ሺሻሺሻฬ ൌ
The area is units .
d. Find the inverse of this transformation.
ቂ ቃ
e. Explain the meaning of the inverse transformation on the unit square.
The inverse is a dilation with a scale factor of .
f. If any matrix produces a dilation with a scale factor of , what would the inverse matrix produce?
It would produce a dilation with a scale factor of .
െ
ඥ ඥ
2. Given൦ ൪
ඥ ඥ
a. Perform this transformation on the unit square, and
sketch the results on graph paper. Label the vertices.
b. Explain the transformation that occurred to the unit
square.
The transformation is a ι rotation counterclockwise
about the origin.
c. Find the area of the image.
ฬ൬ ൰ ൬ ൰ െ ൬െ ൰ ൬ ሻ൰ฬ ൌ ൌ
ξ ξ ξ ξ
The area is units .
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d. Find the inverse of the transformation.
ۍ ې
ێξ ξۑ
ێെ ۑ
ۏξ ξے
e. Explain the meaning of the inverse transformation on the unit square.
The inverse is a െι rotation counterclockwise about the origin.
f. Rewrite the original matrix if it also included a dilation with a scale factor of .
ۍ െ ې
ێξ ξ ۑor ξ െξ൨
ێ ۑ ξ ξ
ۏξ ξ ے
g. What is the inverse of this matrix?
ۍ ې ۍξ ξې
ێ ξ ξ ۑor ێ ۑ
ێ ۑ
ێെ ۑ
ێെ ξ ξۑ
ۏξ ξے ۏ ے
3. Find a transformation that would create a ૢι counterclockwise rotation about the origin. Set up a system of
equations, and solve to find the matrix.
ࢇ ࢉ ࢇ ࢉ െ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃ
࢈ ࢊ ࢈ ࢊ
െ
ቂ ቃ
4.
a. Find a transformation that would create a ૡι counterclockwise rotation about the origin. Set up a system
of equations, and solve to find the matrix.
ࢇ ࢉ െ ࢇ ࢉ
ቂ ቃቂ ቃ ൌ ቂ ቃǡቂ ቃቂ ቃ ൌ ቂ ቃ
࢈ ࢊ ࢈ ࢊ െ
െ
ቂ ቃ
െ
b. Rewrite the matrix to also include a dilation with a scale factor of .
െ
ቂ ቃ
െ
െ
5. For which values of ࢇ does ቂ ቃ have an inverse matrix?
ૢ ࢇ
ࢇ െ ሺെሻሺૢሻ ്
ࢇ ૢ ്
ࢇ ് െ
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ࢇ ࢇ
6. For which values of ࢇ does ቂ ቃ have an inverse matrix?
ࢇ
ሺࢇሻሺࢇሻ െ ሺࢇ ሻሺሻ ്
ࢇ െ ࢇ െ ૡ ്
ሺࢇ െ ሻሺࢇ ሻ ്
ࢇ ് ǡ ࢇ ് െ
ࢇ ࢇെ
7. For which values of ࢇ does ቂ ቃ have an inverse matrix?
ࢇെ ࢇ
ሺࢇ ሻሺࢇ ሻ െ ሺࢇ െ ሻሺࢇ െ ሻ ്
ሺࢇ ࢇ ሻ െ ሺࢇ െ ૠࢇ ሻ ്
ࢇ െ ്
ࢇ്
ܛܗ܋ሺࣂιሻ െ ܖܑܛሺࣂιሻ
8. Chethan says that the matrix ൨ produces a rotation ࣂι counterclockwise. He justifies his work
ܖܑܛሺࣂιሻ ܛܗ܋ሺࣂιሻ
ඥ
ܛܗ܋ሺιሻ െ ܖܑܛሺιሻ െ
by showing that when ࣂ ൌ , the rotation matrix is ൨ ൌ൦ ൪. Shayla disagrees and
ܖܑܛሺιሻ ܛܗ܋ሺιሻ ඥ
െξ
says that the matrix ൨ produces a ι rotation counterclockwise. Tyler says that he has found that the
ξ
െξ
matrix ൨ produces a ι rotation counterclockwise, too.
ξ
a. Who is correct? Explain.
They are all correct. All of the matrices produce a ι rotation counterclockwise, but each has a different
scale factor.
b. Which matrix has the largest scale factor? Explain.
െξ
൨ has the largest scale factor of . The first matrix has a scale factor of , and the second matrix
ξ
a scale factor of Ǥ
c. Create a matrix with a scale factor less than that would produce the same rotation.
ඥ
െ
Answers will vary. ൦ ൪would have a scale factor of .
ඥ
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Closing (2 minutes)
Have a whole-class discussion using the following questions.
ͳ Ͳ
What effect does performing the transformation ቂ ቃ have on the unit square?
Ͳ ͳ
à No effect—it is the multiplicative identity matrix.
Ͷ Ͳ
What effect does performing the transformation ቂ ቃ have on the unit square?
Ͳ Ͷ
à It is a dilation with a scale factor of Ͷ.
ܽ െܾ
What effect does performing the transformation ቂ ቃ have on the unit square?
ܾ ܽ
à It rotates the unit square in a counterclockwise direction about the origin.
ʹ Ͷ
What effect does performing the transformation ቂ ቃ have on the unit square?
Ͷ ͺ
à The unit square collapses onto a line because the determinant is Ͳ.
Exit Ticket (5 minutes)
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Name Date
Lesson 30: When Can We Reverse a Transformation?
Exit Ticket
ܣand ܤare ʹ ൈ ʹ matrices. ܫis the ʹ ൈ ʹ multiplicative identity matrix.
1. If ܤܣൌ ܣ, name the matrix represented by ܤ.
2. If ܣ ܤൌ ܣ, name the matrix represented by ܤ.
3. If ܤܣൌ ܫ, name the matrix represented by ܤ.
4. Do the matrices have inverses? Justify your answer.
െʹ െʹ
a. ቂ ቃ b. ቂ ቃ
െ͵ ͻ ͵ ͻ
5. Find a value of ܽ, such that the given matrix has an inverse.
െͶ ͵ܽ ͷ ܽ
a. ቂ ቃ b. ቂ ቃ
ʹ ͻ െܽ ͷ
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Exit Ticket Sample Solutions
and are ൈ matrices. ࡵ is the ൈ multiplicative identity matrix.
1. If ൌ , name the matrix represented by .
ൌቂ ቃ
2. If ൌ , name the matrix represented by .
ൌቂ ቃ
3. If ൌ ࡵ, name the matrix represented by .
must be the inverse matrix of .
4. Do the matrices have inverses? Justify your answer.
െ െ
a. ቂ ቃ b. ቂ ቃ
െ ૢ ૢ
No. The determinant ሺെሻሺૢሻ െ ሺሻሺെሻ ൌ . Yes. The determinant ሺെሻሺૢሻ െ ሺሻሺሻ ് .
5. Find a value of ࢇ, such that the given matrix has an inverse.
െ ࢇ ࢇ
a. ቂ ቃ b. ቂ ቃ
ૢ െࢇ
ሺെሻሺૢሻ െ ሺࢇሻሺሻ ് ሺሻሺሻ െ ሺࢇሻሺെࢇሻ ്
െ െ ࢇ ് ࢇ ്
ࢇ ് െ For all real values of ࢇ
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Problem Set Sample Solutions
The first seven problems are more procedural. Assign problems based on student abilities. All students do not have to
complete all problems.
1. Find a transformation that would create a ι counterclockwise rotation about the origin and then its inverse.
ۍξ െ ۍ ېξ ې
ێ ۑ, ێ ۑ
ێ ێ ۑ ۑ
ێ ξ ێ ۑ ξۑ
ۏ ے ۏെ
ے
2. Find a transformation that would create a ι counterclockwise rotation about the origin, a dilation with a scale
factor of , and then its inverse.
ۍξ ې
ξ െ ൨ , ێێૡ ૡۑ
ۑ
ξ ێ ξۑ
െ
ۏૡ ૡے
3. Find a transformation that would create a ૠι counterclockwise rotation about the origin. Set up a system of
equations, and solve to find the matrix.
ࢇ ࢉ ࢇ ࢉ
ቂ ቃ ቂ ቃ ൌ ቂ ቃ , ቂ ቃቂ ቃ ൌ ቂ ቃ
࢈ ࢊ െ ࢈ ࢊ
ቂ ቃ
െ
4. Find a transformation that would create a ૠι counterclockwise rotation about the origin, a dilation with a scale
factor of , and its inverse.
െ
൪
ቂ ቃ , ൦
െ
ૡ ࢇ
5. For which values of ࢇ does ቂ ቃ have an inverse matrix?
ࢇ
ሺૡሻሺሻ െ ሺࢇሻሺࢇሻ ്
െ ࢇ ്
ሺ െ ࢇሻሺ ࢇሻ ്
ࢇ ് ǡ ࢇ ് െ
ࢇ ࢇെ
6. For which values of ࢇ does ቂ ቃ have an inverse matrix?
ࢇ ࢇ
ሺࢇሻሺࢇሻ െ ሺࢇ െ ሻሺࢇ ሻ ്
ࢇ െ ሺࢇ െ ሻ ്
്
All real values of ࢇ will produce an inverse matrix.
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ࢇ ࢇ െ
7. For which values of ࢇ does ቂ ቃ have an inverse matrix?
ࢇ ࢇ െ
ሺࢇሻሺࢇ െ ሻ െ ሺࢇ െ ሻሺࢇሻ ്
ሺࢇ െ ࢇሻ െ ሺࢇ െ ࢇሻ ്
്
No real values of ࢇ will produce an inverse matrix.
8. In Lesson 27, we learned the effect of a transformation on a unit square by multiplying a matrix. For example,
ൌቂ ቃ , ቂ ቃቂ ቃ ൌ ቂ ቃ,ቂ ቃ ቂ ቃ ൌ ቂ ቃ ,andቂ ቃ ቂ ቃ ൌ ቂ ቃǤ
a. Sasha says that we can multiply the inverse of to those resultants of the square after the transformation to
get back to the unit square. Is her conjecture correct? Justify your answer.
െ െ
Yes, she is correct. ି ൌ ቂ ቃൌ
െ
െ
െ െ െ െ
൪ ቂቃ ൌ ቂቃ
൦ ൪ ቂ ቃ ൌ ቂ ቃ , ൦ ൪ ቂ ቃ ൌ ቂ ቃ , ൦ ൪ ቂ ቃ ൌ ቂ ቃ , ൦
െ െ െ െ
b. From part (a), what would you say about the inverse matrix with regard to the geometric effect of
transformations?
Multiplying the inverse matrix, ି , will “undo” the transformation that was done by multiplying matrix Ǥ
ܛܗ܋ሺࣂሻ െܖܑܛሺࣂሻ ࣊
c. A pure rotation matrix is ൨. Prove the inverse matrix for a pure rotation of radians
ܖܑܛሺࣂሻ ܛܗ܋ሺࣂሻ
࣊ ࣊ ࢊ െࢉ
ܛܗ܋ቀെ ቁ െ ܖܑܛቀെ ቁ
ࢇࢊെ࢈ࢉ ࢇࢊെ࢈ࢉ.
counterclockwise is ࣊ ࣊ , which is the same as
ܖܑܛቀെ ቁ ܛܗ܋ቀെ ቁ െ࢈ ࢊ
ࢇࢊെ࢈ࢉ ࢇࢊെ࢈ࢉ
ି࣊ ି࣊ ඥ ඥ
࣊ ܛܗ܋െܖܑܛ
radians counterclockwise rotation is ൪,
The matrix for a pure rotation
ି࣊ ି࣊ ൌ ൦ ඥ
ܖܑܛ ܛܗ܋ ି ඥ
࣊ ࣊ ඥ ିඥ
ܛܗ܋െܖܑܛ
ൌ ൪ . Det ൌ ࢇࢊ െ ࢈ࢉ ൌ ඥ ή ඥ െ ඥ ൬െ ඥ൰ ൌ ൌ , and
࣊ ࣊ ൌ ൦ඥ ඥ
ܖܑܛ ܛܗ܋
ඥ ඥ
ି ൌ ൦ ൪.
ିඥ ඥ
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d. Prove that the inverse matrix of a pure dilation with a factor of is , which is the same as
ࢊ െࢉ
ࢇࢊെ࢈ࢉ ࢇࢊെ࢈ࢉ.
െ࢈ ࢊ
ࢇࢊെ࢈ࢉ ࢇࢊെ࢈ࢉ
The matrix for a pure dilation with a factor of is ൌ ቂ ቃ ǤDet ൌ െ ൌ , and
ି ൌ ቂ ቃ ൌ .
࣊
e. Prove that the matrix used to undo a radians clockwise rotation and a dilation of a factor of is
࣊ ࣊
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ࢊ െࢉ
࣊ ࣊
, which is the same as ࢇࢊെ࢈ࢉ
െ࢈
ࢇࢊെ࢈ࢉ.
ࢊ
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ࢇࢊെ࢈ࢉ ࢇࢊെ࢈ࢉ
࣊
The matrix for undoing the rotation of a radians clockwise and dilating a factor of is
࣊ ࣊ ۍ െξ ۍ ې െξې
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ ێ ۑൌێ ۑ.
൦ ൪ൌ ێ
ܖܑܛቀ࣊ቁ ܛܗ܋ቀ࣊ቁ ێξ ێێ ۑۑξ ۑۑ
ۏ ۏ ے ے
࣊
The matrix for a radians clockwise rotation and a dilation of a factor of is
െ࣊ െ࣊ ۍ ξې
ܛܗ܋ቀ ቁ െ ܖܑܛቀ ቁ
ൌ ൦ ێ ۑ ξ
െ࣊ െ࣊ ൪ ൌ ێെξ ۑൌ െξ ൨ ,Det ൌ ൌ , and
ܖܑܛቀ ቁ ܛܗ܋ቀ ቁ ێ ۑ
ۏ ے
ۍ െξ ې
ି ൌ ൌ ێێ ۑ.
െξ ۑۑ
൨ ێξ
ξ ۏ ے
f. Prove that any matrix whose determinant is not will have an inverse matrix to “undo” a transformation.
ࢇ ࢉ ࢞
For example, use the matrix ൌ ቂ ቃ and the point ቂ࢟ቃ.
࢈ ࢊ
ࢊ െࢉ
ࢇ ࢉ ࢇ ࢉ ࢞ ࢇ࢞ ࢉ࢟ ି ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ൪
ൌቂ ቃǡቂ ቃቂ ቃ ൌ ൨ ൌ ൦
࢈ ࢊ ࢈ ࢊ ࢟ ࢈࢞ ࢊ࢟ െ࢈ ࢇ
ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ
ࢊ െࢉ ࢊሺࢇ࢞ ࢉ࢟ሻ ࢉሺ࢈࢞ ࢊ࢟ሻ ࢇࢊ࢞ ࢉࢊ࢟ െ ࢈ࢉ࢞ െ ࢉࢊ࢟
ࢇ࢞ ࢉ࢟ െ
൦ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ ൪ ൨ൌ൦ ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ ൪ ൌ൦ ࢇࢊ െ ࢈ࢉ ൪
െ࢈ ࢇ ࢈࢞ ࢊ࢟ െ࢈ሺࢇ࢞ ࢉ࢟ሻ ࢇሺ࢈࢞ ࢊ࢟ሻ െࢇ࢈࢞ െ ࢈ࢉ࢟ ࢇ࢈࢞ ࢇࢊ࢟
ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ ࢇࢊ െ ࢈ࢉ
ሺࢇࢊ െ ࢈ࢉሻ࢞
࢞
ൌ ൦ ࢇࢊ െ ࢈ࢉ ൪ ൌ ቂ࢟ቃ
ሺࢇࢊ െ ࢈ࢉሻ࢟
ࢇࢊ െ ࢈ࢉ
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9. Perform the transformation ቂ ቃ on the unit square.
a. Can you find the inverse matrix that will “undo” the transformation? Explain
your reasons arithmetically.
No. The determinant of the matrix is . Therefore, there is no inverse matrix
that can be found to undo the transformation.
b. When all four vertices of the unit square are transformed and collapsed onto
a straight line, what can be said about the inverse?
The determinant of the matrix is .
c. Find the equation of the line that all four vertices of the unit square collapsed onto.
࢟ൌ࢞
d. Find the equation of the line that all four vertices of the unit square collapsed onto using the matrix ቂ ቃ.
࢟ ൌ ࢞
e. A function has an inverse function if and only if it is a one-to-one function. By applying this concept, explain
why we do not have an inverse matrix when the transformation is collapsed onto a straight line.
When doing transformations, we are mapping the four vertices to new coordinates; however, when we
reverse this process, there should be a one-to-one property. However, we see that the point ቀ ቁ will map
onto two different points. Because there is no one-to-one property, this means there is no inverse matrix.
10. The determinants of the following matrices are . Describe what pattern you can find among them.
െ
a. ቂ ቃ , ቂ ቃ , ቂ ቃ ,and ቂ ቃ
ૡ െ
If one column is the multiple of the other column, or one row is the multiple of the other row, then the
determinant is , and there is no inverse matrix.
b. ቂ ቃ , ቂ ቃ , ቂ ቃ , ቂ ቃ , and ቂ ቃ
If either column or row is , then the determinant is , and there is no inverse matrix.
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Name Date
ͳ ݇
1. Consider the transformation on the plane given by the ʹ ൈ ʹ matrix ቀ ቁ for a fixed positive number
Ͳ ݇
݇ ͳ.
a. Draw a sketch of the image of the unit square under this transformation (the unit square has
verticesሺͲǡͲሻ, ሺͳǡͲሻ, ሺͲǡͳሻ, ሺͳǡͳሻ). Be sure to label all four vertices of the image figure.
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b. What is the area of the image parallelogram?
ݔ ʹ
c. Find the coordinates of a point ቀݕቁ whose image under the transformation is ቀ ቁ.
͵
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ͳ ݇ ͳ
d. The transformation ቀ ቁ is applied once to the point ቀ ቁ, then once to the image point, then
Ͳ ݇ ͳ
once to the image of the image point, and then once to the image of the image of the image point,
ͳ
and so on. What are the coordinates of a tenfold image of the point ቀ ቁ, that is, the image of the
ͳ
point after the transformation has been applied ͳͲ times?
ሺͳሻ െሺͳሻ
2. Consider the transformation given by ൬ ൰.
ሺͳሻ ሺͳሻ
ݔ
a. Describe the geometric effect of applying this transformation to a point ቀݕቁ in the plane.
ݔ
b. Describe the geometric effect of applying this transformation to a point ቀݕቁ in the plane twice: once
to the point and then once to its image.
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c. Use part (b) to prove ሺʹሻ ൌ ଶሺͳሻ െ ଶ ሺͳሻ a nd ሺʹሻ ൌ ʹ ሺͳሻ ሺͳሻ.
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3.
a. Explain the geometric representation of multiplying a complex number by ͳ ݅.
b. Write ሺͳ ݅ሻଵ as a complex number of the form ܽ ܾ݅ for real numbers ܽ and ܾ.
c. Find a complex number ܽ ܾ݅, with ܽ and ܾ positive real numbers, such that ሺܽ ܾ݅ሻଷ ൌ ݅.
d. If ݖis a complex number, is there sure to exist, for any positive integer ݊, a complex number ݓsuch
that ݓ ൌ ?ݖExplain your answer.
e. If ݖis a complex number, is there sure to exist, for any negative integer ݊, a complex number ݓsuch
that ݓ ൌ ?ݖExplain your answer.
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Ͳ Ͳ Ͳ Ͳ
4. Let ܲ ൌ ቀ ቁ and ܱ ൌ ቀ ቁ.
ͳ Ͳ Ͳ Ͳ
a. Give an example of a ʹ ൈ ʹ matrix ܣ, not with all entries equal to zero, such that ܲ ܣൌ ܱ.
b. Give an example of a ʹ ൈ ʹ matrix ܤwith ܱܲ ് ܤ.
c. Give an example of a ʹ ൈ ʹ matrix ܥsuch that ܴܥൌ ܴ for all ʹ ൈ ʹ matrices ܴ.
d. If a ʹ ൈ ʹ matrix ܦhas the property that ܦ ܴ ൌ ܴ for all ʹ ൈ ʹ matrices ܴ, must ܦbe the zero
matrix ܱ? Explain.
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ʹ Ͷ ͳ Ͳ ͳ Ͳ
e. Let ܧൌ ቀ ቁ. Is there a ʹ ൈ ʹ matrix ܨso that ܨܧൌ ቀ ቁ and ܧܨൌ ቀ ቁ? If so, find one.
͵ Ͳ ͳ Ͳ ͳ
If not, explain why no such matrix ܨcan exist.
5. In programming a computer video game, Mavis coded the changing location of a space rocket as follows:
ݔ
At a time ݐseconds between ݐൌ Ͳ seconds and ݐൌ ʹ seconds, the location ቀݕቁ of the rocket is given by
ߨ ߨ
ቀ ݐቁ െ ቀ ݐቁ
ʹ ʹ െͳ
ቌ ߨ ߨ ቍ ቀെͳቁ .
ቀ ݐቁ ቀ ݐቁ
ʹ ʹ
At a time of ݐseconds between ݐൌ ʹ seconds and ݐൌ Ͷ seconds, the location of the rocket is given by
͵െݐ
ቀ ቁ.
͵െݐ
a. What is the location of the rocket at time ݐൌ Ͳ? What is its location at time ݐൌ Ͷ?
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b. Petrich is worried that Mavis may have made a mistake and the location of the rocket is unclear at
time ݐൌ ʹ seconds. Explain why there is no inconsistency in the location of the rocket at this time.
c. What is the area of the region enclosed by the path of the rocket from time ݐൌ Ͳ to time ݐൌ Ͷ?
d. Mavis later decided that the moving rocket should be shifted five places farther to the right. How
should she adjust her formulations above to accomplish this translation?
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A Progression Toward Mastery
STEP 1 STEP 2 STEP 3 STEP 4
Assessment Missing or Missing or incorrect A correct answer A correct answer
Task Item incorrect answer answer but with some evidence supported by
and little evidence of some of reasoning or substantial
evidence of reasoning or application of evidence of solid
reasoning or application of mathematics to reasoning or
application of mathematics to solve the problem, application of
mathematics to solve the problem. OR an incorrect mathematics to
solve the answer with solve the problem.
problem. substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem.
1 a Student provides a Student computes two Student computes Student applies matrix
solution that does not or more coordinates of coordinates of the image multiplication to each
apply matrix the image incorrectly, correctly, but the sketch coordinate of the unit
multiplication or and the sketch of the of the image may be square to get the image
transformations to image is incomplete or slightly inaccurate. Work coordinates and draws a
determine the poorly labeled, or the to support the fairly accurate sketch of a
coordinates of the image is a parallelogram calculation of the image parallelogram with
resulting image. The with no work shown and coordinates is limited. vertices correctly
sketch is missing. no vertices labeled. OR labeled. Values for ݇
Student computes three vary, but the resulting
out of four coordinates image should look like a
correctly, and the sketch parallelogram, and the
accurately reflects the distance ݇ in the vertical
student’s coordinates. and horizontal direction
should appear equal.
b Student does not Student computes the Student computes the Student computes the
compute the area of a area of his sketched area of his figure using area of the parallelogram
parallelogram or his figure correctly but does the determinant of the correctly using a
sketched figure not use determinant of ʹ ൈ ʹ matrix, but the determinant. Work
correctly. the ʹ ൈ ʹ matrix in his solution may contain shows understanding
calculation. minor errors. that the area of the
image is the product of
the area of the original
figure and the absolute
value of the determinant
of the transformation
matrix.
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c Student does not Student computes an Student creates a correct Student creates a correct
provide a solution. incorrect solution or matrix equation to solve matrix equation to solve
OR setup of the original for the point and for the point. Student
Student provides work matrix equation. Limited translates the equation translates the equation
that is unrelated to evidence is evident that to a system of linear to a system of linear
the standards the student understands equations. Work shown equations and solves the
addressed in this that the solution to the may be incomplete, and system correctly. Work
problem. matrix equation finds the final answer may contain shown is organized in a
point in question. minor errors. manner that is easy to
OR OR follow and uses proper
Student creates a correct Student has the correct mathematical notation.
matrix equation, and no solution, but the matrix
additional work is given. equation or the system
OR of equations is missing
Student creates the from the solution. Very
correct system of linear little work is shown to
equations, and no provide evidence of
additional work Is given. student thinking.
d Student provides a Student provides a Student provides a Student gives correct
solution that does not solution that does not solution that includes solution for the tenfold
correctly apply the correctly apply the evidence that the image. Student solution
transformation one transformation more student understood the provides enough
time. than one time. Student problem and observed evidence and
AND may attempt to patterns, but minor explanation to clearly
Student does not generalize to the tenfold errors prevent a correct illustrate how she
attempt a image, but the answer solution for the tenfold observed and extended
generalization for the contains major image. the pattern.
tenfold image. conceptual errors. OR
Student provides a
solution that shows
correct repeated
application of the
transformation at least
three times, but the
student is unable to
extend the pattern to
the tenfold image.
2 a Student does not Student identifies the Student correctly Student correctly
recognize the transformation as a identifies the identifies the
transformation as a rotation but cannot transformation as a transformation as a
rotation of the point correctly state the rotation about the counterclockwise
about the origin. direction or the angle origin, but the answer rotation about the origin
measure. contains an error, such through an angle of ͳ
as the wrong direction or radian.
the wrong angle
measurement.
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b Student does not Student identifies the Student correctly Student correctly
identify the repeated transformation as a identifies the repeated identifies the repeated
transformation as a rotation, but the solution transformation as an transformation as a
rotation. does not make it clear additional rotation, but rotation of the image of
that the second rotation the answer contains no the point another ͳ
applies to the image of more than one error. radian clockwise about
the original point. the origin for a total of ʹ
OR radians.
Student identifies the
transformation as an
additional rotation, but
the answer contains two
or more errors.
c Student makes little or Student sets up and Student provides a Student provides a
no attempt at attempts the necessary solution that includes solution that details
multiplying the point matrix multiplications, multiplication of ሺݔǡ ݕሻ multiplication by the
ሺݔǡ ݕሻ by either of the but solution has too by the original rotation original rotation matrix
rotation matrices. many major errors. matrix twice and twice, compares that
OR multiplication of ሺݔǡ ݕሻ result to multiplication
Student provides too by the ʹ-radian rotation by the ʹ-radian rotation
little work to make matrix. Student fails to matrix, and equates the
significant progress on equate the two answers two answers to verify the
the proof. to finish the proof. The identities. Student uses
solution may contain correct notation, and the
minor computation solution illustrates his
errors. thinking clearly. The
solution is free from
minor errors.
3 a Student makes little or Student attempts to Student attempts to Student fully explains the
no attempt to explain explain the geometric explain the geometric geometric relationship of
the geometric relationship of relationship of multiplying by ͳ ݅ in
relationship of multiplying by ͳ ݅ but multiplying by ͳ ݅ but terms of a dilation and a
multiplying by ͳ ݅. makes mistakes. mentions either the rotation.
dilation or rotation, not
both.
b Student makes little or Student attempts to find Student has the correct Student writes the
no attempt to find the the modulus and answer, but it may not correct answer in the
modulus and argument, but solution be in proper form, or proper form and
argument. has major errors that student makes minor correctly solves for the
lead to an incorrect computational errors in modulus and argument
answer. finding the modulus and of the expression,
argument. showing all steps.
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c Student makes little or Student attempts to find Student may find a Student correctly finds a
no attempt to solve a complex number but correct answer but does complex number in the
for a complex number. lacks the proper steps in not show any steps form ܽ ܾ݅, where ܽ
order to do so, resulting taken to solve the and ܾ are positive real
in an incorrect answer. problem. numbers, that satisfies
OR the given equation and
Student has an answer shows all steps such as
that does not have ܽ and finding the modulus and
ܾ as positive real argument of ݅.
numbers.
d Student does not give Student answers Student answers Student answers
any explanation as to incorrectly but gives an correctly but does not correctly and provides
whether a complex explanation that has give an accurate written correct reasoning as to
number, ݓ, exists for somewhat valid points and algebraic why ݓis sure to exist,
the given equation but is lacking proper explanation such as including stating the
and conditions and information. stating the modulus and modulus and argument
answers incorrectly. argument of ݖand ݓfor of ݖand ݓif they are
both zero and nonzero nonzero.
cases.
e Student does not give Student answers Student answers Student answers
any explanation as to incorrectly but gives an correctly but lacks correctly and provides
whether a complex explanation that has proper reasoning to correct reasoning as to
number, ݓ, exists for somewhat valid points support the answer. why ݓis sure to exist,
the given equation but is lacking proper including an algebraic
and conditions and information. solution.
answers incorrectly.
4 a Student makes little to Student sets up a matrix Student correctly sets up Student correctly sets up
no attempt to find equation but does not the matrix equation, but, and solves the matrix
matrix. use the correct matrices due to errors in equation leading to the
in order to solve the calculations, fails to find correct matrix.
problem. the correct matrix.
b Student makes little to Student sets up a matrix Student correctly sets up Student correctly sets up
no attempt to find equation but does not the matrix equation, but, and solves the matrix
matrix. use the correct matrices due to errors in equation leading to the
in order to solve the calculations, fails to find correct matrix.
problem. the correct matrix.
c Student makes little to Student sets up a matrix Student identifies the Student identifies the
no attempt to find equation but does not identity matrix as the identity matrix as the
matrix. use the identity matrix in answer but writes the answer and writes it
order to solve the matrix incorrectly. correctly.
problem.
d Student makes little to Student sets up a matrix Student correctly sets up Student correctly sets up
no attempt to find equation but does not the matrix equation, but, and solves the matrix
matrix. use the correct matrices due to errors in equation leading to the
in order to solve the calculations, fails to find correct matrix.
problem. the correct matrix.
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e Student makes little to Student sets up a matrix Student correctly sets up Student correctly sets up
no attempt to find equation but does not one or both matrix and solves both matrix
matrix. use the correct matrices equations, but, due to equations leading to the
in order to answer the errors in calculations, correct answer.
question. fails to arrive at the
correct answer.
5 a Student makes little to Student sets up a matrix Student correctly sets up Student correctly solves
no attempt to solve equation but does not the matrix equation, but, for the location of the
for the location of the use the correct matrices due to errors in rocket at both times
rocket at either time in order to solve the calculations, fails to given, using the correct
given. problem. reach a correct final matrix equation.
answer for the location
of the rocket at both
times.
b Student makes little to Student sets up matrix Student correctly finds Student correctly gives
no attempt to find the equations to solve for the location of the the location of the rocket
location of the rocket the location of the rocket rocket for one set of for the given time for
at the given time for but fails to properly instructions but fails to both sets of instructions
either set of solve the equations and verify that the location and correctly makes the
instructions and gives produce an accurate of the rocket for the correlations between the
no explanation. explanation. other set of instructions two.
is consistent with the
first.
c Student makes little to Student attempts to find Student correctly finds Student correctly finds
no attempt to solve the area of the region that the path traversed is the area of the enclosed
for the area. enclosed by the path of a semicircle but has path of the rocket
the rocket but does not minor errors in including finding the
make the correct calculations that prevent radius of the traversed
conclusion that it travels the correct area from path.
in a semicircle. being found.
d Student makes little to Student sets up Student correctly sets up Student correctly sets up
no attempt to adjust matrix/matrices for one the shifted matrix for the matrices for both
the matrix five places or both sets of one set of instructions sets of instructions that
farther right. instructions but but fails to correctly set results in a shift of the
incorrectly translates the up the shifted matrices rocket five places to the
points ͷ units to the for both sets of right.
right. instructions.
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Name Date
ͳ ݇
1. Consider the transformation on the plane given by the ʹ ൈ ʹ matrix ቀ ቁ for a fixed positive number
Ͳ ݇
݇ ͳ.
a. Draw a sketch of the image of the unit square under this transformation (the unit square has vertices
ሺͲǡͲሻ, ሺͳǡͲሻ, ሺͲǡͳሻ, and ሺͳǡͳሻ). Be sure to label all four vertices of the image figure.
To find the coordinates of the image, multiply the vertices of the unit square by the
matrix.
൬1 k ൰ ቀ0ቁ ൌ ቀ0ቁ
0 k 0 0
൬1 k ൰ ቀ 0 ቁ ൌ ൬k ൰
0 k 1 k
൬1 k൰ ቀ1ቁ ൌ ൬1 + k൰
0 k 1 k
൬1 k൰ ቀ1ቁ ൌ ቀ1ቁ
0 k 0 0
The image is a parallelogram with base = 1 and height = k.
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b. What is the area of the image parallelogram?
To find the area of the image figure, multiply the area of the unit square by the
absolute value of 1 k൨.
0 k
ฬ1 kฬ ൌ ሺ1ൈkሻ െ ሺ0ൈkሻ ൌ k
0 k
Areaൌ1ൈȁkȁൌk since k0
ݔ ʹ
c. Find the coordinates of a point ቀݕቁ whose image under the transformation is ቀ ቁ.
͵
x
Solve the equation to find the coordinates of ൬y൰.
൬1 k൰ ൬x൰ ൌ ቀ2ቁ
0 k y 3
Converting the matrix equation to a system of linear equations gives us
xky ൌ 2
ky ൌ 3Ǥ
Solve this system.
3
y ൌ
k
3
xk ൬ ൰ ൌ 2
k
x3 ൌ 2
x ൌ െ 1
x -1
The point is ൬y൰ ൌ ቌ 3 ቍ.
k
Module 1: Complex Numbers and Transformations 425
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ͳ ݇ ͳ
d. The transformation ቀ ቁ is applied once to the point ቀ ቁ, then once to the image point, then
Ͳ ݇ ͳ
once to the image of the image point, and then once to the image of the image of the image point,
ͳ
and so on. What are the coordinates of a tenfold image of the point ቀ ቁ, that is, the image of the
ͳ
point after the transformation has been applied 10 times?
Multiply to apply the transformation once: ൬1 k൰ ቀ1ቁ ൌ ൬1+k൰.
0 k 1 k
2
Multiply again by the 2 ൈ 2 matrix: ൬1 k൰ ൬1 k൰ ൌ ቆ1 k k ቇ
0 k k k
2
2 2 3
Multiply again by the 2 ൈ 2 matrix: ൬1 k൰ ቆ1 k k ቇ ൌ ቆ1 k k k ቇ.
0 k k
2
k
3
By observing the patterns, we can see that the result of n multiplications is a
n n
2 ൈ 1 matrix whose top row is the previous row plus k and whose bottom row is k .
2 3 10
The tenfold image would be ቆ1+k+k +k
10
+…k ቇ.
k
ሺͳሻ െሺͳሻ
2. Consider the transformation given by ൬ ൰.
ሺͳሻ ሺͳሻ
ݔ
a. Describe the geometric effect of applying this transformation to a point ቀݕቁ in the plane.
x
This transformation will rotate the point ൬y൰ counterclockwise about the origin through
an angle of 1 radian.
ݔ
b. Describe the geometric effect of applying this transformation to a point ቀݕቁ in the plane twice: once
to the point and then once to its image.
x
This transformation will rotate the point ൬y൰ counterclockwise about the origin an
additional 1 radian for a total rotation of 2 radians.
426 Module 1: Complex Numbers and Transformations
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c. Use part (b) to prove ሺʹሻ ൌ ଶ ሺͳሻ െ ଶ ሺͳሻ and ሺʹሻ ൌ ʹ ሺͳሻ ሺͳሻ.
x
To prove this, multiply ൬y൰ by the transformation matrix:
cosሺ1ሻ െsinሺ1ሻ x x cosሺ1ሻ െ y sinሺ1ሻ
൬ ൰ ൬y ൰ ൌ ቆ ቇ
sinሺ1ሻ cosሺ1ሻ x sinሺ1ሻ y cosሺ1ሻ
Then, multiply this answer by the transformation matrix:
cosሺ1ሻ െsinሺ1ሻ x cosሺ1ሻ െ y sinሺ1ሻ
൬ ൰ቆ ቇ
sinሺ1ሻ cosሺ1ሻ x sinሺ1ሻ y cosሺ1ሻ
Apply matrix multiplication:
cosሺ1ሻ൫x cosሺ1ሻ െ y sinሺ1ሻ൯ െ sinሺ1ሻ൫x sinሺ1ሻ y cosሺ1ሻ൯
ቆ ቇ
sinሺ1ሻ൫x cosሺ1ሻ െ y sinሺ1ሻ൯ cosሺ1ሻ൫x sinሺ1ሻ y cosሺ1ሻ൯
Distribute:
xሺcosሺ1ሻሻ2 െ y cosሺ1ሻ sinሺ1ሻ െ x sinሺ1ሻ2 െ y sinሺ1ሻ cosሺ1ሻ
ቆ ቇ
x sinሺ1ሻ cosሺ1ሻ െ y sinሺ1ሻ2 x cosሺ1ሻ sinሺ1ሻ yሺcosሺ1ሻሻ2
Rearrange and factor:
xሺሺcosሺ1ሻሻ2 െ sinሺ1ሻ2 ሻ െ yሺ2 sinሺ1ሻ cosሺ1ሻሻ
ቆ ቇ
xሺ2sinሺ1ሻ cosሺ1ሻሻ yሺcosሺ1ሻ2 െ sinሺ1ሻ2 ሻ
This matrix is equal to the matrix resulting from the 2-radian rotation.
cosሺ2ሻ െsinሺ2ሻ x x cosሺ2ሻ െ y sinሺ2ሻ
cosሺ2ሻ y
൬ ൰൬ ൰ ൌ ቆ ቇ
sinሺ2ሻ x sinሺ2ሻ y cosሺ2ሻ
When you equate the answers and compare the coefficients of x and y, you can see that
cosሺ2ሻ ൌ cosሺ1ሻ2 െ sinሺ1ሻ2 and sinሺ2ሻ ൌ 2 sinሺ1ሻ cosሺ1ሻ.
The matrices are equal because they represent the same transformation.
xሺሺcosሺ1ሻሻ2 െ sinሺ1ሻ2 ሻ െ yሺ2 sinሺ1ሻ cosሺ1ሻሻ x cosሺ2ሻ െ y sinሺ2ሻ
ቆ 2ሻ ቇ ൌ ቆ ቇ
xሺ2sinሺ1ሻ cosሺ1ሻሻ yሺcosሺ1ሻ െ sinሺ1ሻ
2
x sinሺ2ሻ y cosሺ2ሻ
Module 1: Complex Numbers and Transformations 427
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3.
a. Explain the geometric representation of multiplying by ͳ ݅.
ü
1+i has argument and modulus ξ2, so geometrically this represents a dilation with a
4
ü
scale factor of ξ2 and a counterclockwise rotation of about the origin.
4
b. Write ሺͳ ݅ሻଵ as a complex number of the form ܽ ܾ݅ for real numbers ܽ and ܾ.
ü ü ü
1 + i has argument and modulus ξ2, and so ሺ1 + iሻ10 has argument 10 × = + ü
4 4 2
10 5
and modulus ൫ξ2൯ =2 =32. Thus, ሺ1+iሻ =32i.
10
c. Find a complex number ܽ ܾ݅, with ܽ and ܾ positive real numbers, such that ሺܽ ܾ݅ሻଷ ൌ ݅.
ü ü
i has argument and modulus 1. Thus, a complex number a+bi of argument and
2 6
3 ඥ3 1
modulus 1 will satisfy ൫a+bi൯ =i. We have a+bi= +i .
2 2
d. If ݖis a complex number, is there sure to exist, for any positive integer ݊, a complex number ݓsuch
that ݓ ൌ ?ݖExplain your answer.
Yes. If z=0, then w=0 works. If, on the other hand, z is not zero and has argument Ż
Ż 1
and modulus m, then let w be the complex number with argument and modulus m n :
n
1 Ż Ż
w= mn ቆcos ቆ ቇ + i sin ቆ ቇቇ .
n n
e. If ݖis a complex number, is there sure to exist, for any negative integer ݊, a complex number ݓsuch
that ݓ ൌ ?ݖExplain your answer.
1
If z=0, then there is no such complex number w. If z0, then , with w as given in
w
-n
1
part (c), satisfies ቀ ቁ =z, showing that the answer to the question is yes in this case.
w
428 Module 1: Complex Numbers and Transformations
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Ͳ Ͳ Ͳ Ͳ
4. Let ܲ ൌ ቀ ቁ and ܱ ൌ ቀ ቁ.
ͳ Ͳ Ͳ Ͳ
a. Give an example of a ʹ ൈ ʹ matrix ܣ, not with all entries equal to zero, such that ܲ ܣൌ ܱ.
Notice that for any matrix A = ൬a b ൰, we have PA= ቀ
0 0 a
ቁ൬ b ൰ ቀ0 0
ቁ.
c d 1 0 c d a b
0 0
If we choose A= ቀ ቁ, for example, then PA=O.
1 1
b. Give an example of a ʹ ൈ ʹ matrix ܤwith ܱܲ ് ܤ.
1 1 0 0
Following the discussion in part (a), we see that choosing A= ቀ ቁ gives PA= ቀ ቁ,
0 0 1 1
which is different from O.
c. Give an example of a ʹ ൈ ʹ matrix ܥsuch that ܴܥൌ ܴ for all ʹ ൈ ʹ matrices ܴ.
1 0
Choose C= ቀ ቁ. The identity matrix has this property.
0 1
d. If a ʹ ൈ ʹ matrix ܦhas the property that ܦ ܴ ൌ ܴ for all ʹ ൈ ʹ matrices ܴ, must ܦbe the zero
matrix ܱ? Explain.
x y a +x b+y x y
Write D= ൬a b ൰ and R= ቀ ቁ. Then, for D + R = ൬ ൰ to equal ቀ ቁ no
c d z w c+z d+ w z w
matter the values of x, y, z, and w, we need:
a +x=x
b+y=y
c+z=z
d+w=w
to hold for all values x, y, z, and w. Thus, we need a = 0, b = 0, c = 0, and d = 0.
That is, D must indeed be the zero matrix.
Module 1: Complex Numbers and Transformations 429
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ʹ Ͷ ͳ Ͳ ͳ Ͳ
e. Let ܧൌ ቀ ቁ. Is there a ʹ ൈ ʹ matrix ܨso that ܨܧൌ ቀ ቁ and ܧܨൌ ቀ ቁ? If so, find one.
͵ Ͳ ͳ Ͳ ͳ
If not, explain why no such matrix ܨcan exist.
The determinant of E is ቚ2Â6-3Â4ቚ =0, and so no inverse matrix like F can exist.
Alternatively:
Write F= ൬a b ൰. Then, EF= ൬2a+4c 2b+4d൰. For this to equal ቀ1 0
ቁ, we need, at the
c d 3a+6c 3b+3d 0 1
very least:
2a+4c=1
3a+6c=0
1
The first of these equations gives a+2c= and the second a+2c=0. There is no solution
2
to this system of equations, and so there can be no matrix F with the desired property.
5. In programming a computer video game, Mavis coded the changing location of a space rocket as follows:
ݔ
At a time seconds between ݐൌ Ͳ seconds and ݐൌ ʹ seconds, the location ቀݕቁ of the rocket is given by:
ߨ ߨ
ቀ ݐቁ െ ቀ ݐቁ
ʹ ʹ െͳ
ቌ ߨ ߨ ቍ ቀെͳቁ .
ቀ ݐቁ ቀ ݐቁ
ʹ ʹ
At a time of ݐseconds between ݐൌ ʹ seconds and ݐൌ Ͷ seconds, the location of the rocket is given by
͵െݐ
ቀ ቁ.
͵െݐ
a. What is the location of the rocket at time ݐൌ Ͳ? What is its location at time ݐൌ Ͷ?
At time t=0, the location of the rocket is
cosሺ0ሻ െ sinሺ0ሻ െ1 1 0 െ1 െ1
൬ ൰ቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ.
sinሺ0ሻ cosሺ0ሻ െ1 0 1 െ1 െ1
At time t=4, the location of the rocket is
3െ4 െ1
ቀ ቁ ൌ ቀ ቁ,
3െ4 െ1
the same as at the start.
430 Module 1: Complex Numbers and Transformations
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A STORY OF FUNCTIONS End-of-Module Assessment Task M1
PRECALCULUS AND ADVANCED TOPICS
b. Petrich is worried that Mavis may have made a mistake and the location of the rocket is unclear at
time ݐൌ ʹ seconds. Explain why there is no inconsistency in the location of the rocket at this time.
According to the first set of instructions, the location of the rocket at time t=2 is
cosሺüሻ െ sinሺüሻ െ1 െ1 0 െ1 1
൬ ൰ቀ ቁ ൌ ቀ ቁቀ ቁ ൌ ቀ ቁ.
sinሺüሻ cosሺüሻ െ1 0 െ1 െ1 1
According to the second set of instructions, its location at this time is
3െ2 1
ቀ ቁ ൌ ቀ ቁ.
3െ2 1
These are consistent.
c. What is the area of the region enclosed by the path of the rocket from time ݐൌ Ͳ to time ݐൌ Ͷ?
The path traversed is a semicircle with a radius of ξ2. The area enclosed is
1
ൈ ü ൌ ü squared units.
2
d. Mavis later decided that the moving rocket should be shifted five places farther to the right. How
should she adjust her formulations above to accomplish this translation?
Notice that:
ü ü ü ü
cos ቀ tቁ െ sin ቀ tቁ െ cos ቀ tቁ sin ቀ tቁ
2 2 ൲ ቀെ1ቁ ൌ ൮ 2 2
൮ ü ü െ1 ü ü ൲.
sin ቀ tቁ cos ቀ tቁ െ sin ቀ tቁ െ cos ቀ tቁ
2 2 2 2
To translate these points 5 units to the right, use
ü ü
െ cos ቀ tቁ sin ቀ tቁ 5
2 2
ቌ ü ü ቍ for 0 t 2.
െ sin ቀ tቁ െ cos ቀ tቁ
2 2
Also, use
3െt5 8െt
ቀ ቁൌቀ ቁ for 2 t 4.
3െt 3െt
Module 1: Complex Numbers and Transformations 431
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A STORY OF FUNCTIONS Credits M1
PRECALCULUS AND ADVANCED TOPICS
Credits
'ƌĞĂƚDŝŶĚƐΠŚĂƐŵĂĚĞĞǀĞƌLJĞīŽƌƚƚŽŽďƚĂŝŶƉĞƌŵŝƐƐŝŽŶĨŽƌƚŚĞƌĞƉƌŝŶƟŶŐŽĨĂůůĐŽƉLJƌŝŐŚƚĞĚŵĂƚĞƌŝĂů͘
/ĨĂŶLJŽǁŶĞƌŽĨĐŽƉLJƌŝŐŚƚĞĚŵĂƚĞƌŝĂůŝƐŶŽƚĂĐŬŶŽǁůĞĚŐĞĚŚĞƌĞŝŶ͕ƉůĞĂƐĞĐŽŶƚĂĐƚ'ƌĞĂƚDŝŶĚƐĨŽƌƉƌŽƉĞƌ
ĂĐŬŶŽǁůĞĚŐŵĞŶƚŝŶĂůůĨƵƚƵƌĞĞĚŝƟŽŶƐĂŶĚƌĞƉƌŝŶƚƐŽĨƚŚŝƐŵŽĚƵůĞ͘
ඵ ůůŵĂƚĞƌŝĂůĨƌŽŵƚŚĞŽŵŵŽŶŽƌĞ^ƚĂƚĞ^ƚĂŶĚĂƌĚƐĨŽƌDĂƚŚĞŵĂƟĐƐΞŽƉLJƌŝŐŚƚϮϬϭϬEĂƟŽŶĂů
'ŽǀĞƌŶŽƌƐƐƐŽĐŝĂƟŽŶĞŶƚĞƌĨŽƌĞƐƚWƌĂĐƟĐĞƐĂŶĚŽƵŶĐŝůŽĨŚŝĞĨ^ƚĂƚĞ^ĐŚŽŽůKĸĐĞƌƐ͘ůůƌŝŐŚƚƐ
ƌĞƐĞƌǀĞĚ͘
Module 1: Credits 433
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