NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M1

PRECALCULUS AND ADVANCED TOPICS

Lesson 1: Wishful Thinking—Does Linearity Hold?

Classwork
Exercises
Look at these common mistakes that students make, and answer the questions that follow.
1. If 𝑓𝑓(𝑥𝑥) = √𝑥𝑥, does 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏), when 𝑎𝑎 and 𝑏𝑏 are not negative?
a. Can we find a counterexample to refute the claim that 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏) for all nonnegative values of 𝑎𝑎
and 𝑏𝑏?

b. Find some nonnegative values for 𝑎𝑎 and 𝑏𝑏 for which the statement, by coincidence, happens to be true.

c. Find all values of 𝑎𝑎 and 𝑏𝑏 for which the statement is true. Explain your work and the results.

Lesson 1: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M1
PRECALCULUS AND ADVANCED TOPICS

d. Why was it necessary for us to consider only nonnegative values of 𝑎𝑎 and 𝑏𝑏?

e. Does 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 display ideal linear properties? Explain.

2. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 3 , does 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏)?

a. Substitute in some values of 𝑎𝑎 and 𝑏𝑏 to show this statement is not true in general.

b. Find some values for 𝑎𝑎 and 𝑏𝑏 for which the statement, by coincidence, happens to work.

c. Find all values of 𝑎𝑎 and 𝑏𝑏 for which the statement is true. Explain your work and the results.

Lesson 1: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M1
PRECALCULUS AND ADVANCED TOPICS

d. Is this true for all positive and negative values of 𝑎𝑎 and 𝑏𝑏? Explain and prove by choosing positive and negative
values for the variables.

e. Does 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 3 display ideal linear properties? Explain.

Lesson 1: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

Study the statements given in Problems 1–3. Prove that each statement is false, and then find all values of 𝑎𝑎 and 𝑏𝑏 for
which the statement is true. Explain your work and the results.
1. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 , does 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏)?

1
2. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 3 , does 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏)?

3. If 𝑓𝑓(𝑥𝑥) = √4𝑥𝑥, does 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏)?

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.

Lesson 1: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 2: Wishful Thinking—Does Linearity Hold?

Classwork
Exercises
1. Let 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥). Does 𝑓𝑓(2𝑥𝑥) = 2𝑓𝑓(𝑥𝑥) for all values of 𝑥𝑥? Is it true for any values of 𝑥𝑥? Show work to justify your
answer.

2. Let 𝑓𝑓(𝑥𝑥) = log(𝑥𝑥). Find a value for 𝑎𝑎 such that 𝑓𝑓(2𝑎𝑎) = 2𝑓𝑓(𝑎𝑎). Is there one? Show work to justify your answer.

3. Let 𝑓𝑓(𝑥𝑥) = 10𝑥𝑥 . Show that 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏) is true for 𝑎𝑎 = 𝑏𝑏 = log(2) and that it is not true for 𝑎𝑎 = 𝑏𝑏 = 2.

Lesson 2: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M1
PRECALCULUS AND ADVANCED TOPICS

1
4. Let 𝑓𝑓(𝑥𝑥) = . Are there any real numbers 𝑎𝑎 and 𝑏𝑏 so that 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏)? Explain.
𝑥𝑥

5. What do your findings from these exercises illustrate about the linearity of these functions? Explain.

Lesson 2: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

Examine the equations given in Problems 1–4, and show that the functions 𝑓𝑓(𝑥𝑥) = cos(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) = tan(𝑥𝑥) are not
linear transformations by demonstrating that they do not satisfy the conditions indicated for all real numbers. Then, find
values of 𝑥𝑥 and/or 𝑦𝑦 for which the statement holds true.
1. cos(𝑥𝑥 + 𝑦𝑦) = cos(𝑥𝑥) + cos(𝑦𝑦) 2. cos(2𝑥𝑥) = 2 cos(𝑥𝑥)

3. tan(𝑥𝑥 + 𝑦𝑦) = tan(𝑥𝑥) + tan(𝑦𝑦) 4. tan(2𝑥𝑥) = 2 tan(𝑥𝑥)

1
5. Let 𝑓𝑓(𝑥𝑥) = . Are there any real numbers 𝑎𝑎 and 𝑏𝑏 so that 𝑓𝑓(𝑎𝑎 + 𝑏𝑏) = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓(𝑏𝑏)? Explain.
𝑥𝑥2

6. Let 𝑓𝑓(𝑥𝑥) = log(𝑥𝑥). Find values of 𝑎𝑎 such that 𝑓𝑓(3𝑎𝑎) = 3𝑓𝑓(𝑎𝑎).

7. Let 𝑓𝑓(𝑥𝑥) = log(𝑥𝑥). Find values of 𝑎𝑎 such that 𝑓𝑓(𝑘𝑘𝑘𝑘) = 𝑘𝑘𝑘𝑘(𝑎𝑎).

8. Based on your results from the previous two problems, form a conjecture about whether 𝑓𝑓(𝑥𝑥) = log 𝑥𝑥 represents a
linear transformation.

9. Let 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐.

a. Describe the set of all values for 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 that make 𝑓𝑓(𝑥𝑥 + 𝑦𝑦) = 𝑓𝑓(𝑥𝑥) + 𝑓𝑓(𝑦𝑦) valid for all real numbers 𝑥𝑥
and 𝑦𝑦.
b. What does your result indicate about the linearity of quadratic functions?

Trigonometry Table

Angle Measure Angle Measure

𝐬𝐬𝐬𝐬𝐬𝐬(𝒙𝒙) 𝐜𝐜𝐜𝐜𝐜𝐜(𝒙𝒙)
(𝒙𝒙 Degrees) (𝒙𝒙 Radians)

30

𝜋𝜋
4

𝜋𝜋
3

90

Lesson 2: Wishful Thinking—Does Linearity Hold?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 3: Which Real Number Functions Define a Linear

Transformation?

Classwork
Opening Exercise
Recall from the previous two lessons that a linear transformation is a function 𝑓𝑓 that satisfies two conditions:
(1) 𝑓𝑓(𝑥𝑥 + 𝑦𝑦) = 𝑓𝑓(𝑥𝑥) + 𝑓𝑓(𝑦𝑦) and (2) 𝑓𝑓(𝑘𝑘𝑘𝑘) = 𝑘𝑘𝑘𝑘(𝑥𝑥). Here, 𝑘𝑘 refers to any real number, and 𝑥𝑥 and 𝑦𝑦 represent arbitrary
elements in the domain of 𝑓𝑓.
a. Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 . Is 𝑓𝑓 a linear transformation? Explain why or why not.

b. Let 𝑔𝑔(𝑥𝑥) = √𝑥𝑥. Is 𝑔𝑔 a linear transformation? Explain why or why not.

Lesson 3: Which Real Number Functions Define a Linear Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Suppose you have a linear transformation 𝑓𝑓: ℝ → ℝ, where 𝑓𝑓(2) = 1 and 𝑓𝑓(4) = 2.
a. Use the addition property to compute 𝑓𝑓(6), 𝑓𝑓(8), 𝑓𝑓(10), and 𝑓𝑓(12).
b. Find 𝑓𝑓(20), 𝑓𝑓(24), and 𝑓𝑓(30). Show your work.
c. Find 𝑓𝑓(−2), 𝑓𝑓(−4), and 𝑓𝑓(−8). Show your work.
d. Find a formula for 𝑓𝑓(𝑥𝑥).
e. Draw the graph of the function 𝑓𝑓(𝑥𝑥).

2. The symbol ℤ represents the set of integers, and so 𝑔𝑔: ℤ → ℤ represents a function that takes integers as inputs and
produces integers as outputs. Suppose that a function 𝑔𝑔: ℤ → ℤ satisfies 𝑔𝑔(𝑎𝑎 + 𝑏𝑏) = 𝑔𝑔(𝑎𝑎) + 𝑔𝑔(𝑏𝑏) for all integers 𝑎𝑎
and 𝑏𝑏. Is there necessarily an integer 𝑘𝑘 such that 𝑔𝑔(𝑛𝑛) = 𝑘𝑘𝑘𝑘 for all integer inputs 𝑛𝑛?
a. Let 𝑘𝑘 = 𝑔𝑔(1). Compute 𝑔𝑔(2) and 𝑔𝑔(3).
b. Let 𝑛𝑛 be any positive integer. Compute 𝑔𝑔(𝑛𝑛).
c. Now, consider 𝑔𝑔(0). Since 𝑔𝑔(0) = 𝑔𝑔(0 + 0), what can you conclude about 𝑔𝑔(0)?
d. Lastly, use the fact that 𝑔𝑔(𝑛𝑛 + −𝑛𝑛) = 𝑔𝑔(0) to learn something about 𝑔𝑔(−𝑛𝑛), where 𝑛𝑛 is any positive integer.
e. Use your work above to prove that 𝑔𝑔(𝑛𝑛) = 𝑘𝑘𝑘𝑘 for every integer 𝑛𝑛. Be sure to consider the fact that 𝑛𝑛 could be
positive, negative, or 0.

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 𝑓𝑓: ℝ → ℝ that satisfies 𝑓𝑓(𝑥𝑥 ∙ 𝑦𝑦) = 𝑓𝑓(𝑥𝑥) + 𝑓𝑓(𝑦𝑦).
b. Give an example of a function 𝑔𝑔: ℝ → ℝ that satisfies 𝑔𝑔(𝑥𝑥 + 𝑦𝑦) = 𝑔𝑔(𝑥𝑥) ∙ 𝑔𝑔(𝑦𝑦).
c. Give an example of a function ℎ: ℝ → ℝ that satisfies ℎ(𝑥𝑥 ∙ 𝑦𝑦) = ℎ(𝑥𝑥) ∙ ℎ(𝑦𝑦).

Lesson 3: Which Real Number Functions Define a Linear Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 4: An Appearance of Complex Numbers

Classwork
Opening Exercise
1
Is 𝑅𝑅(𝑥𝑥) = a linear transformation? Explain how you know.
𝑥𝑥

Exercises
1. Solve 5𝑥𝑥 2 − 3𝑥𝑥 + 17 = 9.

Lesson 4: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M1
PRECALCULUS AND ADVANCED TOPICS

2. Use the fact that 𝑖𝑖2 = −1 to show that 𝑖𝑖3 = −𝑖𝑖. Interpret this statement geometrically.

3. Calculate 𝑖𝑖6 .

4. Calculate 𝑖𝑖5 .

Lesson 4: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 4 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Solve the equation below.

5𝑥𝑥 2 − 7𝑥𝑥 + 8 = 2

2. Consider the equation 𝑥𝑥3 = 8.

a. What is the first solution that comes to mind?
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 𝑥𝑥3 − 8 = 0, which can be rewritten as (𝑥𝑥 − 2)�𝑥𝑥2 + 2𝑥𝑥 + 4� = 0 (check this for yourself).
Find all of the solutions to this equation.

3. Make a drawing that shows the first 5 powers of 𝑖𝑖 (i.e., 𝑖𝑖1 , 𝑖𝑖2 , …, 𝑖𝑖5 ), and then confirm your results algebraically.

4. What is the value of 𝑖𝑖99 ? Explain your answer using words or drawings.

5. What is the geometric effect of multiplying a number by −𝑖𝑖? Does your answer make sense to you? Give an
explanation using words or drawings.

Lesson 4: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 5: An Appearance of Complex Numbers

Classwork
Opening Exercise
Write down two fundamental facts about 𝑖𝑖 that you learned in the previous lesson.

Discussion: Visualizing Complex Numbers

Lesson 5: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M1
PRECALCULUS AND ADVANCED TOPICS

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.

2. In the complex plane, what is the horizontal axis used for? What is the vertical axis used for?

3. How would you represent −4 + 3𝑖𝑖 in the complex plane?

Lesson 5: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M1
PRECALCULUS AND ADVANCED TOPICS

For Exercises 4–7, let 𝑎𝑎 = 1 + 3𝑖𝑖 and 𝑏𝑏 = 2 − 𝑖𝑖.

4. Find 𝑎𝑎 + 𝑏𝑏. Then, plot 𝑎𝑎, 𝑏𝑏, and 𝑎𝑎 + 𝑏𝑏 in the
complex plane.

5. Find 𝑎𝑎 − 𝑏𝑏. Then, plot 𝑎𝑎, 𝑏𝑏, and 𝑎𝑎 − 𝑏𝑏 in the

complex plane.

6. Find 2𝑎𝑎. Then, plot 𝑎𝑎 and 2𝑎𝑎 in the complex plane.

7. Find 𝑎𝑎 ∙ 𝑏𝑏. Then, plot 𝑎𝑎, 𝑏𝑏, and 𝑎𝑎 ∙ 𝑏𝑏 in the complex plane.

Lesson 5: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. The number 5 is a real number. Is it also a complex number? Try to find values of 𝑎𝑎 and 𝑏𝑏 so that 5 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.

2. The number 3𝑖𝑖 is an imaginary number and a multiple of 𝑖𝑖. Is it also a complex number? Try to find values of 𝑎𝑎 and
𝑏𝑏 so that 3𝑖𝑖 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.

3. Daria says that every real number is a complex number. Do you agree with her? Why or why not?

4. Colby says that every imaginary number is a complex number. Do you agree with him? Why or why not?

In Problems 5–9, perform the indicated operations. Report each answer as a complex number 𝑤𝑤 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏, and graph it
in a complex plane.
5. Given 𝑧𝑧1 = −9 + 5𝑖𝑖, 𝑧𝑧2 = −10 − 2𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 + 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

6. Given 𝑧𝑧1 = −4 + 10𝑖𝑖, 𝑧𝑧2 = −7 − 6𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 − 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

7. Given 𝑧𝑧1 = 3√2 + 2𝑖𝑖, 𝑧𝑧2 = √2 − 𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 − 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

8. Given 𝑧𝑧1 = 3, 𝑧𝑧2 = −4 + 8𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

1
9. Given 𝑧𝑧1 = , 𝑧𝑧2 = 12 − 4𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.
4

10. Given 𝑧𝑧1 = −1, 𝑧𝑧2 = 3 + 4𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

11. Given 𝑧𝑧1 = 5 + 3𝑖𝑖, 𝑧𝑧2 = −4 − 2𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

12. Given 𝑧𝑧1 = 1 + 𝑖𝑖, 𝑧𝑧2 = 1 + 𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

13. Given 𝑧𝑧1 = 3, 𝑧𝑧2 = 𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

14. Given 𝑧𝑧1 = 4 + 3𝑖𝑖, 𝑧𝑧2 = 𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

15. Given 𝑧𝑧1 = 2√2 + 2√2𝑖𝑖, 𝑧𝑧2 = −√2 + √2𝑖𝑖, find 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 , and graph 𝑧𝑧1 , 𝑧𝑧2 , and 𝑤𝑤.

16. Represent 𝑤𝑤 = −4 + 3𝑖𝑖 as a point in the complex plane.

17. Represent 2𝑤𝑤 as a point in the complex plane. 2𝑤𝑤 = 2(−4 + 3𝑖𝑖) = −8 + 6𝑖𝑖

18. Compare the positions of 𝑤𝑤 and 2𝑤𝑤 from Problems 10 and 11. Describe what you see. (Hint: Draw a segment from
the origin to each point.)

Lesson 5: An Appearance of Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 6: Complex Numbers as Vectors

Classwork
Opening Exercise
Perform the indicated arithmetic operations for complex numbers 𝑧𝑧 = −4 + 5𝑖𝑖 and 𝑤𝑤 = −1 − 2𝑖𝑖.
a. 𝑧𝑧 + 𝑤𝑤

b. 𝑧𝑧 − 𝑤𝑤

c. 𝑧𝑧 + 2𝑤𝑤

d. 𝑧𝑧 − 𝑧𝑧

e. Explain how you add and subtract complex numbers.

Lesson 6: Complex Numbers as Vectors

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises
1. The length of the vector that represents 𝑧𝑧1 = 6 − 8𝑖𝑖 is 10 because �62 + (−8)2 = √100 = 10.
a. Find at least seven other complex numbers that can be represented as vectors that have length 10.

b. Draw the vectors on the coordinate axes provided below.

c. What do you observe about all of these vectors?

Lesson 6: Complex Numbers as Vectors

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1
PRECALCULUS AND ADVANCED TOPICS

2. In the Opening Exercise, we computed 𝑧𝑧 + 2𝑤𝑤. Calculate this sum using vectors.

3. In the Opening Exercise, we also computed 𝑧𝑧 − 𝑧𝑧. Calculate this sum using vectors.

4. For the vectors 𝐮𝐮 and 𝐯𝐯 pictured below, draw the specified sum or difference on the coordinate axes provided.
a. 𝐮𝐮 + 𝐯𝐯
b. 𝐯𝐯 − 𝐮𝐮
c. 2𝐮𝐮 − 𝐯𝐯
d. −𝐮𝐮 − 3𝐯𝐯

Lesson 6: Complex Numbers as Vectors

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1
PRECALCULUS AND ADVANCED TOPICS

5. Find the sum of 4 + 𝑖𝑖 and −3 + 2𝑖𝑖 geometrically.

6. Show that (7 + 2𝑖𝑖) − (4 − 𝑖𝑖) = 3 + 3𝑖𝑖 by representing the complex numbers as vectors.

Lesson 6: Complex Numbers as Vectors

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Let 𝑧𝑧 = 1 + 𝑖𝑖 and 𝑤𝑤 = 1 − 3𝑖𝑖. Find the following. Express your answers in 𝑎𝑎 + 𝑏𝑏𝑏𝑏 form.
a. 𝑧𝑧 + 𝑤𝑤
b. 𝑧𝑧 − 𝑤𝑤
c. 4𝑤𝑤
d. 3𝑧𝑧 + 𝑤𝑤
e. −𝑤𝑤 − 2𝑧𝑧
f. What is the length of the vector representing 𝑧𝑧?
g. What is the length of the vector representing 𝑤𝑤?

2. Let 𝑢𝑢 = 3 + 2𝑖𝑖, 𝑣𝑣 = 1 + 𝑖𝑖, and 𝑤𝑤 = −2 − 𝑖𝑖. Find

the following. Express your answer in 𝑎𝑎 + 𝑏𝑏𝑏𝑏
form, and represent the result in the plane.
a. 𝑢𝑢 − 2𝑣𝑣
b. 𝑢𝑢 − 2𝑤𝑤
c. 𝑢𝑢 + 𝑣𝑣 + 𝑤𝑤
d. 𝑢𝑢 − 𝑣𝑣 + 𝑤𝑤
e. What is the length of the vector
representing 𝑢𝑢?
f. What is the length of the vector
representing 𝑢𝑢 − 𝑣𝑣 + 𝑤𝑤?

3. Find the sum of −2 − 4𝑖𝑖 and 5 + 3𝑖𝑖 geometrically.

4. Show that (−5 − 6𝑖𝑖) − (−8 − 4𝑖𝑖) = 3 − 2𝑖𝑖 by representing the complex numbers as vectors.

5. Let 𝑧𝑧1 = 𝑎𝑎1 + 𝑏𝑏1 𝑖𝑖, 𝑧𝑧2 = 𝑎𝑎2 + 𝑏𝑏2 𝑖𝑖, and 𝑧𝑧3 = 𝑎𝑎3 + 𝑏𝑏3 𝑖𝑖. Prove the following using algebra or by showing with vectors.
a. 𝑧𝑧1 + 𝑧𝑧2 = 𝑧𝑧2 + 𝑧𝑧1
b. 𝑧𝑧1 + (𝑧𝑧2 + 𝑧𝑧3 ) = (𝑧𝑧1 + 𝑧𝑧2 ) + 𝑧𝑧3

Lesson 6: Complex Numbers as Vectors

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M1
PRECALCULUS AND ADVANCED TOPICS

6. Let 𝑧𝑧 = −3 − 4𝑖𝑖 and 𝑤𝑤 = −3 + 4𝑖𝑖.

a. Draw vectors representing 𝑧𝑧 and 𝑤𝑤 on the same set of axes.
b. What are the lengths of the vectors representing 𝑧𝑧 and 𝑤𝑤?
c. Find a new vector, 𝑢𝑢𝑧𝑧 , such that 𝑢𝑢𝑧𝑧 is equal to 𝑧𝑧 divided by the length of the vector representing 𝑧𝑧.
d. Find 𝑢𝑢𝑤𝑤 , such that 𝑢𝑢𝑤𝑤 is equal to 𝑤𝑤 divided by the length of the vector representing 𝑤𝑤.
e. Draw vectors representing 𝑢𝑢𝑧𝑧 and 𝑢𝑢𝑤𝑤 on the same set of axes as part (a).
f. What are the lengths of the vectors representing 𝑢𝑢𝑧𝑧 and 𝑢𝑢𝑤𝑤 ?
g. Compare the vectors representing 𝑢𝑢𝑧𝑧 to 𝑧𝑧 and 𝑢𝑢𝑤𝑤 to 𝑤𝑤. What do you notice?
h. What is the value of 𝑢𝑢𝑧𝑧 times 𝑢𝑢𝑤𝑤 ?
i. What does your answer to part (h) tell you about the relationship between 𝑢𝑢𝑧𝑧 and 𝑢𝑢𝑤𝑤 ?

7. Let 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.
a. Let 𝑢𝑢𝑧𝑧 be represented by the vector in the direction of 𝑧𝑧 with length 1. How can you find 𝑢𝑢𝑧𝑧 ? What is the
value of 𝑢𝑢𝑧𝑧 ?
b. Let 𝑢𝑢𝑤𝑤 be the complex number that when multiplied by 𝑢𝑢𝑧𝑧 , the product is 1. What is the value of 𝑢𝑢𝑤𝑤 ?
c. What number could we multiply 𝑧𝑧 by to get a product of 1?

8. Let 𝑧𝑧 = −3 + 5𝑖𝑖.
a. Draw a picture representing 𝑧𝑧 + 𝑤𝑤 = 8 + 2𝑖𝑖.
b. What is the value of 𝑤𝑤?

Lesson 6: Complex Numbers as Vectors

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 7: Complex Number Division

Classwork
Opening Exercise
Perform the indicated operations. Write your answer in 𝑎𝑎 + 𝑏𝑏𝑏𝑏 form. Identify the real part of your answer and the
imaginary part of your answer.
a. (2 + 3𝑖𝑖) + (−7 − 4𝑖𝑖)

b. 𝑖𝑖2 (−4𝑖𝑖)

c. 3𝑖𝑖 − (−2 + 5𝑖𝑖)

d. (3 − 2𝑖𝑖)(−7 + 4𝑖𝑖)

e. (−4 − 5𝑖𝑖)(−4 + 5𝑖𝑖)

Lesson 7: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises
1. What is the multiplicative inverse of 2𝑖𝑖?

2. Find the multiplicative inverse of 5 + 3𝑖𝑖.

State the conjugate of each number, and then using the general formula for the multiplicative inverse of 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏, find
the multiplicative inverse.
3. 3 + 4𝑖𝑖

4. 7 − 2𝑖𝑖

Lesson 7: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS

5. 𝑖𝑖

6. 2

1 1 1
7. Show that 𝑎𝑎 = −1 + √3𝑖𝑖 and 𝑏𝑏 = 2 satisfy = + .
𝑎𝑎+𝑏𝑏 𝑎𝑎 𝑏𝑏

Lesson 7: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. State the conjugate of each complex number. Then, find the multiplicative inverse of each number, and verify by
multiplying by 𝑎𝑎 + 𝑏𝑏𝑏𝑏 and solving a system of equations.
a. −5𝑖𝑖
b. 5 − √3𝑖𝑖

2. Find the multiplicative inverse of each number, and verify using the general formula to find multiplicative inverses of
numbers of the form 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.
a. 𝑖𝑖3
1
b.
3
√3 − 𝑖𝑖
c.
4
d. 1 + 2𝑖𝑖
e. 4 − 3𝑖𝑖
f. 2 + 3𝑖𝑖
g. −5 − 4𝑖𝑖
h. −3 + 2𝑖𝑖
i. �2 + 𝑖𝑖

j. 3 − √2 ∙ 𝑖𝑖
k. �5 + �3 ∙ 𝑖𝑖

3. Given 𝑧𝑧1 = 1 + 𝑖𝑖 and 𝑧𝑧2 = 2 + 3𝑖𝑖.

a. Let 𝑤𝑤 = 𝑧𝑧1 ⋅ 𝑧𝑧2 . Find 𝑤𝑤 and the multiplicative inverse of 𝑤𝑤.
b. Show that the multiplicative inverse of 𝑤𝑤 is the same as the product of the multiplicative inverses of 𝑧𝑧1 and 𝑧𝑧2 .

Lesson 7: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 8: Complex Number Division

Classwork
Opening Exercise
Use the general formula to find the multiplicative inverse of each complex number.
a. 2 + 3𝑖𝑖

b. −7 − 4𝑖𝑖

c. −4 + 5𝑖𝑖

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. 𝐴𝐴: 3 + 4𝑖𝑖

2. 𝐵𝐵: −2 − 𝑖𝑖

3. 𝐶𝐶: 7

4. 𝐷𝐷: 4𝑖𝑖

Lesson 8: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1
PRECALCULUS AND ADVANCED TOPICS

Find the modulus.

5. 3 + 4𝑖𝑖

6. −2 − 𝑖𝑖

7. 7

8. 4𝑖𝑖

Given 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.
9. Show that for all complex numbers 𝑧𝑧, |𝑖𝑖𝑖𝑖| = |𝑧𝑧|.

Lesson 8: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1
PRECALCULUS AND ADVANCED TOPICS

10. Show that for all complex numbers 𝑧𝑧, 𝑧𝑧 ∙ 𝑧𝑧̅ = |𝑧𝑧|2 .

1 𝑧𝑧̅
11. Explain the following: Every nonzero complex number 𝑧𝑧 has a multiplicative inverse. It is given by = .
𝑧𝑧 |𝑧𝑧|

Example
2 − 6𝑖𝑖
2 + 5𝑖𝑖

Exercises 12–13
Divide.
3 + 2𝑖𝑖
12.
−2 − 7𝑖𝑖

3
13.
3 − 𝑖𝑖

Lesson 8: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Let 𝑧𝑧 = 4 − 3𝑖𝑖 and 𝑤𝑤 = 2 − 𝑖𝑖. Show that

a. |𝑧𝑧| = |𝑧𝑧
�|
1 1
b. � � = | �|
𝑧𝑧 𝑧𝑧
c. If |𝑧𝑧| = 0, must it be that 𝑧𝑧 = 0?
d. Give a specific example to show that |𝑧𝑧 + 𝑤𝑤| usually does not equal |𝑧𝑧| + |𝑤𝑤|.

2. Divide.
1 − 2𝑖𝑖
a.
2𝑖𝑖
5 − 2𝑖𝑖
b.
5 + 2𝑖𝑖
√3 − 2𝑖𝑖
c.
−2 − √3𝑖𝑖

3. Prove that |𝑧𝑧𝑧𝑧| = |𝑧𝑧| ∙ |𝑤𝑤| for complex numbers 𝑧𝑧 and 𝑤𝑤.

4. Given 𝑧𝑧 = 3 + 𝑖𝑖, 𝑤𝑤 = 1 + 3.
a. Find 𝑧𝑧 + 𝑤𝑤, and graph 𝑧𝑧, 𝑤𝑤, and 𝑧𝑧 + 𝑤𝑤 on the same complex plane. Explain what you discover if you draw line
segments from the origin to those points 𝑧𝑧, 𝑤𝑤, and 𝑧𝑧 + 𝑤𝑤. Then, draw line segments to connect 𝑤𝑤 to 𝑧𝑧 + 𝑤𝑤
and 𝑧𝑧 + 𝑤𝑤 to 𝑧𝑧.
b. Find −𝑤𝑤, and graph 𝑧𝑧, 𝑤𝑤, and 𝑧𝑧 − 𝑤𝑤 on the same complex plane. Explain what you discover if you draw line
segments from the origin to those points 𝑧𝑧, 𝑤𝑤, and 𝑧𝑧 − 𝑤𝑤. Then, draw line segments to connect 𝑤𝑤 to 𝑧𝑧 − 𝑤𝑤
and 𝑧𝑧 − 𝑤𝑤 to 𝑧𝑧.

5. Explain why |𝑧𝑧 + 𝑤𝑤| ≤ |𝑧𝑧| + |𝑤𝑤| and |𝑧𝑧 − 𝑤𝑤| ≤ |𝑧𝑧| + |𝑤𝑤| geometrically. (Hint: Triangle inequality theorem)

Lesson 8: Complex Number Division

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 9: The Geometric Effect of Some Complex Arithmetic

Classwork
Exercises
1. Taking the conjugate of a complex number corresponds to reflecting a complex number about the real axis. What
operation on a complex number induces a reflection across the imaginary axis?

2. Given the complex numbers 𝑤𝑤 = −4 + 3𝑖𝑖 and

𝑧𝑧 = 2 − 5𝑖𝑖, graph each of the following:
a. 𝑤𝑤
b. 𝑧𝑧
c. 𝑤𝑤 + 2
d. 𝑧𝑧 + 2
e. 𝑤𝑤 − 1
f. 𝑧𝑧 − 1

3. Describe in your own words the geometric effect adding or subtracting a real number has on a complex number.

Lesson 9: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1
PRECALCULUS AND ADVANCED TOPICS

4. Given the complex numbers 𝑤𝑤 = −4 + 3𝑖𝑖 and

𝑧𝑧 = 2 − 5𝑖𝑖, graph each of the following:
a. 𝑤𝑤
b. 𝑧𝑧
c. 𝑤𝑤 + 𝑖𝑖
d. 𝑧𝑧 + 𝑖𝑖
e. 𝑤𝑤 − 2𝑖𝑖
f. 𝑧𝑧 − 2𝑖𝑖

5. Describe in your own words the geometric effect adding or subtracting an imaginary number has on a complex
number.

Example

Given the complex number 𝑧𝑧, find a complex number 𝑤𝑤 such that 𝑧𝑧 + 𝑤𝑤 is shifted √2 units in a southwest direction.

Lesson 9: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
 The conjugate, 𝑧𝑧̅, of a complex number 𝑧𝑧 reflects the point across the real axis.
 The negative conjugate, −𝑧𝑧̅, of a complex number 𝑧𝑧 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.

Problem Set

1. Given the complex numbers 𝑤𝑤 = 2 − 3𝑖𝑖 and

𝑧𝑧 = −3 + 2𝑖𝑖, graph each of the following:
a. 𝑤𝑤 − 2
b. 𝑧𝑧 + 2
c. 𝑤𝑤 + 2𝑖𝑖
d. 𝑧𝑧 − 3𝑖𝑖
e. 𝑤𝑤 + 𝑧𝑧
f. 𝑧𝑧 − 𝑤𝑤

2. Let 𝑧𝑧 = 5 − 2𝑖𝑖. Find 𝑤𝑤 for each case.

a. 𝑧𝑧 is a 90° counterclockwise rotation about
the origin of 𝑤𝑤.
b. 𝑧𝑧 is reflected about the imaginary axis from
𝑤𝑤.
c. 𝑧𝑧 is reflected about the real axis from 𝑤𝑤.

3. Let 𝑧𝑧 = −1 + 2𝑖𝑖, 𝑤𝑤 = 4 − 𝑖𝑖. Simplify the following expressions.

a. 𝑧𝑧 + 𝑤𝑤
�
b. |𝑤𝑤 − 𝑧𝑧̅|
c. 2𝑧𝑧 − 3𝑤𝑤
𝑧𝑧
d.
𝑤𝑤

4. Given the complex number 𝑧𝑧, find a complex number 𝑤𝑤 where 𝑧𝑧 + 𝑤𝑤 is shifted:
a. 2√2 units in a northeast direction.
b. 5√2 units in a southeast direction.

Lesson 9: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 10: The Geometric Effect of Some Complex Arithmetic

Classwork
Opening Exercises
1. Given 𝑧𝑧 = 3 − 2𝑖𝑖, plot and label the following, and describe the geometric effect of the operation.
a. 𝑧𝑧

b. 𝑧𝑧 − 2

c. 𝑧𝑧 + 4𝑖𝑖

d. 𝑧𝑧 + (−2 + 4𝑖𝑖)

2. Describe the geometric effect of the following:

a. Multiplying by 𝑖𝑖

b. Taking the complex conjugate

c. What operation reflects a complex number across the imaginary axis?

Lesson 10: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1
PRECALCULUS AND ADVANCED TOPICS

Example 1
Plot the given points, and then plot the image 𝐿𝐿(𝑧𝑧) = 2𝑧𝑧.
a. 𝑧𝑧1 = 3

b. 𝑧𝑧2 = 2𝑖𝑖

c. 𝑧𝑧3 = 1 + 𝑖𝑖

d. 𝑧𝑧4 = −4 + 3𝑖𝑖

e. 𝑧𝑧5 = 2 − 5𝑖𝑖

Lesson 10: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises
Plot the given points, and then plot the image 𝐿𝐿(𝑧𝑧) = 𝑖𝑖𝑖𝑖.
1. 𝑧𝑧1 = 3

2. 𝑧𝑧2 = 2𝑖𝑖

3. 𝑧𝑧3 = 1 + 𝑖𝑖

4. 𝑧𝑧4 = −4 + 3𝑖𝑖

5. 𝑧𝑧5 = 2 − 5𝑖𝑖

Lesson 10: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1
PRECALCULUS AND ADVANCED TOPICS

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.

7. Is 𝐿𝐿(𝑧𝑧) a linear transformation? Explain how you know.

Example 2
Describe the geometric effect of 𝐿𝐿(𝑧𝑧) = (1 + 𝑖𝑖)𝑧𝑧 given the following. Plot the images on graph paper, and describe the
geometric effect in words.
a. 𝑧𝑧1 = 1

b. 𝑧𝑧2 = 𝑖𝑖

c. 𝑧𝑧3 = 1 + 𝑖𝑖

d. 𝑧𝑧4 = 4 + 6𝑖𝑖

Lesson 10: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Let 𝑧𝑧 = −4 + 2𝑖𝑖. Simplify the following, and describe the geometric effect of the operation. Plot the result in the
complex plane.
a. 𝑧𝑧 + 2 − 3𝑖𝑖
b. 𝑧𝑧 − 2 − 3𝑖𝑖
c. 𝑧𝑧 − (2 − 3𝑖𝑖)
d. 2𝑧𝑧
𝑧𝑧
e.
2

2. Let 𝑧𝑧 = 1 + 2𝑖𝑖. Simplify the following, and describe the geometric effect of the operation.
a. 𝑖𝑖𝑖𝑖
b. 𝑖𝑖 2 𝑧𝑧
c. 𝑧𝑧̅
d. −𝑧𝑧̅
e. 𝑖𝑖𝑧𝑧̅
f. 2𝑖𝑖𝑖𝑖
g. 𝑖𝑖𝑖𝑖 + 5 − 3𝑖𝑖

3. Simplify the following expressions.

a. (4 − 2𝑖𝑖)(5 − 3𝑖𝑖)
b. (−2 + 3𝑖𝑖)(−2 − 3𝑖𝑖)
c. (1 + 𝑖𝑖)2
d. (1 + 𝑖𝑖)10 (Hint: 𝑏𝑏 𝑛𝑛𝑛𝑛 = (𝑏𝑏 𝑛𝑛 )𝑚𝑚 )
−1 + 2𝑖𝑖
e.
1 − 2𝑖𝑖
𝑥𝑥2 + 4
f.  , provided 𝑥𝑥 ≠ 2𝑖𝑖
𝑥𝑥 − 2𝑖𝑖

Lesson 10: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1
PRECALCULUS AND ADVANCED TOPICS

4. Given 𝑧𝑧 = 2 + 𝑖𝑖, describe the geometric effect of the following. Plot the result.
a. 𝑧𝑧(1 + 𝑖𝑖)
�3 1
b. 𝑧𝑧 � + 𝑖𝑖�
2 2

5. We learned that multiplying by 𝑖𝑖 produces a 90° counterclockwise rotation about the origin. What do we need to
multiply by to produce a 90° clockwise rotation about the origin?

6. Given 𝑧𝑧 is a complex number 𝑎𝑎 + 𝑏𝑏𝑏𝑏, determine if 𝐿𝐿(𝑧𝑧) is a linear transformation. Explain why or why not.
a. 𝐿𝐿(𝑧𝑧) = 𝑖𝑖 3 𝑧𝑧
b. 𝐿𝐿(𝑧𝑧) = 𝑧𝑧 + 4𝑖𝑖

Lesson 10: The Geometric Effect of Some Complex Arithmetic

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 11: Distance and Complex Numbers

Classwork
Opening Exercise
a. Plot the complex number 𝑧𝑧 = 2 + 3𝑖𝑖 on the complex plane. Plot the ordered pair (2, 3) on the coordinate
plane.

b. In what way are complex numbers points?

c. What point on the coordinate plane corresponds to the complex number −1 + 8𝑖𝑖?

d. What complex number corresponds to the point located at coordinate (0, −9)?

Lesson 11: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1
PRECALCULUS AND ADVANCED TOPICS

Exercise 1
The endpoints of ����
𝐴𝐴𝐴𝐴 are 𝐴𝐴(1, 8) and 𝐵𝐵(−5, 3). What is the midpoint of ����
𝐴𝐴𝐴𝐴?

Exercise 2
a. What is the midpoint of 𝐴𝐴 = 1 + 8𝑖𝑖 and 𝐵𝐵 = −5 + 3𝑖𝑖?

𝐴𝐴+𝐵𝐵
b. Using 𝐴𝐴 = 𝑥𝑥1 + 𝑦𝑦1 𝑖𝑖 and 𝐵𝐵 = 𝑥𝑥2 + 𝑦𝑦2 𝑖𝑖, show that, in general, the midpoint of points 𝐴𝐴 and 𝐵𝐵 is , the
2
arithmetic average of the two numbers.

Exercise 3
The endpoints of ����
𝐴𝐴𝐴𝐴 are 𝐴𝐴(1, 8) and 𝐵𝐵(−5, 3). What is the length of ����
𝐴𝐴𝐴𝐴?

Lesson 11: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1
PRECALCULUS AND ADVANCED TOPICS

Exercise 4
a. What is the distance between 𝐴𝐴 = 1 + 8𝑖𝑖 and 𝐵𝐵 = −5 + 3𝑖𝑖?

b. Show that, in general, the distance between 𝐴𝐴 = 𝑥𝑥1 + 𝑦𝑦1 𝑖𝑖 and 𝐵𝐵 = 𝑥𝑥2 + 𝑦𝑦2 𝑖𝑖 is the modulus of 𝐴𝐴 − 𝐵𝐵.

Exercise 5
Suppose 𝑧𝑧 = 2 + 7𝑖𝑖 and 𝑤𝑤 = −3 + 𝑖𝑖.
a. Find the midpoint 𝑚𝑚 of 𝑧𝑧 and 𝑤𝑤.

b. Verify that |𝑧𝑧 − 𝑚𝑚| = |𝑤𝑤 − 𝑚𝑚|.

Lesson 11: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1
PRECALCULUS AND ADVANCED TOPICS

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.
𝐴𝐴+𝐵𝐵
 For two complex numbers 𝐴𝐴 = 𝑥𝑥1 + 𝑦𝑦1 𝑖𝑖 and 𝐵𝐵 = 𝑥𝑥2 + 𝑦𝑦2 𝑖𝑖, the midpoint of points 𝐴𝐴 and 𝐵𝐵 is .
2
 The distance between two complex numbers 𝐴𝐴 = 𝑥𝑥1 + 𝑦𝑦1 𝑖𝑖 and 𝐵𝐵 = 𝑥𝑥2 + 𝑦𝑦2 𝑖𝑖 is equal to |𝐴𝐴 − 𝐵𝐵|.

Problem Set

1. Find the midpoint between the two given points in the rectangular coordinate plane.
a. 2 + 4𝑖𝑖 and 4 + 8𝑖𝑖
b. −3 + 7𝑖𝑖 and 5 − 𝑖𝑖
c. −4 + 3𝑖𝑖 and 9 − 4𝑖𝑖
d. 4 + 𝑖𝑖 and −12 − 7𝑖𝑖
e. −8 − 3𝑖𝑖 and 3 − 4𝑖𝑖
2 5
f. − 𝑖𝑖 and −0.2 + 0.4𝑖𝑖
3 2

2. Let 𝐴𝐴 = 2 + 4𝑖𝑖, 𝐵𝐵 = 14 + 8𝑖𝑖, and suppose that 𝐶𝐶 is the midpoint of 𝐴𝐴 and 𝐵𝐵 and that 𝐷𝐷 is the midpoint of 𝐴𝐴 and 𝐶𝐶.
a. Find points 𝐶𝐶 and 𝐷𝐷.
b. Find the distance between 𝐴𝐴 and 𝐵𝐵.
c. Find the distance between 𝐴𝐴 and 𝐶𝐶.
d. Find the distance between 𝐶𝐶 and 𝐷𝐷.
e. Find the distance between 𝐷𝐷 and 𝐵𝐵.
f. Find a point one-quarter of the way along the line segment connecting segment 𝐴𝐴 and 𝐵𝐵, closer to 𝐴𝐴 than to 𝐵𝐵.
g. Terrence thinks the distance from 𝐵𝐵 to 𝐶𝐶 is the same as the distance from 𝐴𝐴 to 𝐵𝐵. Is he correct? Explain why
or why not.
h. Using your answer from part (g), if 𝐸𝐸 is the midpoint of 𝐶𝐶 and 𝐵𝐵, can you find the distance from 𝐸𝐸 to 𝐶𝐶?
Explain.
i. Without doing any more work, can you find point 𝐸𝐸? Explain.

Lesson 11: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 12: Distance and Complex Numbers

Classwork
Opening Exercise
a. Let 𝐴𝐴 = 2 + 3𝑖𝑖 and 𝐵𝐵 = −4 − 8𝑖𝑖. Find a complex number 𝐶𝐶 so that 𝐵𝐵 is the midpoint of 𝐴𝐴 and 𝐶𝐶.

b. Given two complex numbers 𝐴𝐴 and 𝐵𝐵, find a formula for a complex number 𝐶𝐶 in terms of 𝐴𝐴 and 𝐵𝐵 so that 𝐵𝐵 is
the midpoint of 𝐴𝐴 and 𝐶𝐶.

c. Verify that your formula is correct by using the result of part (a).

Exercise 1
Let 𝑧𝑧 = −100 + 100𝑖𝑖 and 𝑤𝑤 = 1000 − 1000𝑖𝑖.
a. Find a point one-quarter of the way along the line segment connecting 𝑧𝑧 and 𝑤𝑤 closer to 𝑧𝑧 than to 𝑤𝑤.

b. Write this point in the form 𝛼𝛼𝛼𝛼 + 𝛽𝛽𝛽𝛽 for some real numbers 𝛼𝛼 and 𝛽𝛽. Verify that this does in fact represent
the point found in part (a).

2 3
c. Describe the location of the point 𝑧𝑧 + 𝑤𝑤 on this line segment.
5 5

Lesson 12: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M1
PRECALCULUS AND ADVANCED TOPICS

Exploratory Challenge 1
a. Draw three points 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶 in the plane.

b. Start at any position 𝑃𝑃0 , and leapfrog over 𝐴𝐴 to a new position 𝑃𝑃1 so that 𝐴𝐴 is the midpoint of ������
𝑃𝑃0 𝑃𝑃1 .

c. ������
From 𝑃𝑃1 , leapfrog over 𝐵𝐵 to a new position 𝑃𝑃2 so that 𝐵𝐵 is the midpoint 𝑃𝑃1 𝑃𝑃2 .

d. From 𝑃𝑃2 , leapfrog over 𝐶𝐶 to a new position 𝑃𝑃3 so that 𝐶𝐶 is the midpoint ������
𝑃𝑃2 𝑃𝑃3 .

e. Continue alternately leapfrogging over 𝐴𝐴, then 𝐵𝐵, and then 𝐶𝐶.

f. What eventually happens?

g. Using the formula from the Opening Exercise, part (b), show why this happens.

Lesson 12: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M1
PRECALCULUS AND ADVANCED TOPICS

Exploratory Challenge 2
a. Plot a single point 𝐴𝐴 in the plane.

b. What happens when you repeatedly jump over 𝐴𝐴?

c. Using the formula from the Opening Exercise, part (b), show why this happens.

d. Make a conjecture about what will happen if you leapfrog over two points, 𝐴𝐴 and 𝐵𝐵, in the coordinate plane.

e. Test your conjecture by using the formula from the Opening Exercise, part (b).

f. Was your conjecture correct? If not, what is your new conjecture about what happens when you leapfrog over
two points, 𝐴𝐴 and 𝐵𝐵, in the coordinate plane?

g. Test your conjecture by actually conducting the experiment.

Lesson 12: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Find the distance between the following points.

a. Point 𝐴𝐴(2, 3) and point 𝐵𝐵(6, 6)
b. 𝐴𝐴 = 2 + 3𝑖𝑖 and 𝐵𝐵 = 6 + 6𝑖𝑖
c. 𝐴𝐴 = −1 + 5𝑖𝑖 and 𝐵𝐵 = 5 + 11𝑖𝑖
d. 𝐴𝐴 = 1 − 2𝑖𝑖 and 𝐵𝐵 = −2 + 3𝑖𝑖
1 1 2 1
e. 𝐴𝐴 = − 𝑖𝑖 and 𝐵𝐵 = − + 𝑖𝑖
2 2 3 3

2. Given three points 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, where 𝐶𝐶 is the midpoint of 𝐴𝐴 and 𝐵𝐵
a. If 𝐴𝐴 = −5 + 2𝑖𝑖 and 𝐶𝐶 = 3 + 4𝑖𝑖, find 𝐵𝐵.
b. If 𝐵𝐵 = 1 + 11𝑖𝑖 and 𝐶𝐶 = −5 + 3𝑖𝑖, find 𝐴𝐴.

3. Point 𝐶𝐶 is the midpoint between 𝐴𝐴 = 4 + 3𝑖𝑖 and 𝐵𝐵 = −6 − 5𝑖𝑖. Find the distance between points 𝐶𝐶 and 𝐷𝐷 for each
point 𝐷𝐷 provided below.
a. 2𝐷𝐷 = −6 + 8𝑖𝑖
b. 𝐷𝐷 = −𝐵𝐵�

4. The distance between points 𝐴𝐴 = 1 + 1𝑖𝑖 and 𝐵𝐵 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 is 5. Find the point 𝐵𝐵 for each value provided below.
a. 𝑎𝑎 = 4
b. 𝑏𝑏 = 6

5. Draw five points in the plane 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷, 𝐸𝐸. Start at any position, 𝑃𝑃0 , and leapfrog over 𝐴𝐴 to a new position, 𝑃𝑃1 (so,
������
𝐴𝐴 is the midpoint of 𝑃𝑃0 𝑃𝑃1 ). Then, leapfrog over 𝐵𝐵, then 𝐶𝐶, then 𝐷𝐷, then 𝐸𝐸, then 𝐴𝐴, then 𝐵𝐵, then 𝐶𝐶, then 𝐷𝐷, then 𝐸𝐸,
then 𝐴𝐴 again, and so on. How many jumps will it take to get back to the starting position, 𝑃𝑃0 ?

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 𝐴𝐴 = 2, 𝐵𝐵 = 2 + 𝑖𝑖, and 𝑃𝑃0 = 𝑖𝑖. Will
𝑃𝑃𝑛𝑛 ever return to the starting position, 𝑃𝑃0 ? Explain how you know.

Lesson 12: Distance and Complex Numbers

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 13: Trigonometry and Complex Numbers

Classwork
Opening Exercise
For each complex number shown below, answer the following questions. Record your answers in the table.
a. What are the coordinates (𝑎𝑎, 𝑏𝑏) 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 1 is rotated 𝜃𝜃° so it passes through the point
(𝑎𝑎, 𝑏𝑏). What is a value of 𝜃𝜃?

Complex Number (𝒂𝒂, 𝒃𝒃) Modulus Degrees of Rotation 𝜽𝜽°

𝑧𝑧1 = −3 + 0𝑖𝑖

𝑧𝑧2 = 0 + 2𝑖𝑖

𝑧𝑧3 = 3 + 3𝑖𝑖

𝑧𝑧4 = 2 − 2√3𝑖𝑖

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Exercises 1–2
1. Can you find at least two additional rotations that would map a ray from the origin through the real number 1 to a
ray from the origin passing through the point (3, 3)?

2. How are the rotations you found in Exercise 1 related?

Every complex number 𝑧𝑧 = 𝑥𝑥 + 𝑦𝑦𝑦𝑦 appears as a point on the complex plane with coordinates (𝑥𝑥, 𝑦𝑦) as a point in the
coordinate plane.

In the diagram above, notice that each complex number 𝑧𝑧 has a distance 𝑟𝑟 from the origin to the point (𝑥𝑥, 𝑦𝑦) and a
rotation of 𝜃𝜃° that maps the ray from the origin along the positive real axis to the ray passing through the point (𝑥𝑥, 𝑦𝑦).
ARGUMENT OF THE COMPLEX NUMBER 𝒛𝒛: The argument of the complex number 𝑧𝑧 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 𝑧𝑧 in the complex plane. The argument of 𝑧𝑧
is denoted arg(𝑧𝑧).
MODULUS OF A COMPLEX NUMBER 𝒛𝒛: The modulus of a complex number 𝑧𝑧, denoted |𝑧𝑧|, is the distance from the origin to the
point corresponding to 𝑧𝑧 in the complex plane. If 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏, then |𝑧𝑧| = √𝑎𝑎2 + 𝑏𝑏 2.

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PRECALCULUS AND ADVANCED TOPICS

Example 1: The Polar Form of a Complex Number

Derive a formula for a complex number in terms of its modulus 𝑟𝑟 and argument 𝜃𝜃.

Suppose that 𝑧𝑧 has coordinates (𝑥𝑥, 𝑦𝑦) that lie on the unit circle as shown.
a. What is the value of 𝑟𝑟, and what are the coordinates of the point (𝑥𝑥, 𝑦𝑦) in terms of 𝜃𝜃? Explain how you know.

b. If 𝑟𝑟 = 2, what would be the coordinates of the point (𝑥𝑥, 𝑦𝑦)? Explain how you know.

c. If 𝑟𝑟 = 20, what would be the coordinates of the point (𝑥𝑥, 𝑦𝑦)? Explain how you know.

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d. Use the definitions of sine and cosine to write coordinates of the point (𝑥𝑥, 𝑦𝑦) in terms of cosine and sine for
any 𝑟𝑟 ≥ 0 and real number 𝜃𝜃.

e. Use your answer to part (d) to express 𝑧𝑧 = 𝑥𝑥 + 𝑦𝑦𝑦𝑦 in terms of 𝑟𝑟 and 𝜃𝜃.

POLAR FORM OF A COMPLEX NUMBER: The polar form of a complex number 𝑧𝑧 is 𝑟𝑟(cos(𝜃𝜃) + 𝑖𝑖 sin(𝜃𝜃)), where 𝑟𝑟 = |𝑧𝑧| and
𝜃𝜃 = arg(𝑧𝑧).
RECTANGULAR FORM OF A COMPLEX NUMBER: The rectangular form of a complex number 𝑧𝑧 is 𝑎𝑎 + 𝑏𝑏𝑏𝑏, where 𝑧𝑧 corresponds to
the point (𝑎𝑎, 𝑏𝑏) in the complex plane, and 𝑖𝑖 is the imaginary unit. The number 𝑎𝑎 is called the real part of 𝑎𝑎 + 𝑏𝑏𝑏𝑏, and the
number 𝑏𝑏 is called the imaginary part of 𝑎𝑎 + 𝑏𝑏𝑏𝑏.

General Form Polar Form Rectangular Form

𝑧𝑧 = 𝑟𝑟(cos(𝜃𝜃) + 𝑖𝑖 sin(𝜃𝜃)) 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏
Examples
3(cos(60°) + 𝑖𝑖 sin(60°))

0 + 2𝑖𝑖

Key Features Modulus Modulus

Argument Coordinate

Coordinate

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Exercises 3–6
3. Write each complex number from the Opening Exercise in polar form.

Rectangular Polar Form

𝑧𝑧1 = −3 + 0𝑖𝑖

𝑧𝑧2 = 0 + 2𝑖𝑖

𝑧𝑧3 = 3 + 3𝑖𝑖

𝑧𝑧4 = 2 − 2√3𝑖𝑖

4. Use a graph to help you answer these questions.

a. What is the modulus of the complex number 2 − 2𝑖𝑖?

b. What is the argument of the number 2 − 2𝑖𝑖?

c. Write the complex number in polar form.

d. Arguments can be measured in radians. Express your answer to part (c) using radians.

Lesson 13: Trigonometry and Complex Numbers

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e. Explain why the polar and rectangular forms of a complex number represent the same number.

5. State the modulus and argument of each complex number, and then graph it using the modulus and argument.
a. 4(cos(120°) + 𝑖𝑖 sin(120°))

𝜋𝜋 𝜋𝜋
b. 5 �cos � � + 𝑖𝑖 sin � ��
4 4

c. 3(cos(190°) + 𝑖𝑖 sin(190°))

6. Evaluate the sine and cosine functions for the given values of 𝜃𝜃, and then express each complex number in
rectangular form, 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏. Explain why the polar and rectangular forms represent the same number.
a. 4(cos(120°) + 𝑖𝑖 sin(120°))

𝜋𝜋 𝜋𝜋
b. 5 �cos � � + 𝑖𝑖 sin � ��
4 4

Lesson 13: Trigonometry and Complex Numbers

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c. 3(cos(190°) + 𝑖𝑖 sin(190°))

Example 2: Writing a Complex Number in Polar Form

a. Convert 3 + 4𝑖𝑖 to polar form.

b. Convert 3 − 4𝑖𝑖 to polar form.

Exercise 7
7. Express each complex number in polar form. State the arguments in radians rounded to the nearest thousandth.
a. 2 + 5𝑖𝑖

b. −6 + 𝑖𝑖

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M1
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Lesson Summary
The polar form of a complex number 𝑧𝑧 = 𝑟𝑟 (cos(𝜃𝜃) + 𝑖𝑖 sin(𝜃𝜃)) where 𝜃𝜃 is the argument of 𝑧𝑧 and 𝑟𝑟 is the modulus
of 𝑧𝑧. The rectangular form of a complex number is 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.
The polar and rectangular forms of a complex number are related by the formulas 𝑎𝑎 = 𝑟𝑟 cos(𝜃𝜃), 𝑏𝑏 = 𝑟𝑟 sin(𝜃𝜃), and
𝑟𝑟 = √𝑎𝑎2 + 𝑏𝑏 2.
The notation for modulus is |𝑧𝑧|, and the notation for argument is arg(𝑧𝑧).

Problem Set

1. Explain why the complex numbers 𝑧𝑧1 = 1 − √3𝑖𝑖, 𝑧𝑧2 = 2 − 2√3𝑖𝑖, and 𝑧𝑧3 = 5 − 5√3𝑖𝑖 can all have the same
argument. Draw a diagram to support your answer.

2. What is the modulus of each of the complex numbers 𝑧𝑧1 , 𝑧𝑧2 , and 𝑧𝑧3 given in Problem 1 above?

3. Write the complex numbers from Exercise 1 in polar form.

4. Explain why 1 − √3𝑖𝑖 and 2(cos(300°) + 𝑖𝑖 sin(300°)) represent the same number.

5. Julien stated that a given modulus and a given argument uniquely determine a complex number. Confirm or refute
Julien’s reasoning.

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. 0° ≤ arg(𝑧𝑧) ≤ 360° or 0 ≤ arg(𝑧𝑧) ≤ 2𝜋𝜋 (radians)
a. 𝑧𝑧 = cos(30°) + 𝑖𝑖 sin(30°)
𝜋𝜋 𝜋𝜋
b. 𝑧𝑧 = 2 �cos � � + 𝑖𝑖 sin � ��
4 4
𝜋𝜋 𝜋𝜋
c. 𝑧𝑧 = 4 �cos � � + 𝑖𝑖 sin � ��
3 3
5𝜋𝜋 5𝜋𝜋
d. 𝑧𝑧 = 2√3 �cos � � + 𝑖𝑖 sin � ��
6 6
e. 𝑧𝑧 = 5(cos(5.637) + 𝑖𝑖 sin(5.637))
f. 𝑧𝑧 = 5(cos(2.498) + 𝑖𝑖 sin(2.498))

g. 𝑧𝑧 = √34(cos(3.682) + 𝑖𝑖 sin(3.682))
5𝜋𝜋 5𝜋𝜋
h. 𝑧𝑧 = 4√3 �cos � � + 𝑖𝑖 sin � ��
3 3

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M1
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𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋
7. Convert the complex numbers in rectangular form to polar form. If the argument is a multiple of , , , or ,
6 4 3 2
𝑏𝑏
express your answer exactly. If not, use arctan � � to find arg(𝑧𝑧) rounded to the nearest thousandth,
𝑎𝑎
0 ≤ arg(𝑧𝑧) ≤ 2𝜋𝜋 (radians).
a. 𝑧𝑧 = √3 + 𝑖𝑖
b. 𝑧𝑧 = −3 + 3𝑖𝑖
c. 𝑧𝑧 = 2 − 2√3𝑖𝑖
d. 𝑧𝑧 = −12 − 5𝑖𝑖
e. 𝑧𝑧 = 7 − 24𝑖𝑖

8. Show that the following complex numbers have the same argument.
a. 𝑧𝑧1 = 3 + 3√3𝑖𝑖 and 𝑧𝑧2 = 1 + √3𝑖𝑖
b. 𝑧𝑧1 = 1 + 𝑖𝑖 and 𝑧𝑧2 = 4 + 4𝑖𝑖

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.

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10. Determine complex numbers in polar form whose coordinates are the vertices of the square shown below.

11. How do the modulus and argument of coordinate 𝐴𝐴 in Problem 9 correspond to the modulus and argument of point
𝐴𝐴′ in Problem 10? Does a similar relationship exist when you compare 𝐵𝐵 to 𝐵𝐵′, 𝐶𝐶 to 𝐶𝐶′, and 𝐷𝐷 to 𝐷𝐷′? Explain why
you think this relationship exists.

12. Describe the transformations that map 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 to 𝐴𝐴′𝐵𝐵′𝐶𝐶′𝐷𝐷′.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M1
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General Form Polar Form Rectangular Form

𝑧𝑧 = 𝑟𝑟(cos(𝜃𝜃) + 𝑖𝑖 sin(𝜃𝜃)) 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏
Examples

Key Features Modulus Modulus

Argument Coordinate

Coordinate

Examples

Key Features Modulus Modulus

Argument Coordinate

Coordinate

Examples

Key Features Modulus Modulus

Argument Coordinate

Coordinate

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 14: Discovering the Geometric Effect of Complex

Multiplication

Classwork
Exercises
The vertices 𝐴𝐴(0,0), 𝐵𝐵(1,0), 𝐶𝐶(1,1), and 𝐷𝐷(0,1) of a unit square can be represented by the complex numbers 𝐴𝐴 = 0,
𝐵𝐵 = 1, 𝐶𝐶 = 1 + 𝑖𝑖, and 𝐷𝐷 = 𝑖𝑖.
1. Let 𝐿𝐿1 (𝑧𝑧) = −𝑧𝑧.
a. Calculate 𝐴𝐴′ = 𝐿𝐿1 (𝐴𝐴), 𝐵𝐵′ = 𝐿𝐿1 (𝐵𝐵), 𝐶𝐶′ = 𝐿𝐿1 (𝐶𝐶),
and 𝐷𝐷′ = 𝐿𝐿1 (𝐷𝐷). Plot these four points on the
axes.

b. Describe the geometric effect of the linear

transformation 𝐿𝐿1 (𝑧𝑧) = −𝑧𝑧 on the square
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M1
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2. Let 𝐿𝐿2 (𝑧𝑧) = 2𝑧𝑧.

a. Calculate 𝐴𝐴′ = 𝐿𝐿2 (𝐴𝐴), 𝐵𝐵′ = 𝐿𝐿2 (𝐵𝐵), 𝐶𝐶′ = 𝐿𝐿2 (𝐶𝐶),
and 𝐷𝐷′ = 𝐿𝐿2 (𝐷𝐷). Plot these four points on the
axes.

b. Describe the geometric effect of the linear

transformation 𝐿𝐿2 (𝑧𝑧) = 2𝑧𝑧 on the square
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

3. Let 𝐿𝐿3 (𝑧𝑧) = 𝑖𝑖𝑖𝑖.

a. Calculate 𝐴𝐴′ = 𝐿𝐿3 (𝐴𝐴), 𝐵𝐵′ = 𝐿𝐿3 (𝐵𝐵), 𝐶𝐶′ = 𝐿𝐿3 (𝐶𝐶),
and 𝐷𝐷′ = 𝐿𝐿3 (𝐷𝐷). Plot these four points on the
axes.

b. Describe the geometric effect of the linear

transformation 𝐿𝐿3 (𝑧𝑧) = 𝑖𝑖𝑖𝑖 on the square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

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4. Let 𝐿𝐿4 (𝑧𝑧) = (2𝑖𝑖)𝑧𝑧.

a. Calculate 𝐴𝐴′ = 𝐿𝐿4 (𝐴𝐴), 𝐵𝐵′ = 𝐿𝐿4 (𝐵𝐵), 𝐶𝐶′ = 𝐿𝐿4 (𝐶𝐶),
and 𝐷𝐷′ = 𝐿𝐿4 (𝐷𝐷). Plot these four points on the
axes.

b. Describe the geometric effect of the linear

transformation 𝐿𝐿4 (𝑧𝑧) = (2𝑖𝑖)𝑧𝑧 on the square
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

5. Explain how transformations 𝐿𝐿2 , 𝐿𝐿3 , and 𝐿𝐿4 are related.

6. We will continue to use the unit square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 with 𝐴𝐴 = 0, 𝐵𝐵 = 1, 𝐶𝐶 = 1 + 𝑖𝑖, 𝐷𝐷 = 𝑖𝑖 for this exercise.
a. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = 5𝑧𝑧 on the unit square?

b. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (5𝑖𝑖)𝑧𝑧 on the unit square?

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c. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (5𝑖𝑖 2 )𝑧𝑧 on the unit square?

d. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (5𝑖𝑖 3 )𝑧𝑧 on the unit square?

e. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (5𝑖𝑖 4 )𝑧𝑧 on the unit square?

f. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (5𝑖𝑖 5 )𝑧𝑧 on the unit square?

g. What is the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (5𝑖𝑖 𝑛𝑛 )𝑧𝑧 on the unit square, for some integer
𝑛𝑛 ≥ 0?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M1
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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:
𝐿𝐿1 (𝑧𝑧) = (3 + 4𝑖𝑖)𝑧𝑧
𝐿𝐿2 (𝑧𝑧) = (−3 + 4𝑖𝑖)𝑧𝑧
𝐿𝐿3 (𝑧𝑧) = (−3 − 4𝑖𝑖)𝑧𝑧
𝐿𝐿4 (𝑧𝑧) = (3 − 4𝑖𝑖)𝑧𝑧.
The unit square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 with 𝐴𝐴 = 0, 𝐵𝐵 = 1, 𝐶𝐶 = 1 + 𝑖𝑖, 𝐷𝐷 = 𝑖𝑖 is shown below. Apply your transformation to the vertices
of the square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, and plot the transformed points 𝐴𝐴′, 𝐵𝐵′, 𝐶𝐶′, and 𝐷𝐷′ on the same coordinate axes.

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For the 1-team: For the 2-team:

a. Why is 𝐵𝐵′ = 3 + 4𝑖𝑖? a. Why is 𝐵𝐵′ = −3 + 4𝑖𝑖?

b. What is the argument of 3 + 4𝑖𝑖? b. What is the argument of −3 + 4𝑖𝑖?

c. What is the modulus of 3 + 4𝑖𝑖? c. What is the modulus of −3 + 4𝑖𝑖?

For the 3-team: For the 4-team:

a. Why is 𝐵𝐵′ = −3 − 4𝑖𝑖? a. Why is 𝐵𝐵′ = 3 − 4𝑖𝑖?

b. What is the argument of −3 − 4𝑖𝑖? b. What is the argument of 3 − 4𝑖𝑖?

c. What is the modulus of −3 − 4𝑖𝑖? c. What is the modulus of 3 − 4𝑖𝑖?

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All groups should also answer the following:

a. Describe the amount the square has been rotated counterclockwise.

b. What is the dilation factor of the square? Explain how you know.

c. What is the geometric effect of your transformation 𝐿𝐿1 , 𝐿𝐿2 , 𝐿𝐿3 , or 𝐿𝐿4 on the unit square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴?

d. Make a conjecture: What do you expect to be the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (2 + 𝑖𝑖)𝑧𝑧 on
the unit square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴?

e. Test your conjecture with the unit square on the axes below.

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PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Find the modulus and argument for each of the following complex numbers.
�3 1
a. 𝑧𝑧1 = + 𝑖𝑖
2 2
b. 𝑧𝑧2 = 2 + 2√3𝑖𝑖
c. 𝑧𝑧3 = −3 + 5𝑖𝑖
d. 𝑧𝑧4 = −2 − 2𝑖𝑖
e. 𝑧𝑧5 = 4 − 4𝑖𝑖
f. 𝑧𝑧6 = 3 − 6𝑖𝑖

2. For parts (a)–(c), determine the geometric effect of the specified transformation.
a. 𝐿𝐿(𝑧𝑧) = −3𝑧𝑧
b. 𝐿𝐿(𝑧𝑧) = −100𝑧𝑧
1
c. 𝐿𝐿(𝑧𝑧) = − 𝑧𝑧
3
d. Describe the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = 𝑎𝑎𝑎𝑎 for any negative real number 𝑎𝑎.

3. For parts (a)–(c), determine the geometric effect of the specified transformation.
a. 𝐿𝐿(𝑧𝑧) = (−3𝑖𝑖)𝑧𝑧
b. 𝐿𝐿(𝑧𝑧) = (−100𝑖𝑖)𝑧𝑧
1
c. 𝐿𝐿(𝑧𝑧) = �− 𝑖𝑖� 𝑧𝑧
3
d. Describe the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (𝑏𝑏𝑏𝑏)𝑧𝑧 for any negative real number 𝑏𝑏.

4. Suppose that we have two linear transformations, 𝐿𝐿1 (𝑧𝑧) = 3𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = (5𝑖𝑖)𝑧𝑧.
a. What is the geometric effect of first performing transformation 𝐿𝐿1 and then performing transformation 𝐿𝐿2 ?
b. What is the geometric effect of first performing transformation 𝐿𝐿2 and then performing transformation 𝐿𝐿1 ?
c. Are your answers to parts (a) and (b) the same or different? Explain how you know.

5. Suppose that we have two linear transformations, 𝐿𝐿1 (𝑧𝑧) = (4 + 3𝑖𝑖)𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = −𝑧𝑧. What is the geometric effect
of first performing transformation 𝐿𝐿1 and then performing transformation 𝐿𝐿2 ?

6. Suppose that we have two linear transformations, 𝐿𝐿1 (𝑧𝑧) = (3 − 4𝑖𝑖)𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = −𝑧𝑧. What is the geometric effect
of first performing transformation 𝐿𝐿1 and then performing transformation 𝐿𝐿2 ?

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7. Explain the geometric effect of the linear transformation 𝐿𝐿(𝑧𝑧) = (𝑎𝑎 − 𝑏𝑏𝑏𝑏)𝑧𝑧, where 𝑎𝑎 and 𝑏𝑏 are positive real
numbers.

8. In Geometry, we learned the special angles of a right triangle whose hypotenuse is 1 unit. The figures are shown
above. Describe the geometric effect of the following transformations.
�3 1
a. 𝐿𝐿1 (𝑧𝑧) = � + 𝑖𝑖� 𝑧𝑧
2 2
b. 𝐿𝐿2 (𝑧𝑧) = �2 + 2√3𝑖𝑖�𝑧𝑧
�2 �2
c. 𝐿𝐿3 (𝑧𝑧) = � + 𝑖𝑖� 𝑧𝑧
2 2
d. 𝐿𝐿4 (𝑧𝑧) = (4 + 4𝑖𝑖)𝑧𝑧

9. Recall that a function 𝐿𝐿 is a linear transformation if all 𝑧𝑧 and 𝑤𝑤 in the domain of 𝐿𝐿 and all constants 𝑎𝑎 meet the
following two conditions:
i. 𝐿𝐿(𝑧𝑧 + 𝑤𝑤) = 𝐿𝐿(𝑧𝑧) + 𝐿𝐿(𝑤𝑤)
ii. 𝐿𝐿(𝑎𝑎𝑎𝑎) = 𝑎𝑎𝑎𝑎(𝑧𝑧)
Show that the following functions meet the definition of a linear transformation.
a. 𝐿𝐿1 (𝑧𝑧) = 4𝑧𝑧
b. 𝐿𝐿2 (𝑧𝑧) = 𝑖𝑖𝑖𝑖
c. 𝐿𝐿3 (𝑧𝑧) = (4 + 𝑖𝑖)𝑧𝑧

10. The vertices 𝐴𝐴(0, 0), 𝐵𝐵(1, 0), 𝐶𝐶(1, 1), 𝐷𝐷(0, 1) of a unit square can be represented by the complex numbers 𝐴𝐴 = 0,
𝐵𝐵 = 1, 𝐶𝐶 = 1 + 𝑖𝑖, 𝐷𝐷 = 𝑖𝑖. We learned that multiplication of those complex numbers by 𝑖𝑖 rotates the unit square by
90° counterclockwise. What do you need to multiply by so that the unit square will be rotated by 90° clockwise?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 15: Justifying the Geometric Effect of Complex

Multiplication

Classwork
Opening Exercise
For each exercise below, compute the product 𝑤𝑤𝑤𝑤. Then, plot the complex numbers 𝑧𝑧, 𝑤𝑤, and 𝑤𝑤𝑤𝑤 on the axes provided.
a. 𝑧𝑧 = 3 + 𝑖𝑖, 𝑤𝑤 = 1 + 2𝑖𝑖

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b. 𝑧𝑧 = 1 + 2𝑖𝑖, 𝑤𝑤 = −1 + 4𝑖𝑖

c. 𝑧𝑧 = −1 + 𝑖𝑖, 𝑤𝑤 = −2 − 𝑖𝑖

d. For each part (a), (b), and (c), draw line segments connecting each point 𝑧𝑧, 𝑤𝑤, and 𝑤𝑤𝑤𝑤 to the origin. Determine
a relationship between the arguments of the complex numbers 𝑧𝑧, 𝑤𝑤, and 𝑤𝑤𝑤𝑤.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises
1. Let 𝑤𝑤 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 and 𝑧𝑧 = 𝑐𝑐 + 𝑑𝑑𝑑𝑑.
a. Calculate the product 𝑤𝑤𝑤𝑤.

b. Calculate the moduli |𝑤𝑤|, |𝑧𝑧|, and |𝑤𝑤𝑤𝑤|.

c. What can you conclude about the quantities |𝑤𝑤|, |𝑧𝑧|, and |𝑤𝑤𝑤𝑤|?

2. What does the result of Exercise 1 tell us about the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = 𝑤𝑤𝑤𝑤?

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3. If 𝑧𝑧 and 𝑤𝑤 are the complex numbers with the specified arguments and moduli, locate the point that represents the
product 𝑤𝑤𝑤𝑤 on the provided coordinate axes.
𝜋𝜋
a. |𝑤𝑤| = 3, arg(𝑤𝑤) =
4
2 𝜋𝜋
|𝑧𝑧| = , arg(𝑧𝑧) = −
3 2

b. |𝑤𝑤| = 2, arg(𝑤𝑤) = 𝜋𝜋
𝜋𝜋
|𝑧𝑧| = 1, arg(𝑧𝑧) =
4

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M1
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1 4𝜋𝜋
c. |𝑤𝑤| = , arg(𝑤𝑤) =
2 3
𝜋𝜋
|𝑧𝑧| = 4, arg(𝑧𝑧) = −
6

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
For complex numbers 𝑧𝑧 and 𝑤𝑤,
 The modulus of the product is the product of the moduli:
|𝑤𝑤𝑤𝑤| = |𝑤𝑤| ⋅ |𝑧𝑧| .
 The argument of the product is the sum of the arguments:
arg(𝑤𝑤𝑤𝑤) = arg(𝑤𝑤) + arg(𝑧𝑧).

Problem Set

1. In the lesson, we justified our observation that the geometric effect of a transformation 𝐿𝐿(𝑧𝑧) = 𝑤𝑤𝑤𝑤 is a rotation by
arg(𝑤𝑤) and a dilation by |𝑤𝑤| for a complex number 𝑤𝑤 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 𝑤𝑤. For a
complex number 𝑤𝑤 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏, we only considered the case where 𝑎𝑎 > 0 and 𝑏𝑏 > 0. There are eight additional
possibilities we need to consider.
a. Case 1: The point representing 𝑤𝑤 is the origin. That is, 𝑎𝑎 = 0 and 𝑏𝑏 = 0.
In this case, explain why 𝐿𝐿(𝑧𝑧) = (𝑎𝑎 + 𝑏𝑏𝑏𝑏)𝑧𝑧 has the geometric effect of rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) and dilation by
|𝑎𝑎 + 𝑏𝑏𝑏𝑏|.
b. Case 2: The point representing 𝑤𝑤 lies on the positive real axis. That is, 𝑎𝑎 > 0 and 𝑏𝑏 = 0.
In this case, explain why 𝐿𝐿(𝑧𝑧) = (𝑎𝑎 + 𝑏𝑏𝑏𝑏)𝑧𝑧 has the geometric effect of rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) and dilation by
|𝑎𝑎 + 𝑏𝑏𝑏𝑏|.
c. Case 3: The point representing 𝑤𝑤 lies on the negative real axis. That is, 𝑎𝑎 < 0 and 𝑏𝑏 = 0.
In this case, explain why 𝐿𝐿(𝑧𝑧) = (𝑎𝑎 + 𝑏𝑏𝑏𝑏)𝑧𝑧 has the geometric effect of rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) and dilation by
|𝑎𝑎 + 𝑏𝑏𝑏𝑏|.
d. Case 4: The point representing 𝑤𝑤 lies on the positive imaginary axis. That is, 𝑎𝑎 = 0 and 𝑏𝑏 > 0.
In this case, explain why 𝐿𝐿(𝑧𝑧) = (𝑎𝑎 + 𝑏𝑏𝑏𝑏)𝑧𝑧 has the geometric effect of rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) and dilation by
|𝑎𝑎 + 𝑏𝑏𝑏𝑏|.
e. Case 5: The point representing 𝑤𝑤 lies on the negative imaginary axis. That is, 𝑎𝑎 = 0 and 𝑏𝑏 < 0.
In this case, explain why 𝐿𝐿(𝑧𝑧) = (𝑎𝑎 + 𝑏𝑏𝑏𝑏)𝑧𝑧 has the geometric effect of rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) and dilation by
|𝑎𝑎 + 𝑏𝑏𝑏𝑏|.

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f. Case 6: The point representing 𝑤𝑤 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 lies in the second quadrant. That is, 𝑎𝑎 < 0 and 𝑏𝑏 > 0.
Points representing 𝑤𝑤, 𝑧𝑧, 𝑎𝑎𝑎𝑎, (𝑏𝑏𝑏𝑏)𝑧𝑧, and 𝑤𝑤𝑤𝑤 = 𝑎𝑎𝑎𝑎 + (𝑏𝑏𝑏𝑏)𝑧𝑧 are shown in the figure below.

For convenience, rename the origin 𝑂𝑂, and let 𝑃𝑃 = 𝑤𝑤, 𝑄𝑄 = 𝑎𝑎, 𝑅𝑅 = 𝑤𝑤𝑤𝑤, 𝑆𝑆 = 𝑎𝑎𝑎𝑎, and 𝑇𝑇 = 𝑧𝑧, as shown below.
Let 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜃𝜃.

i. Argue that △ 𝑂𝑂𝑂𝑂𝑂𝑂 ∼ △ 𝑂𝑂𝑂𝑂𝑂𝑂.

ii. Express the argument of 𝑎𝑎𝑎𝑎 in terms of arg(𝑧𝑧).
iii. Express arg(𝑤𝑤) in terms of 𝜃𝜃, where 𝜃𝜃 = 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃.
iv. Explain why arg(𝑤𝑤𝑤𝑤) = arg(𝑎𝑎𝑎𝑎) − 𝜃𝜃.
v. Combine your responses from parts (ii), (iii), and (iv) to express arg(𝑤𝑤𝑤𝑤) in terms of arg(𝑧𝑧) and arg(𝑤𝑤).

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g. Case 7: The point representing 𝑤𝑤 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 lies in the third quadrant. That is, 𝑎𝑎 < 0 and 𝑏𝑏 < 0.
Points representing 𝑤𝑤, 𝑧𝑧, 𝑎𝑎𝑎𝑎, (𝑏𝑏𝑏𝑏)𝑧𝑧, and 𝑤𝑤𝑤𝑤 = 𝑎𝑎𝑎𝑎 + (𝑏𝑏𝑏𝑏)𝑧𝑧 are shown in the figure below.

For convenience, rename the origin 𝑂𝑂, and let 𝑃𝑃 = 𝑤𝑤, 𝑄𝑄 = 𝑎𝑎, 𝑅𝑅 = 𝑤𝑤𝑤𝑤, 𝑆𝑆 = 𝑎𝑎𝑎𝑎, and 𝑇𝑇 = 𝑧𝑧, as shown below.
Let 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜃𝜃.

i. Argue that △ 𝑂𝑂𝑂𝑂𝑂𝑂 ∼ △ 𝑂𝑂𝑂𝑂𝑂𝑂.

ii. Express the argument of 𝑎𝑎𝑎𝑎 in terms of arg(𝑧𝑧).
iii. Express arg(𝑤𝑤) in terms of 𝜃𝜃, where 𝜃𝜃 = 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃.
iv. Explain why arg(𝑤𝑤𝑤𝑤) = arg(𝑎𝑎𝑎𝑎) + 𝜃𝜃.
v. Combine your responses from parts (ii), (iii), and (iv) to express arg(𝑤𝑤𝑤𝑤) in terms of arg(𝑧𝑧) and arg(𝑤𝑤).

Lesson 15: Justifying the Geometric Effect of Complex Multiplication

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PRECALCULUS AND ADVANCED TOPICS

h. Case 8: The point representing 𝑤𝑤 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 lies in the fourth quadrant. That is, 𝑎𝑎 > 0 and 𝑏𝑏 < 0.
Points representing 𝑤𝑤, 𝑧𝑧, 𝑎𝑎𝑎𝑎, (𝑏𝑏𝑏𝑏)𝑧𝑧, and 𝑤𝑤𝑤𝑤 = 𝑎𝑎𝑎𝑎 + (𝑏𝑏𝑏𝑏)𝑧𝑧 are shown in the figure below.

For convenience, rename the origin 𝑂𝑂, and let 𝑃𝑃 = 𝑤𝑤, 𝑄𝑄 = 𝑎𝑎, 𝑅𝑅 = 𝑤𝑤𝑤𝑤, 𝑆𝑆 = 𝑎𝑎𝑎𝑎, and 𝑇𝑇 = 𝑧𝑧, as shown below.
Let 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 𝜃𝜃.

i. Argue that △ 𝑂𝑂𝑂𝑂𝑂𝑂 ∼ △ 𝑂𝑂𝑂𝑂𝑂𝑂.

ii. Express arg(𝑤𝑤) in terms of 𝜃𝜃, where 𝜃𝜃 = 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃.
iii. Explain why 𝑚𝑚∠𝑄𝑄𝑄𝑄𝑄𝑄 = 𝜃𝜃 − arg(𝑧𝑧).
iv. Express arg(𝑤𝑤𝑤𝑤) in terms of 𝑚𝑚∠𝑄𝑄𝑄𝑄𝑄𝑄.
v. Combine your responses from parts (ii), (iii), and (iv) to express arg(𝑤𝑤𝑤𝑤) in terms of arg(𝑧𝑧) and arg(𝑤𝑤).

2. Summarize the results of Problem 1, parts (a)–(h), and the lesson.

Lesson 15: Justifying the Geometric Effect of Complex Multiplication

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M1
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3. Find a linear transformation 𝐿𝐿 that will have the geometric effect of rotation by the specified amount without
dilating.
a. 45° counterclockwise
b. 60° counterclockwise
c. 180° counterclockwise
d. 120° counterclockwise
e. 30° clockwise
f. 90° clockwise
g. 180° clockwise
h. 135° clockwise

4. Suppose that we have linear transformations 𝐿𝐿1 and 𝐿𝐿2 as specified below. Find a formula for 𝐿𝐿2 �𝐿𝐿1 (𝑧𝑧)� for
complex numbers 𝑧𝑧.
a. 𝐿𝐿1 (𝑧𝑧) = (1 + 𝑖𝑖)𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = (1 − 𝑖𝑖)𝑧𝑧
b. 𝐿𝐿1 (𝑧𝑧) = (3 − 2𝑖𝑖)𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = (2 + 3𝑖𝑖)𝑧𝑧
c. 𝐿𝐿1 (𝑧𝑧) = (−4 + 3𝑖𝑖)𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = (−3 − 𝑖𝑖)𝑧𝑧
d. 𝐿𝐿1 (𝑧𝑧) = (𝑎𝑎 + 𝑏𝑏𝑏𝑏)𝑧𝑧 and 𝐿𝐿2 (𝑧𝑧) = (𝑐𝑐 + 𝑑𝑑𝑑𝑑)𝑧𝑧 for real numbers 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 and 𝑑𝑑.

Lesson 15: Justifying the Geometric Effect of Complex Multiplication

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 16: Representing Reflections with Transformations

Classwork
Opening Exercise
a. Find a transformation 𝑅𝑅(0,45°) : ℂ → ℂ that rotates a point represented by the complex number 𝑧𝑧 by 45°
counterclockwise in the coordinate plane but does not produce a dilation.

b. Find a transformation 𝑅𝑅(0,−45°) : ℂ → ℂ that rotates a point represented by the complex number 𝑧𝑧 by 45°
clockwise in the coordinate plane but does not produce a dilation.

c. Find a transformation 𝑟𝑟𝑥𝑥-axis : ℂ → ℂ that reflects a point represented by the complex number 𝑧𝑧 across the
𝑥𝑥-axis.

Discussion
We want to find a transformation 𝑟𝑟ℓ : ℂ → ℂ that reflects a point representing a complex number 𝑧𝑧 across the diagonal
line ℓ with equation 𝑦𝑦 = 𝑥𝑥.

Lesson 16: Representing Reflections with Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises
1. The number 𝑧𝑧 in the figure used in the discussion above is the complex number 1 + 5𝑖𝑖. Compute 𝑟𝑟ℓ (1 + 5𝑖𝑖), and
plot it below.

2. We know from previous courses that the reflection of a point (𝑥𝑥, 𝑦𝑦) across the line with equation 𝑦𝑦 = 𝑥𝑥 is the point
(𝑦𝑦, 𝑥𝑥). Does this agree with our result from the previous discussion?

3. We now want to find a formula for the transformation of reflection across the line ℓ that makes a 60° angle with the
positive 𝑥𝑥-axis. Find formulas to represent each component of the transformation, and use them to find one
formula that represents the overall transformation.

Lesson 16: Representing Reflections with Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 16 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
Let ℓ be a line through the origin that contains the terminal ray of a rotation of the 𝑥𝑥-axis by 𝜃𝜃. Then, reflection
across line ℓ can be done by the following sequence of transformations:
 Rotation by −𝜃𝜃 about the origin.
 Reflection across the 𝑥𝑥-axis.
 Rotation by 𝜃𝜃 about the origin.

Problem Set

1. Find a formula for the transformation of reflection across the line ℓ with equation 𝑦𝑦 = −𝑥𝑥.

2. Find the formula for the sequence of transformations comprising reflection across the line with equation 𝑦𝑦 = 𝑥𝑥 and
then rotation by 180° about the origin.

3. Compare your answers to Problems 1 and 2. Explain what you find.

4. Find a formula for the transformation of reflection across the line ℓ that makes a −30° angle with the positive
𝑥𝑥-axis.

5. Max observed that when reflecting a complex number, 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 about the line 𝑦𝑦 = 𝑥𝑥, 𝑎𝑎 and 𝑏𝑏 are reversed, which
is similar to how we learned to find an inverse function. Will Max’s observation also be true when the line
𝑦𝑦 = −𝑥𝑥 is used, where 𝑎𝑎 = −𝑏𝑏 and 𝑏𝑏 = −𝑎𝑎? Give an example to show his assumption is either correct or incorrect.

6. For reflecting a complex number, 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 about the line 𝑦𝑦 = 2𝑥𝑥, will Max’s idea work if he makes 𝑏𝑏 = 2𝑎𝑎 and
𝑏𝑏
𝑎𝑎 = ? Use 𝑧𝑧 = 1 + 4𝑖𝑖 as an example to show whether or not it works.
2

7. What would the formula look like if you want to reflect a complex number about the line 𝑦𝑦 = 𝑚𝑚𝑚𝑚, where 𝑚𝑚 > 0?

Lesson 16: Representing Reflections with Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 17: The Geometric Effect of Multiplying by a Reciprocal

Classwork
Opening Exercise
Given 𝑤𝑤 = 1 + 𝑖𝑖, what is arg(𝑤𝑤) and |𝑤𝑤|? Explain how you got your answer.

Exploratory Challenge 1/Exercises 1–9

1. Describe the geometric effect of the transformation 𝐿𝐿(𝑧𝑧) = (1 + 𝑖𝑖)𝑧𝑧.

2. Describe a way to undo the effect of the transformation 𝐿𝐿(𝑧𝑧) = (1 + 𝑖𝑖)𝑧𝑧.

3. Given that 0 ≤ arg(𝑧𝑧) < 2𝜋𝜋 for any complex number, how could you describe any clockwise rotation of 𝜃𝜃 as an
argument of a complex number?

4. Write a complex number in polar form that describes a rotation and dilation that will undo multiplication by (1 + 𝑖𝑖),
and then convert it to rectangular form.

Lesson 17: The Geometric Effect of Multiplying by a Reciprocal

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1
PRECALCULUS AND ADVANCED TOPICS

1
5. In a previous lesson, you learned that to undo multiplication by 1 + 𝑖𝑖, you would multiply by the reciprocal .
1+𝑖𝑖
1
Write the complex number in rectangular form 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are real numbers.
1+𝑖𝑖

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?

7. Jimmy states the following:

1
Multiplication by has the reverse geometric effect of multiplication 𝑏𝑏𝑏𝑏 + 𝑏𝑏𝑏𝑏.
𝑎𝑎+𝑏𝑏𝑏𝑏
Do you agree or disagree? Use your work on the previous exercises to support your reasoning.

8. Show that the following statement is true when 𝑧𝑧 = 2 − 2√3𝑖𝑖:

1 1
The reciprocal of a complex number 𝑧𝑧 with modulus 𝑟𝑟 and argument 𝜃𝜃 is with modulus and argument 2𝜋𝜋 − 𝜃𝜃.
𝑧𝑧 𝑟𝑟

1
9. Explain using transformations why 𝑧𝑧 ∙ = 1.
𝑧𝑧

Lesson 17: The Geometric Effect of Multiplying by a Reciprocal

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1
PRECALCULUS AND ADVANCED TOPICS

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 𝒛𝒛 = 𝟑𝟑 − 𝟑𝟑𝟑𝟑 and 𝒘𝒘 = −𝟐𝟐𝟐𝟐.
Addition
𝑧𝑧 + 𝑤𝑤

Subtraction
𝑧𝑧 − 𝑤𝑤

Conjugate of
𝑧𝑧

Multiplication
𝑤𝑤 ∙ 𝑧𝑧

Reciprocal of
𝑧𝑧

Division
𝑤𝑤
𝑧𝑧

Lesson 17: The Geometric Effect of Multiplying by a Reciprocal

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises 11–13
Let 𝑧𝑧 = −1 + 𝑖𝑖, and let 𝑤𝑤 = 2𝑖𝑖. Describe each complex number as a transformation of 𝑧𝑧, and then write the number in
rectangular form.
11. 𝑤𝑤𝑧𝑧̅

1
12.
𝑧𝑧̅

13. ��������
𝑤𝑤 + 𝑧𝑧

Lesson 17: The Geometric Effect of Multiplying by a Reciprocal

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Describe the geometric effect of multiplying 𝑧𝑧 by the reciprocal of each complex number listed below.
a. 𝑤𝑤1 = 3𝑖𝑖 b. 𝑤𝑤2 = −2
c. 𝑤𝑤3 = √3 + 𝑖𝑖 d. 𝑤𝑤4 = 1 − √3𝑖𝑖

2. Let 𝑧𝑧 = −2 − 2√3𝑖𝑖. 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.
−2−2√3𝑖𝑖 −2−2√3𝑖𝑖
a. b.
𝑤𝑤1 𝑤𝑤2

−2−2√3𝑖𝑖 −2−2√3𝑖𝑖
c. d.
𝑤𝑤3 𝑤𝑤4

1
3. In Exercise 12 of this lesson, you described the complex number as a transformation of 𝑧𝑧 for a specific complex
𝑧𝑧̅
number 𝑧𝑧. Show that this transformation always produces a dilation of 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏.

1
4. Does 𝐿𝐿(𝑧𝑧) = satisfy the conditions that 𝐿𝐿(𝑧𝑧 + 𝑤𝑤) = 𝐿𝐿(𝑧𝑧) + 𝐿𝐿(𝑤𝑤) and 𝐿𝐿(𝑚𝑚𝑚𝑚) = 𝑚𝑚𝑚𝑚(𝑧𝑧), which makes it a linear
𝑧𝑧
transformation? Justify your answer.

����
1
5. Show that 𝐿𝐿(𝑧𝑧) = 𝑤𝑤 � 𝑧𝑧� describes a reflection of 𝑧𝑧 about the line containing the origin and 𝑤𝑤 for 𝑧𝑧 = 3𝑖𝑖 and
𝑤𝑤
𝑤𝑤 = 1 + 𝑖𝑖.

6. Describe the geometric effect of each transformation function on 𝑧𝑧 where 𝑧𝑧, 𝑤𝑤, and 𝑎𝑎 are complex numbers.
𝑧𝑧−𝑤𝑤 ��������
𝑧𝑧−𝑤𝑤
a. 𝐿𝐿1 (𝑧𝑧) = b. 𝐿𝐿2 (𝑧𝑧) = � �
𝑎𝑎 𝑎𝑎
��������
𝑧𝑧−𝑤𝑤 ��������
𝑧𝑧−𝑤𝑤
c. 𝐿𝐿3 (𝑧𝑧) = 𝑎𝑎 � � d. 𝐿𝐿3 (𝑧𝑧) = 𝑎𝑎 � � + 𝑤𝑤
𝑎𝑎 𝑎𝑎

7. Verify your answers to Problem 6 if 𝑧𝑧 = 1 − 𝑖𝑖, 𝑤𝑤 = 2𝑖𝑖, and 𝑎𝑎 = −1 − 𝑖𝑖.

𝑧𝑧−𝑤𝑤 ��������
𝑧𝑧−𝑤𝑤
a. 𝐿𝐿1 (𝑧𝑧) = b. 𝐿𝐿2 (𝑧𝑧) = � �
𝑎𝑎 𝑎𝑎

��������
𝑧𝑧−𝑤𝑤 ��������
𝑧𝑧−𝑤𝑤
c. 𝐿𝐿3 (𝑧𝑧) = 𝑎𝑎 � � d. 𝐿𝐿3 (𝑧𝑧) = 𝑎𝑎 � � + 𝑤𝑤
𝑎𝑎 𝑎𝑎

Lesson 17: The Geometric Effect of Multiplying by a Reciprocal

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 18: Exploiting the Connection to Trigonometry

Classwork
Opening Exercise
a. Identify the modulus and argument of each complex number, and then rewrite it in rectangular form.
𝜋𝜋 𝜋𝜋
i. 2 �cos � � + 𝑖𝑖 sin � ��
4 4

2𝜋𝜋 2𝜋𝜋
ii. 5 �cos � � + 𝑖𝑖 sin � ��
3 3

7𝜋𝜋 7𝜋𝜋
iii. 3√2 �cos � � + 𝑖𝑖 sin � ��
4 4

7𝜋𝜋 7𝜋𝜋
iv. 3 �cos � � + 𝑖𝑖 sin � ��
6 6

v. 1(cos(𝜋𝜋) + 𝑖𝑖 sin(𝜋𝜋))

b. What is the argument and modulus of each complex number? Explain how you know.
i. 2 − 2𝑖𝑖

Lesson 18: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1
PRECALCULUS AND ADVANCED TOPICS

ii. 3√3 + 3𝑖𝑖

iii. −1 − √3𝑖𝑖

iv. −5𝑖𝑖

v. 1

Exploratory Challenge/Exercises 1–12

1. Rewrite each expression as a complex number in rectangular form.
a. (1 + 𝑖𝑖)2

b. (1 + 𝑖𝑖)3

c. (1 + 𝑖𝑖)4

Lesson 18: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1
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2. Complete the table below showing the rectangular form of each number and its modulus and argument.

Power of (𝟏𝟏 + 𝒊𝒊) Rectangular Form Modulus Argument

(1 + 𝑖𝑖)0

(1 + 𝑖𝑖)1

(1 + 𝑖𝑖)2

(1 + 𝑖𝑖)3

(1 + 𝑖𝑖)4

3. What patterns do you notice each time you multiply by another factor of (1 + 𝑖𝑖)?

4. Graph each power of 1 + 𝑖𝑖 shown in the table on the same coordinate grid. Describe the location of these numbers
in relation to one another using transformations.

Lesson 18: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1
PRECALCULUS AND ADVANCED TOPICS

5. Predict what the modulus and argument of (1 + 𝑖𝑖)5 would be without actually performing the multiplication.
Explain how you made your prediction.

6. Graph (1 + 𝑖𝑖)5 in the complex plane using the transformations you described in Exercise 5.

7. Write each number in polar form using the modulus and argument you calculated in Exercise 4.
(1 + 𝑖𝑖)0

(1 + 𝑖𝑖)1

(1 + 𝑖𝑖)2

(1 + 𝑖𝑖)3

(1 + 𝑖𝑖)4

8. Use the patterns you have observed to write (1 + 𝑖𝑖)5 in polar form, and then convert it to rectangular form.

9. What is the polar form of (1 + 𝑖𝑖)20 ? What is the modulus of (1 + 𝑖𝑖)20 ? What is its argument? Explain why
(1 + 𝑖𝑖)20 is a real number.

Lesson 18: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1
PRECALCULUS AND ADVANCED TOPICS

10. If 𝑧𝑧 has modulus 𝑟𝑟 and argument 𝜃𝜃, what are the modulus and argument of 𝑧𝑧2 ? Write the number 𝑧𝑧2 in polar form.

11. If 𝑧𝑧 has modulus 𝑟𝑟 and argument 𝜃𝜃, what are the modulus and argument of 𝑧𝑧 𝑛𝑛 where 𝑛𝑛 is a nonnegative integer?
Write the number 𝑧𝑧 𝑛𝑛 in polar form. Explain how you got your answer.

1 1
12. Recall that = (cos(−𝜃𝜃) + 𝑖𝑖 sin(−𝜃𝜃)). Explain why it would make sense that the formula holds for all integer
𝑧𝑧 𝑟𝑟
values of 𝑛𝑛.

Exercises 13–14
7
1−𝑖𝑖
13. Compute � � , and write it as a complex number in the form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are real numbers.
�2

6
14. Write �1 + √3𝑖𝑖� , and write it as a complex number in the form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are real numbers.

Lesson 18: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
Given a complex number 𝑧𝑧 with modulus 𝑟𝑟 and argument 𝜃𝜃, the 𝑛𝑛th power of 𝑧𝑧 is given by
𝑧𝑧 𝑛𝑛 = 𝑟𝑟 𝑛𝑛 (cos(𝑛𝑛𝑛𝑛) + 𝑖𝑖 sin(𝑛𝑛𝑛𝑛)) where 𝑛𝑛 is an integer.

Problem Set

1. Write the complex number in 𝑎𝑎 + 𝑏𝑏𝑏𝑏 form where 𝑎𝑎 and 𝑏𝑏 are real numbers.
5𝜋𝜋 5𝜋𝜋
a. 2 �cos � � + 𝑖𝑖 sin � �� b. 3(cos(210°) + 𝑖𝑖 sin(210°))
3 3
10 15𝜋𝜋 15𝜋𝜋
c. �√2� �cos � � + 𝑖𝑖 sin � �� d. cos(9𝜋𝜋) + 𝑖𝑖 sin(9𝜋𝜋)
4 4
3𝜋𝜋 3𝜋𝜋
e. 43 �cos � � +   𝑖𝑖 sin � �� f. 6(cos(480°) + 𝑖𝑖 sin(480°))
4 4

2. Use the formula discovered in this lesson to compute each power of 𝑧𝑧. Verify that the formula works by expanding
and multiplying the rectangular form and rewriting it in the form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are real numbers.
3
a. �1 + √3𝑖𝑖� b. (−1 + 𝑖𝑖)4
c. (2 + 2𝑖𝑖)5 d. (2 − 2𝑖𝑖)−2
4 6
e. �√3 − 𝑖𝑖� f. �3√3 − 3𝑖𝑖�

3. Given 𝑧𝑧 = −1 − 𝑖𝑖, graph the first five powers of 𝑧𝑧 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.

4. Use your work from Problem 3 to determine three values of 𝑛𝑛 for which (−1 − 𝑖𝑖)𝑛𝑛 is a multiple of −1 − 𝑖𝑖.

5. Find the indicated power of the complex number, and write your answer in form 𝑎𝑎 + 𝑏𝑏𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are real
numbers.
3𝜋𝜋 3𝜋𝜋 3 𝜋𝜋 𝜋𝜋 10
a. �2 �cos � � + 𝑖𝑖 sin � ��� b. �√2 �cos � � + 𝑖𝑖 sin � ���
4 4 4 4
5𝜋𝜋 5𝜋𝜋 6 1 3𝜋𝜋 3𝜋𝜋 4
c. �cos � � + 𝑖𝑖 sin � �� d. � �cos � � + 𝑖𝑖 sin � ���
6 6 3 2 2
4𝜋𝜋 4𝜋𝜋 −4
e. �4 �cos � � + 𝑖𝑖 sin � ���
3 3

Lesson 18: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 19: Exploiting the Connection to Trigonometry

Classwork
Opening Exercise
A polar grid is shown below. The grid is formed by rays from the origin at equal rotation intervals and concentric circles
centered at the origin. The complex number 𝑧𝑧 = √3 + 𝑖𝑖 is graphed on this polar grid.

a. Use the polar grid to identify the modulus and argument of 𝑧𝑧.

b. Graph the next three powers of 𝑧𝑧 on the polar grid. Explain how you got your answers.

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

c. Write the polar form of the number in the table below, and then rewrite it in rectangular form.

Power of 𝒛𝒛 Polar Form Rectangular Form

√3 + 𝑖𝑖
2
�√3 + 𝑖𝑖�
3
�√3 + 𝑖𝑖�
4
�√3 + 𝑖𝑖�

Exercises 1–3
2
The complex numbers 𝑧𝑧2 = �−1 + √3𝑖𝑖� and 𝑧𝑧1 are graphed below.

1. Use the graph to help you write the numbers in polar and rectangular form.

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

2. Describe how the modulus and argument of 𝑧𝑧1 = −1 + √3𝑖𝑖 are related to the modulus and argument of
2
𝑧𝑧2 = �−1 + √3𝑖𝑖� .

3. Why could we call −1 + √3𝑖𝑖 a square root of −2 − 2√3𝑖𝑖?

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

Example: Find the Two Square Roots of a Complex Number

Find both of the square roots of −2 − 2√3𝑖𝑖.

Exercises 4–6
4. Find the cube roots of −2 = 2√3𝑖𝑖.

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

5. Find the square roots of 4𝑖𝑖.

6. Find the cube roots of 8.

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
Given a complex number 𝑧𝑧 with modulus 𝑟𝑟 and argument 𝜃𝜃, the 𝑛𝑛th roots of 𝑧𝑧 are given by
𝑛𝑛 𝜃𝜃 2𝜋𝜋𝜋𝜋 𝜃𝜃 2𝜋𝜋𝜋𝜋
√𝑟𝑟 �cos � + � + 𝑖𝑖 sin � + ��
𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛
for integers 𝑘𝑘 and 𝑛𝑛 such that 𝑛𝑛 > 0 and 0 ≤ 𝑘𝑘 < 𝑛𝑛.

Problem Set

1. For each complex number, what is 𝑧𝑧 2 ?

a. 1 + √3𝑖𝑖
b. 3 − 3𝑖𝑖
c. 4𝑖𝑖
�3 1
d. − + 𝑖𝑖
2 2
1 1
e. + 𝑖𝑖
9 9
f. −1

2. For each complex number, what are the square roots of 𝑧𝑧?
a. 1 + √3𝑖𝑖
b. 3 − 3𝑖𝑖
c. 4𝑖𝑖
�3 1
d. − + 𝑖𝑖
2 2
1 1
e. + 𝑖𝑖
9 9
f. −1

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1
PRECALCULUS AND ADVANCED TOPICS

3. Graph each complex number on a polar grid.

𝜋𝜋 𝜋𝜋
a. 2 �cos � � + 𝑖𝑖 sin � ��
3 3
b. 3(cos(210°) + 𝑖𝑖 sin(210°))
𝜋𝜋 𝜋𝜋
c. 2 �cos � � + 𝑖𝑖 sin � ��
4 4
d. cos(𝜋𝜋) + 𝑖𝑖 sin(𝜋𝜋)
3𝜋𝜋 3𝜋𝜋
e. 4 �cos � � + 𝑖𝑖 sin � ��
4 4
1
f. (cos(60°) +   𝑖𝑖 sin(60°))
2

4. What are the cube roots of −3𝑖𝑖?

5. What are the fourth roots of 64?

6. What are the square roots of −4 − 4𝑖𝑖?

7. Find the square roots of −5. Show that the square roots satisfy the equation 𝑥𝑥 2 + 5 = 0.

8. Find the cube roots of 27. Show that the cube roots satisfy the equation 𝑥𝑥 3 − 27 = 0.

Lesson 19: Exploiting the Connection to Trigonometry

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 20: Exploiting the Connection to Cartesian Coordinates

Classwork
Opening Exercise
a. Find a complex number 𝑤𝑤 so that the transformation 𝐿𝐿1 (𝑧𝑧) = 𝑤𝑤𝑤𝑤 produces a clockwise rotation by 1° about
the origin with no dilation.

b. Find a complex number 𝑤𝑤 so that the transformation 𝐿𝐿2 (𝑧𝑧) = 𝑤𝑤𝑤𝑤 produces a dilation with scale factor 0.1 with
no rotation.

Exercises
1.
a. Find values of 𝑎𝑎 and 𝑏𝑏 so that 𝐿𝐿1 (𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) has the effect of dilation with scale factor 2 and
no rotation.

Lesson 20: Exploiting the Connection to Cartesian Coordinates

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

b. Evaluate 𝐿𝐿1 �𝐿𝐿1 (𝑥𝑥, 𝑦𝑦)�, and identify the resulting transformation.

2.
a. Find values of 𝑎𝑎 and 𝑏𝑏 so that 𝐿𝐿2 (𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) has the effect of rotation about the origin by
180° counterclockwise and no dilation.

b. Evaluate 𝐿𝐿2 �𝐿𝐿2 (𝑥𝑥, 𝑦𝑦)�, and identify the resulting transformation.

3.
a. Find values of 𝑎𝑎 and 𝑏𝑏 so that 𝐿𝐿3 (𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) has the effect of rotation about the origin by 90°
counterclockwise and no dilation.

Lesson 20: Exploiting the Connection to Cartesian Coordinates

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

b. Evaluate 𝐿𝐿3 �𝐿𝐿3 (𝑥𝑥, 𝑦𝑦)�, and identify the resulting transformation.

4.
a. Find values of 𝑎𝑎 and 𝑏𝑏 so that 𝐿𝐿3 (𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) has the effect of rotation about the origin by 45°
counterclockwise and no dilation.

b. Evaluate 𝐿𝐿4 �𝐿𝐿4 (𝑥𝑥, 𝑦𝑦)�, and identify the resulting transformation.

Lesson 20: Exploiting the Connection to Cartesian Coordinates

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

5. The figure below shows a quadrilateral with vertices 𝐴𝐴(0,0), 𝐵𝐵(1,0), 𝐶𝐶(3,3), and 𝐷𝐷(0,3).
a. Transform each vertex under 𝐿𝐿5 = (3𝑥𝑥 + 𝑦𝑦, 3𝑦𝑦 − 𝑥𝑥), and plot the transformed vertices on the figure.

b. Does 𝐿𝐿5 represent a rotation and dilation? If so, estimate the amount of rotation and the scale factor from
your figure.

c. If 𝐿𝐿5 represents a rotation and dilation, calculate the amount of rotation and the scale factor from the formula
for 𝐿𝐿5 . Do your numbers agree with your estimate in part (b)? If not, explain why there are no values of 𝑎𝑎 and
𝑏𝑏 so that 𝐿𝐿5 (𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑥𝑥 + 𝑎𝑎𝑎𝑎).

Lesson 20: Exploiting the Connection to Cartesian Coordinates

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

6. The figure below shows a figure with vertices 𝐴𝐴(0,0), 𝐵𝐵(1,0), 𝐶𝐶(3,3), and 𝐷𝐷(0,3).
a. Transform each vertex under 𝐿𝐿6 = (2𝑥𝑥 + 2𝑦𝑦, 2𝑥𝑥 − 2𝑦𝑦), and plot the transformed vertices on the figure.

b. Does 𝐿𝐿6 represent a rotation and dilation? If so, estimate the amount of rotation and the scale factor from
your figure.

c. If 𝐿𝐿5 represents a rotation and dilation, calculate the amount of rotation and the scale factor from the formula
for 𝐿𝐿6 . Do your numbers agree with your estimate in part (b)? If not, explain why there are no values of 𝑎𝑎 and
𝑏𝑏 so that 𝐿𝐿6 (𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎).

Lesson 20: Exploiting the Connection to Cartesian Coordinates

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
For real numbers 𝑎𝑎 and 𝑏𝑏, the transformation 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) corresponds to a counterclockwise
rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) about the origin and dilation with scale factor √𝑎𝑎2 + 𝑏𝑏 2 .

Problem Set

1. Find real numbers 𝑎𝑎 and 𝑏𝑏 so that the transformation 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) produces the specified rotation
and dilation.
1
a. Rotation by 270° counterclockwise and dilation by scale factor .
2
b. Rotation by 135° counterclockwise and dilation by scale factor √2.
c. Rotation by 45° clockwise and dilation by scale factor 10.
d. Rotation by 540° counterclockwise and dilation by scale factor 4.

2. Determine if the following transformations represent a rotation and dilation. If so, identify the scale factor and the
amount of rotation.

a. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (3𝑥𝑥 + 4𝑦𝑦, 4𝑥𝑥 + 3𝑦𝑦)

b. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (−5𝑥𝑥 + 12𝑦𝑦, −12𝑥𝑥 − 5𝑦𝑦)
c. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (3𝑥𝑥 + 3𝑦𝑦, −3𝑦𝑦 + 3𝑥𝑥)

3. Grace and Lily have a different point of view about the transformation on cube 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 that is shown above.
Grace states that it is a reflection about the imaginary axis and a dilation of factor of 2. However, Lily argues it
should be a 90° counterclockwise rotation about the origin with a dilation of a factor of 2.
a. Who is correct? Justify your answer.
b. Represent the above transformation in the form 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎).

Lesson 20: Exploiting the Connection to Cartesian Coordinates

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 20 M1
PRECALCULUS AND ADVANCED TOPICS

4. Grace and Lily still have a different point of view on this transformation on triangle 𝐴𝐴𝐴𝐴𝐴𝐴 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
2. However, Lily asserts that it is a 180° counterclockwise rotation about the origin with a dilation of a factor of 2.

a. Who is correct? Justify your answer.

b. Represent the above transformation in the form 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎).

5. Given 𝑧𝑧 = √3 + 𝑖𝑖
a. Find the complex number 𝑤𝑤 that will cause a rotation with the same number of degrees as 𝑧𝑧 without a
dilation.
b. Can you come up with a general formula 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎) for any complex number 𝑧𝑧 = 𝑥𝑥 + 𝑦𝑦𝑦𝑦 to
represent this condition?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 21: The Hunt for Better Notation

Classwork
Opening Exercise
Suppose that 𝐿𝐿1 (𝑥𝑥, 𝑦𝑦) = (2𝑥𝑥 − 3𝑦𝑦, 3𝑥𝑥 + 2𝑦𝑦) and 𝐿𝐿2 (𝑥𝑥, 𝑦𝑦) = (3𝑥𝑥 + 4𝑦𝑦, −4𝑦𝑦 + 3𝑥𝑥).
Find the result of performing 𝐿𝐿1 and then 𝐿𝐿2 on a point (𝑝𝑝, 𝑞𝑞). That is, find 𝐿𝐿2 �𝐿𝐿1 (𝑝𝑝, 𝑞𝑞)�.

Exercises
1. Calculate each of the following products.
3 −2 1
a. � �� �
−1 4 5

3 3 4
b. � �� �
3 3 −4

2 −4 3
c. � �� �
5 −1 −2

Lesson 21: The Hunt for Better Notation

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M1
PRECALCULUS AND ADVANCED TOPICS

1 2 3 1
2. Find a value of 𝑘𝑘 so that � � � � = � �.
𝑘𝑘 1 −1 11

𝑎𝑎 𝑏𝑏
3. Find a matrix � � so that we can represent the transformation 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (2𝑥𝑥 − 3𝑦𝑦, 3𝑥𝑥 + 2𝑦𝑦) by
𝑐𝑐 𝑑𝑑
𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥
𝐿𝐿 �𝑦𝑦� = � � � �.
𝑐𝑐 𝑑𝑑 𝑦𝑦

𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥
4. If a transformation 𝐿𝐿 �𝑦𝑦� = � � � � has the geometric effect of rotation and dilation, what do you know about
𝑐𝑐 𝑑𝑑 𝑦𝑦
the values 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, and 𝑑𝑑?

𝑎𝑎 𝑏𝑏 𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥
5. Describe the form of a matrix � � so that the transformation 𝐿𝐿 �𝑦𝑦� = � � � � has the geometric effect of
𝑐𝑐 𝑑𝑑 𝑐𝑐 𝑑𝑑 𝑦𝑦
only dilation by a scale factor 𝑟𝑟.

𝑎𝑎 𝑏𝑏 𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥
6. Describe the form of a matrix � � so that the transformation 𝐿𝐿 �𝑦𝑦� = � � � � has the geometric effect of
𝑐𝑐 𝑑𝑑 𝑐𝑐 𝑑𝑑 𝑦𝑦
only rotation by 𝜃𝜃. Describe the matrix in terms of 𝜃𝜃.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M1
PRECALCULUS AND ADVANCED TOPICS

𝑎𝑎 𝑏𝑏 𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥
7. Describe the form of a matrix � � so that the transformation 𝐿𝐿 �𝑦𝑦� = � � � � has the geometric effect of
𝑐𝑐 𝑑𝑑 𝑐𝑐 𝑑𝑑 𝑦𝑦
rotation by 𝜃𝜃 and dilation with scale factor 𝑟𝑟. Describe the matrix in terms of 𝜃𝜃 and 𝑟𝑟.

𝑥𝑥 1 2 𝑥𝑥
8. Suppose that we have a transformation 𝐿𝐿 �𝑦𝑦� = � � � �.
0 1 𝑦𝑦
a. Does this transformation have the geometric effect of rotation and dilation?

0 1 1 0
b. Transform each of the points 𝐴𝐴 = � �, 𝐵𝐵 = � �, 𝐶𝐶 = � �, and 𝐷𝐷 = � �, and plot the images in the plane
0 0 1 1
shown.

𝑥𝑥 1 0 𝑥𝑥
9. Describe the geometric effect of the transformation 𝐿𝐿 �𝑦𝑦� = � � � �.
0 1 𝑦𝑦

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
For real numbers 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, and 𝑑𝑑, the transformation 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏, 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑑𝑑) can be represented using matrix
𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑥𝑥 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 𝑥𝑥
multiplication by 𝐿𝐿 �𝑦𝑦� = � � � �, where � �� � = � � and the �𝑦𝑦� represents the point (𝑥𝑥, 𝑦𝑦) in
𝑐𝑐 𝑑𝑑 𝑦𝑦 𝑐𝑐 𝑑𝑑 𝑦𝑦 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑑𝑑
the plane.
 The transformation is a counterclockwise rotation by 𝜃𝜃 if and only if the matrix representation is
𝑥𝑥 cos(𝜃𝜃) −sin(𝜃𝜃) 𝑥𝑥
𝐿𝐿 �𝑦𝑦� = � � � �.
sin(𝜃𝜃) cos(𝜃𝜃) 𝑦𝑦
 The transformation is a dilation with scale factor 𝑘𝑘 if and only if the matrix representation is
𝑥𝑥 𝑘𝑘 0 𝑥𝑥
𝐿𝐿 �𝑦𝑦� = � � � �.
0 𝑘𝑘 𝑦𝑦
 The transformation is a counterclockwise rotation by arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏) and dilation with scale factor |𝑎𝑎 + 𝑏𝑏𝑏𝑏|
𝑥𝑥 𝑎𝑎 −𝑏𝑏 𝑥𝑥
if and only if the matrix representation is 𝐿𝐿 �𝑦𝑦� = � � �𝑦𝑦�. If we let 𝑟𝑟 = |𝑎𝑎 + 𝑏𝑏𝑏𝑏| and
𝑏𝑏 𝑎𝑎
𝑥𝑥 𝑟𝑟 cos(𝜃𝜃) −𝑟𝑟 sin(𝜃𝜃) 𝑥𝑥
𝜃𝜃 = arg(𝑎𝑎 + 𝑏𝑏𝑏𝑏), then the matrix representation is 𝐿𝐿 �𝑦𝑦� = � � � �.
𝑟𝑟 sin(𝜃𝜃) 𝑟𝑟 cos(𝜃𝜃) 𝑦𝑦

Problem Set

1. Perform the indicated multiplication.

1 2 3
a. � �� �
4 8 −2
3 5 2
b. � �� �
−2 −6 4
1 1 6
c. � �� �
1 −1 8
5 7 10
d. � �� �
4 9 100
4 2 −3
e. � �� �
3 7 1
6 4 2
f. � �� �
9 6 −3
cos(𝜃𝜃) −sin(𝜃𝜃) 𝑥𝑥
g. � �� �
sin(𝜃𝜃) cos(𝜃𝜃) 𝑦𝑦
𝜋𝜋 1 10
h. � �� �
1 −𝜋𝜋 7

𝑘𝑘 3 4 7
2. Find a value of 𝑘𝑘 so that � � � � = � �.
4 𝑘𝑘 5 6

𝑘𝑘 3 5 7
3. Find values of 𝑘𝑘 and 𝑚𝑚 so that � �� � = � �.
−2 𝑚𝑚 4 −10

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21 M1
PRECALCULUS AND ADVANCED TOPICS

1 2 𝑘𝑘 0
4. Find values of 𝑘𝑘 and 𝑚𝑚 so that � � � � = � �.
−2 5 𝑚𝑚 −9

5. Write the following transformations using matrix multiplication.

a. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (3𝑥𝑥 − 2𝑦𝑦, 4𝑥𝑥 − 5𝑦𝑦)
b. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (6𝑥𝑥 + 10𝑦𝑦, −2𝑥𝑥 + 𝑦𝑦)
c. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (25𝑥𝑥 + 10𝑦𝑦, 8𝑥𝑥 − 64𝑦𝑦)
d. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (𝜋𝜋𝜋𝜋 − 𝑦𝑦, −2𝑥𝑥 + 3𝑦𝑦)
e. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (10𝑥𝑥, 100𝑥𝑥)
f. 𝐿𝐿(𝑥𝑥, 𝑦𝑦) = (2𝑦𝑦, 7𝑥𝑥)

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.
𝑥𝑥 3 −2 𝑥𝑥
a. 𝐿𝐿 �𝑦𝑦� = � �� �
4 −5 𝑦𝑦
𝑥𝑥 42 0 𝑥𝑥
b. 𝐿𝐿 �𝑦𝑦� = � �� �
0 42 𝑦𝑦
𝑥𝑥 −4 −2 𝑥𝑥
c. 𝐿𝐿 �𝑦𝑦� = � �� �
2 −4 𝑦𝑦
𝑥𝑥 5 −1 𝑥𝑥
d. 𝐿𝐿 �𝑦𝑦� = � � �𝑦𝑦�
−1 5
𝑥𝑥 −7 1 𝑥𝑥
e. 𝐿𝐿 �𝑦𝑦� = � �� �
1 7 𝑦𝑦
𝑥𝑥 0 −2 𝑥𝑥
f. 𝐿𝐿 �𝑦𝑦� = � � �𝑦𝑦�
2 0

7. Create a matrix representation of a linear transformation that has the specified geometric effect.
a. Dilation by a factor of 4 and no rotation
b. Rotation by 180° and no dilation
𝜋𝜋
c. Rotation by − rad and dilation by a scale factor of 3
2
d. Rotation by 30° and dilation by a scale factor of 4

8. Identify the geometric effect of the following transformations. Justify your answers.
�2 �2
𝑥𝑥 − −
a. 𝐿𝐿 �𝑦𝑦� = � 2 2 � �𝑥𝑥 �
�2 �2 𝑦𝑦
−
2 2
𝑥𝑥 0 −5 𝑥𝑥
b. 𝐿𝐿 �𝑦𝑦� = � � �𝑦𝑦�
5 0
𝑥𝑥 −10 0 𝑥𝑥
c. 𝐿𝐿 �𝑦𝑦� = � � �𝑦𝑦�
0 −10
𝑥𝑥 6 6√3 𝑥𝑥
d. 𝐿𝐿 �𝑦𝑦� = � � �𝑦𝑦�
−6√3 6

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 22: Modeling Video Game Motion with Matrices

Classwork
Opening Exercise
𝑥𝑥 3 0 𝑥𝑥
Let 𝐷𝐷 �𝑦𝑦� = � � � �.
0 3 𝑦𝑦
2
a. Plot the point � �.
1
2
b. Find 𝐷𝐷 � � and plot it.
1
𝑥𝑥 𝑥𝑥
c. Describe the geometric effect of performing the transformation �𝑦𝑦� → 𝐷𝐷 �𝑦𝑦�.

Exercises
𝑡𝑡 0 2
1. Let 𝑓𝑓(𝑡𝑡) = � � � �, where 𝑡𝑡 represents time, measured in seconds. 𝑃𝑃 = 𝑓𝑓(𝑡𝑡) represents the position of a
0 𝑡𝑡 4
moving object at time 𝑡𝑡. If the object starts at the origin, how long would it take to reach (12, 24)?

𝑘𝑘𝑘𝑘 0 2
2. Let 𝑔𝑔(𝑡𝑡) = � � � �.
0 𝑘𝑘𝑘𝑘 4
a. Find the value of 𝑘𝑘 that moves an object from the origin to (12, 24) in just 2 seconds.

Lesson 22: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1
PRECALCULUS AND ADVANCED TOPICS

b. Find the value of 𝑘𝑘 that moves an object from the origin to (12, 24) in 30 seconds.

2 + 𝑡𝑡 0 5
3. Let 𝑓𝑓(𝑡𝑡) = � � � �, where 𝑡𝑡 represents time, measured in seconds, and 𝑓𝑓(𝑡𝑡) represents the position of
0 2 + 𝑡𝑡 7
a moving object at time 𝑡𝑡.
a. Find the position of the object at 𝑡𝑡 = 0, 𝑡𝑡 = 1, and 𝑡𝑡 = 2.

𝑥𝑥(𝑡𝑡)
b. Write 𝑓𝑓(𝑡𝑡) in the form � �.
𝑦𝑦(𝑡𝑡)

15 + 5𝑡𝑡
4. Write the transformation 𝑔𝑔(𝑡𝑡) = � � as a matrix transformation.
−6 − 2𝑡𝑡

Lesson 22: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1
PRECALCULUS AND ADVANCED TOPICS

5. An object is moving in a straight line from (18,12) to the origin over a 6-second period of time. Find a function 𝑓𝑓(𝑡𝑡)
𝑥𝑥(𝑡𝑡)
that gives the position of the object after 𝑡𝑡 seconds. Write your answer in the form 𝑓𝑓(𝑡𝑡) = � �, and then
𝑦𝑦(𝑡𝑡)
express 𝑓𝑓(𝑡𝑡) as a matrix transformation.

6. Write a rule for the function that shifts every point in the plane 6 units to the left.

7. Write a rule for the function that shifts every point in the plane 9 units upward.

8. Write a rule for the function that shifts every point in the plane 10 units down and 4 units to the right.

𝑥𝑥 𝑥𝑥 − 7
9. Consider the rule �𝑦𝑦� → � �. Describe the effect this transformation has on the plane.
𝑦𝑦 + 2

Lesson 22: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

𝑥𝑥 2 0 𝑥𝑥
1. Let 𝐷𝐷 �𝑦𝑦� = � � � �. Find and plot the following.
0 2 𝑦𝑦
−1 −1
a. Plot the point � �, and find 𝐷𝐷 � � and plot it.
2 2
3 3
b. Plot the point � �, and find 𝐷𝐷 � � and plot it.
4 4
5 5
c. Plot the point � �, and find 𝐷𝐷 � � and plot it.
2 2

𝑡𝑡 0 −1
2. Let 𝑓𝑓(𝑡𝑡) = � � � �. Find 𝑓𝑓(0), 𝑓𝑓(1), 𝑓𝑓(2), 𝑓𝑓(3), and plot them on the same graph.
0 𝑡𝑡 2

𝑡𝑡
0 3
3. Let 𝑓𝑓(𝑡𝑡) = � � � � represent the location of an object at time 𝑡𝑡 that is measured in seconds.
0𝑡𝑡 2
12
a. How long does it take the object to travel from the origin to the point � �?
8
b. Find the speed of the object in the horizontal direction and in the vertical direction.

0.2𝑡𝑡 0 3 2𝑡𝑡 0 3 12
4. Let 𝑓𝑓(𝑡𝑡) = � � � � , ℎ(𝑡𝑡) = � � � �. Which one will reach the point � � first? The time 𝑡𝑡 is
0 0.2𝑡𝑡 2 0 2𝑡𝑡 2 8
measured in seconds.

𝑘𝑘𝑘𝑘 0 3 −45
5. Let 𝑓𝑓(𝑡𝑡) = � � � �. Find the value of 𝑘𝑘 that moves the object from the origin to � � in 5 seconds.
0 𝑘𝑘𝑘𝑘 2 −30

𝑥𝑥(𝑡𝑡)
6. Write 𝑓𝑓(𝑡𝑡) in the form � �
𝑦𝑦(𝑡𝑡)
𝑡𝑡 0 2
a. 𝑓𝑓(𝑡𝑡) = � �� �
0 𝑡𝑡 5
2𝑡𝑡 + 1 0 3
b. 𝑓𝑓(𝑡𝑡) = � �� �
0 2𝑡𝑡 + 1 2
𝑡𝑡
−3 0 4
c. 𝑓𝑓(𝑡𝑡) = � 2 �� �
𝑡𝑡
0 − 3 −6
2

𝑡𝑡0 2
7. Let 𝑓𝑓(𝑡𝑡) = � � � � represent the location of an object after 𝑡𝑡 seconds.
0 𝑡𝑡 5
6 34
a. If the object starts at � �, how long would it take to reach � �?
15 85
b. Write the new function 𝑓𝑓(𝑡𝑡) that gives the position of the object after 𝑡𝑡 seconds.
c. Write 𝑓𝑓(𝑡𝑡) as a matrix transformation.

Lesson 22: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1
PRECALCULUS AND ADVANCED TOPICS

8. Write the following functions as a matrix transformation.

10 + 2𝑡𝑡
a. 𝑓𝑓(𝑡𝑡) = � �
15 + 3𝑡𝑡
−6𝑡𝑡 + 15
b. 𝑓𝑓(𝑡𝑡) = � �
8𝑡𝑡 − 20

𝑥𝑥
9. Write a function rule that represents the change in position of the point �𝑦𝑦� for the following.
a. 5 units to the right and 3 units downward
b. 2 units downward and 3 units to the left
c. 3 units upward, 5 units to the left, and then it dilates by 2.
𝜋𝜋
d. 3 units upward, 5 units to the left, and then it rotates by counterclockwise.
2

𝑥𝑥
10. Annie is designing a video game and wants her main character to be able to move from any given point �𝑦𝑦� in the
following ways: right 1 unit, jump up 1 unit, and both jump up and move right 1 unit each.
a. What function rules can she use to represent each time the character moves?
b. Annie is also developing a ski slope stage for her game and wants to model her character’s position using
−20
matrix transformations. Annie wants the player to start at � � and eventually pass through the origin
10
moving 5 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 𝑥𝑥 = 20, how long until the player moves off the screen traveling this path?

11. Remy thinks that he has developed matrix transformations to model the movements of Annie’s characters in
𝑥𝑥 1
Problem 10 from any given point �𝑦𝑦�, and he has tested them on the point � �. This is the work Remy did on the
1
transformations:
2 0 1 2 1 0 1 1 2 0 1 2
� �� � = � � � �� � = � � � � � � = � �.
0 1 1 1 1 1 1 2 1 1 1 2
Do these matrix transformations accomplish the movements that Annie wants to program into the game? Explain
why or why not.

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 (2,5):
2𝑡𝑡 + 6 𝑡𝑡 + 2
𝑓𝑓(𝑡𝑡) = � � and 𝑔𝑔(𝑡𝑡) = � �.
5𝑡𝑡 + 15 𝑡𝑡 + 5
a. If we simplify these functions algebraically, how does the rule for 𝑓𝑓 differ from the rule for 𝑔𝑔? What does this
say about which function can be expressed with matrix transformations?
b. Nolan has noticed that functions that can be expressed with matrix transformations always pass through the
origin; does either 𝑓𝑓 or 𝑔𝑔 pass through the origin, and does this support or contradict Nolan’s reasoning?
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.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 23: Modeling Video Game Motion with Matrices

Classwork
Opening Exercise
𝜋𝜋 𝜋𝜋
𝑥𝑥 cos � � − sin � � 𝑥𝑥
3 3
Let 𝑅𝑅 �𝑦𝑦� = � 𝜋𝜋 𝜋𝜋
� �𝑦𝑦�.
sin � � cos � �
3 3
𝑥𝑥 𝑥𝑥
a. Describe the geometric effect of performing the transformation �𝑦𝑦� → 𝑅𝑅 �𝑦𝑦�.

1 1
b. Plot the point � �, and then find 𝑅𝑅 � � and plot it.
0 0

1 1
c. If we want to show that 𝑅𝑅 has been applied twice to (1,0), we can write 𝑅𝑅2 � �, which represents 𝑅𝑅 �𝑅𝑅 � ��.
0 0
1 1 1
Find 𝑅𝑅2 � � and plot it. Then, find 𝑅𝑅3 � � = 𝑅𝑅 �𝑅𝑅 �𝑅𝑅 � ���, and plot it.
0 0 0

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1
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𝑥𝑥 𝑥𝑥
d. Describe the matrix transformation �𝑦𝑦� → 𝑅𝑅2 �𝑦𝑦� using a single matrix.

Exercises
𝑡𝑡 𝑡𝑡
cos(2𝑡𝑡) −sin(2𝑡𝑡) 1 cos � � − sin � � 1
2 2
1. Let 𝑓𝑓(𝑡𝑡) = � � � �, and let 𝑔𝑔(𝑡𝑡) = � 𝑡𝑡 𝑡𝑡 � �0�.
sin(2𝑡𝑡) cos(2𝑡𝑡) 0 sin � � cos � �
2 2
a. Suppose 𝑓𝑓(𝑡𝑡) represents the position of a moving object that starts at (1,0). How long does it take for this
object to return to its starting point? When the argument of the trigonometric function changes from 𝑡𝑡 to 2𝑡𝑡,
what effect does this have?

b. If the position is given instead by 𝑔𝑔(𝑡𝑡), how long would it take the object to return to its starting point? When
𝑡𝑡
the argument of the trigonometric functions changes from 𝑡𝑡 to , what effect does this have?
2

𝜋𝜋 𝜋𝜋
cos � ∙ 𝑡𝑡� − sin � ∙ 𝑡𝑡� 0
2 2
2. Let 𝐺𝐺(𝑡𝑡) = � 𝜋𝜋 𝜋𝜋 � � �.
sin � ∙ 𝑡𝑡� cos � ∙ 𝑡𝑡� 1
2 2
a. Draw the path that 𝑃𝑃 = 𝐺𝐺(𝑡𝑡) traces out as 𝑡𝑡 varies within the interval 0 ≤ 𝑡𝑡 ≤ 1.

b. Where will the object be at 𝑡𝑡 = 3 seconds?

Lesson 23: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1
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c. How long will it take the object to reach (0, −1)?

𝜋𝜋 𝜋𝜋
cos � ∙ 𝑡𝑡� − sin � ∙ 𝑡𝑡� 1
2 2
3. Let 𝐻𝐻(𝑡𝑡) = � 𝜋𝜋 𝜋𝜋 � � �.
sin � ∙ 𝑡𝑡� cos � ∙ 𝑡𝑡� 4
2 2
a. Draw the path that 𝑃𝑃 = 𝐻𝐻(𝑡𝑡) traces out as 𝑡𝑡 varies within the interval 0 ≤ 𝑡𝑡 ≤ 2.

b. Where will the object be at 𝑡𝑡 = 1 second?

c. How long will it take the object to return to its starting point?

4. Suppose you want to write a program that takes the point (3, 5) and rotates it about the origin to the point
(−3, −5) over a 1-second interval. Write a function 𝑃𝑃 = 𝑓𝑓(𝑡𝑡) that encodes this rotation.

5. If instead you wanted the rotation to take place over a 1.5-second interval, how would your function change?

Lesson 23: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1
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Problem Set

𝜋𝜋 𝜋𝜋
𝑥𝑥 cos � � − sin � � 𝑥𝑥
4 4
1. Let 𝑅𝑅 �𝑦𝑦� = � 𝜋𝜋 𝜋𝜋 � �𝑦𝑦�. Find the following.
sin � � cos � �
4 4

a. 𝑅𝑅2 �√2�
√2
b. How many transformations do you need to take so that the image returns to where it started?
𝑥𝑥 𝑥𝑥 𝑥𝑥
c. Describe the matrix transformation �𝑦𝑦� → 𝑅𝑅2 �𝑦𝑦� and 𝑅𝑅𝑛𝑛 �𝑦𝑦� using a single matrix.

cos(𝑡𝑡) −sin(𝑡𝑡) 1 1
2. For 𝑓𝑓(𝑡𝑡) = � � � �, it takes 2𝜋𝜋 to transform the object at � � back to where it starts. How long
sin(𝑡𝑡) cos(𝑡𝑡) 1 1
does it take the following functions to return to their starting point?
cos(3𝑡𝑡) −sin(3𝑡𝑡) 1
a. 𝑓𝑓(𝑡𝑡) = � �� �
sin(3𝑡𝑡) cos(3𝑡𝑡) 1
𝑡𝑡 𝑡𝑡
cos � � − sin � � 1
3 3
b. 𝑓𝑓(𝑡𝑡) = � 𝑡𝑡 𝑡𝑡 � �1�
sin � � cos � �
3 3
2𝑡𝑡 2𝑡𝑡
cos � � − sin � � 1
5 5
c. 𝑓𝑓(𝑡𝑡) = � 2𝑡𝑡 2𝑡𝑡
�� �
sin � � cos � � 1
5 5

cos(𝑡𝑡) −sin(𝑡𝑡) 2
3. Let 𝐹𝐹(𝑡𝑡) = � � � �, where 𝑡𝑡 is measured in radians. Find the following:
sin(𝑡𝑡) cos(𝑡𝑡) 1
3𝜋𝜋 7𝜋𝜋
a. 𝐹𝐹 � �, 𝐹𝐹 � �, and the radius of the path
2 6
cos(𝑡𝑡) −sin(𝑡𝑡) 𝑥𝑥
b. Show that the radius is always �𝑥𝑥 2 + 𝑦𝑦 2 for the path of this transformation 𝐹𝐹(𝑡𝑡) = � � � �.
sin(𝑡𝑡) cos(𝑡𝑡) 𝑦𝑦

𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋
cos � � − sin � � 4
2 2
4. Let 𝐹𝐹(𝑡𝑡) = � 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 � �4�, where 𝑡𝑡 is a real number.
sin � � cos � �
2 2
a. Draw the path that 𝑃𝑃 = 𝐹𝐹(𝑡𝑡) traces out as 𝑡𝑡 varies within each of the following intervals:
i. 0 ≤ 𝑡𝑡 ≤ 1
ii. 1 ≤ 𝑡𝑡 ≤ 2
iii. 2 ≤ 𝑡𝑡 ≤ 3
iv. 3 ≤ 𝑡𝑡 ≤ 4
b. Where will the object be located at 𝑡𝑡 = 2.5 seconds?
−8√6
c. How long does it take the object to reach � �?
8√2

Lesson 23: Modeling Video Game Motion with Matrices

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𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋
cos � � − sin � � −1
3 3
5. Let 𝐹𝐹(𝑡𝑡) = � 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 � �−√3�.
sin � � cos � �
3 3
a. Draw the path that 𝑃𝑃 = 𝐹𝐹(𝑡𝑡) traces out as 𝑡𝑡 varies within the interval 0 ≤ 𝑡𝑡 ≤ 1.
b. How long does it take the object to reach �√3, 0�?
c. How long does it take the object to return to its starting point?

6. Find the function that will rotate the point (4, 2) about the origin to the point (−4, −2) over the following time
intervals.
a. Over a 1-second interval
b. Over a 2-second interval
1
c. Over a -second interval
3
4
d. How about rotating it back to where it starts over a -second interval?
5

7. Summarize the geometric effect of the following function at the given point and the time interval.
𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋
5 cos � � −5 sin � � 4
4 4
a. 𝐹𝐹(𝑡𝑡) = � 𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋 � �3�, 0 ≤ 𝑡𝑡 ≤ 1
5 sin � � 5 cos � �
4 4
1 𝜋𝜋𝜋𝜋 1 𝜋𝜋𝜋𝜋
cos � � − sin � � 6
2 6 2 6
b. 𝐹𝐹(𝑡𝑡) = � 1 𝜋𝜋𝜋𝜋 1 𝜋𝜋𝜋𝜋
� � �, 0 ≤ 𝑡𝑡 ≤ 1
sin � � cos � � 2
2 6 2 6

8. In programming a computer video game, Grace coded the changing location of a rocket as follows:
𝑥𝑥
At the time 𝑡𝑡 second between 𝑡𝑡 = 0 seconds and 𝑡𝑡 = 4 seconds, the location �𝑦𝑦� of the rocket is given by
𝜋𝜋 𝜋𝜋
cos � 𝑡𝑡� − sin � 𝑡𝑡�
� 4
𝜋𝜋 𝜋𝜋
4
� �√2�.
sin � 𝑡𝑡� cos � 𝑡𝑡� √2
4 4
At a time of 𝑡𝑡 seconds between 𝑡𝑡 = 4 and 𝑡𝑡 = 8 seconds, the location of the rocket is given by
�2
−√2 + (𝑡𝑡 − 4)
� 2 �.
�2
−√2 + (𝑡𝑡 − 4)
2
a. What is the location of the rocket at time 𝑡𝑡 = 0? What is its location at time 𝑡𝑡 = 8?
b. Mason is worried that Grace may have made a mistake and the location of the rocket is unclear at time 𝑡𝑡 = 4
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 𝑡𝑡 = 0 to 𝑡𝑡 = 8?

Lesson 23: Modeling Video Game Motion with Matrices

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 24: Matrix Notation Encompasses New Transformations!

Classwork
Example 1
Determine the following:
1 0 3
a. � �� �
0 1 −2

1 0 −7
b. � �� �
0 1 12

1 0 3 5
c. � �� �
0 1 −2 1

1 0 −1 −3
d. � �� �
0 1 −7 6

9 12 1 0
e. � �� �
−3 −1 0 1

1 0 𝑎𝑎 𝑏𝑏
f. � �� �
0 1 𝑐𝑐 𝑑𝑑

𝑥𝑥 𝑦𝑦 1 0
g. � �� �
𝑧𝑧 𝑤𝑤 0 1

Lesson 24: Matrix Notation Encompasses New Transformations!

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1
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Example 2
Can the reflection about the real axis 𝐿𝐿(𝑧𝑧) = 𝑧𝑧̅ be expressed in matrix notation?

Exercises
1. Express a reflection about the vertical axis in matrix notation. Prove that it produces the desired reflection by using
matrix multiplication.

2. Express a reflection about both the horizontal and vertical axes in matrix notation. Prove that it produces the
desired reflection by using matrix multiplication.

3. Express a reflection about the vertical axis and a dilation with a scale factor of 6 in matrix notation. Prove that it
produces the desired reflection by using matrix multiplication.

Lesson 24: Matrix Notation Encompasses New Transformations!

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1
PRECALCULUS AND ADVANCED TOPICS

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

1 0
4. � �
0 1

0 1
5. � �
1 0

0 0
6. � �
0 0

1 1
7. � �
0 1

1 0
8. � �
1 1

Lesson 24: Matrix Notation Encompasses New Transformations!

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
𝑎𝑎11 𝑎𝑎12
All matrices in the form �𝑎𝑎 𝑎𝑎22 � correspond to a transformation of some kind.
21

1 0
 The matrix � � reflects all coordinates about the horizontal axis.
0 −1
−1 0
 The matrix � � reflects all coordinates about the vertical axis.
0 1
1 0
 The matrix � � is the identity matrix and corresponds to a transformation that leaves points alone.
0 1
0 0
 The matrix � � is the zero matrix and corresponds to a dilation of scale factor 0.
0 0

Problem Set

1. What matrix do you need to use to reflect the following points about the 𝑦𝑦-axis? What is the resulting matrix when
this is done? Show all work and sketch it.
3 3
a. � � b. � �
0 2
−4 −3
c. � � d. � �
3 −2
3 5
e. � � f. � �
−3 4

2. What matrix do you need to use to reflect the following points about the 𝑥𝑥-axis? What is the resulting matrix when
this is done? Show all work and sketch it.
0 2
a. � � b. � �
3 3
−2 −3
c. � � d. � �
3 −3
3 −3
e. � � f. � �
−3 4

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.
1 3
a. � �, a factor of 3 b. � �, a factor of 2
0 2
1 −4 1
c. � �, a factor of 1 d. � �, a factor of
−2 −6 2
9 1
e. � �, a factor of f. � √3 �, a factor of √2
3 3 √11

Lesson 24: Matrix Notation Encompasses New Transformations!

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1
PRECALCULUS AND ADVANCED TOPICS

4. What matrix will rotate the given point by the angle? What is the resulting matrix when this is done? Show all work
and sketch it.
1 𝜋𝜋 1 𝜋𝜋
a. � �, radians b. � �, radians
0 2 0 3
1 𝜋𝜋 1 𝜋𝜋
c. � �, radians d. � �, radians
0 6 0 4
�3 �2
𝜋𝜋 𝜋𝜋
e. � 2 �, radians f. � 2 �, radians
1 6 �2 4
2 2
�3
1 𝜋𝜋
g. � 2 �, 𝜋𝜋 radians h. � �, − radians
1 0 6
2

5. For the transformation shown below, find the matrix that will transform point 𝐴𝐴 to 𝐴𝐴’, and verify your answer.

0 1
6. In this lesson, we learned � � will produce a reflection about the line 𝑦𝑦 = 𝑥𝑥. What matrix will produce a
1 0
3
reflection about the line 𝑦𝑦 = −𝑥𝑥? Verify your answers by testing the given point � � and graphing them on the
1
coordinate plane.

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?

Lesson 24: Matrix Notation Encompasses New Transformations!

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1
PRECALCULUS AND ADVANCED TOPICS

8. Given the kite figure 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 below, answer the following questions.

a. Explain how you would create the star figure above using only rotations.
b. Explain how to create the star figure above using reflections and rotation.
c. Explain how to create the star figure above using only reflections. Explain your answer.

9. Given the rectangle 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 below, answer the following questions.

a. Can you transform the rectangle 𝐴𝐴′𝐵𝐵′𝐶𝐶′𝐷𝐷′ above using only rotations? Explain your answer.
b. Describe a way to create the rectangle 𝐴𝐴′𝐵𝐵′𝐶𝐶′𝐷𝐷′.
c. Can you make the rectangle 𝐴𝐴′𝐵𝐵′𝐶𝐶′𝐷𝐷′ above using only reflections? Explain your answer.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 25: Matrix Multiplication and Addition

Classwork
Opening Exercise
4
Consider the point � � that undergoes a series of two transformations: a dilation of scale factor 4 followed by a
1
reflection about the horizontal axis.
a. What matrix produces the dilation of scale factor 4? What is the coordinate of the point after the dilation?

b. What matrix produces the reflection about the horizontal axis? What is the coordinate of the point after the
reflection?

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 4 and produce a reflection about the horizontal axis? Show that the matrix you
came up with combines these two matrices.

Example 1: Is Matrix Multiplication Commutative?

2 𝜋𝜋
a. Take the point � � through the following transformations: a rotation of and a reflection across the 𝑦𝑦-axis.
1 2

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
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b. Will the resulting point be the same if the order of the transformations is reversed?

c. Are transformations commutative?

0 −1 −1 0
d. Let 𝐴𝐴 = � � and 𝐵𝐵 = � �. Find 𝐴𝐴𝐴𝐴 and then 𝐵𝐵𝐵𝐵.
1 0 0 1

e. Is matrix multiplication commutative?

2
f. If we apply matrix 𝐴𝐴𝐴𝐴 to the point � �, in what order are the transformations applied?
1

2
g. If we apply matrix 𝐵𝐵𝐵𝐵 to the point � �, in what order are the transformations applied?
1

2
h. Can we apply � � to matrix 𝐵𝐵𝐵𝐵?
1

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises 1–3
1 0 4 −6
1. Let 𝐼𝐼 = � � and 𝑀𝑀 = � �.
0 1 3 −2
a. Find 𝐼𝐼𝐼𝐼.

b. Find 𝑀𝑀𝑀𝑀.

1 0
c. Do these results make sense based on what you know about the matrix � �?
0 1

2. Calculate 𝐴𝐴𝐴𝐴 and then 𝐵𝐵𝐵𝐵. Is matrix multiplication commutative?

2 0 1 5
a. 𝐴𝐴 = � � , 𝐵𝐵 = � �
−2 3 0 1

−10 1 −3 2
b. 𝐴𝐴 = � � , 𝐵𝐵 = � �
3 7 4 −1

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
PRECALCULUS AND ADVANCED TOPICS

3. Write a matrix that would perform the following transformations in this order: a rotation of 180°, a dilation by a
2
scale factor of 4, and a reflection across the horizontal axis. Use the point � � to illustrate that your matrix is
1
correct.

Example 2: More Operations on Matrices

2 0 1 5
Find the sum. � �+� �
−2 3 0 1

2 0 1 5
Find the difference. � �−� �
−2 3 0 1

2 0 0 0
Find the sum. � �+� �
−2 3 0 0

Exercises 4–5
4. Express each of the following as a single matrix.
6 −3 −2 8
a. � �+� �
10 −1 3 −12

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
PRECALCULUS AND ADVANCED TOPICS

−2 7 3 1
b. � �� � + � �
−3 1 4 5

8 5 4 −6
c. � �−� �
0 15 −3 18

5. In arithmetic, the additive identity says that for some number 𝑎𝑎, 𝑎𝑎 + 0 = 0 + 𝑎𝑎 = 0. What would be an additive
identity in matrix arithmetic?

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
𝑎𝑎 𝑐𝑐 𝑝𝑝 𝑟𝑟 𝑥𝑥
 If 𝐿𝐿 is given by � � and 𝑀𝑀 is given by �𝑞𝑞 𝑠𝑠 �, then 𝑀𝑀𝑀𝑀 �𝑦𝑦� is the same as applying the matrix
𝑏𝑏 𝑑𝑑
𝑝𝑝𝑝𝑝 + 𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝 + 𝑟𝑟𝑟𝑟 𝑥𝑥
� � to �𝑦𝑦�.
𝑞𝑞𝑞𝑞 + 𝑠𝑠𝑠𝑠 𝑞𝑞𝑞𝑞 + 𝑠𝑠𝑠𝑠
𝑎𝑎 𝑐𝑐 1 0
 If 𝐿𝐿 is given by � � and 𝐼𝐼 is given by � �, then 𝐼𝐼 acts as a multiplicative identity, and 𝐼𝐼𝐼𝐼 = 𝐿𝐿𝐿𝐿 = 𝐿𝐿.
𝑏𝑏 𝑑𝑑 0 1
𝑎𝑎 𝑐𝑐 0 0
 If 𝐿𝐿 is given by � � and 𝑂𝑂 is given by � �, then 𝑂𝑂 acts as an additive identity, and
𝑏𝑏 𝑑𝑑 0 0
𝑂𝑂 + 𝐿𝐿 = 𝐿𝐿 + 𝑂𝑂 = 𝐿𝐿.

Problem Set

1. What type of transformation is shown in the following examples? What is the resulting matrix?
cos (𝜋𝜋) −sin (𝜋𝜋) 3
a. � �� �
sin (𝜋𝜋) cos (𝜋𝜋) 2
0 −1 3
b. � �� �
1 0 2
3 0 3
c. � �� �
0 3 2
−1 0 3
d. � �� �
0 1 2
1 0 3
e. � �� �
0 −1 2
cos (2𝜋𝜋) −sin (2𝜋𝜋) 3
f. � �� �
sin (2𝜋𝜋) cos (2𝜋𝜋) 2
�2 −�2
g. �2 2 � � 3�
�2 �2 2
2 2
�3
− 12 3
h. � 2 �� �
1 �3 2
2 2

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1
PRECALCULUS AND ADVANCED TOPICS

2. Calculate each of the following products.

2 3 2
a. � �� �
5 4 3
3 2 3 2
b. � �� �
4 5 1 0
−1 −3 −3
c. � �� �
−2 −4 −1
−3 −1 −2 −1
d. � �� �
−4 −2 0 −3
5 0 0
e. � �� �
−1 2 3
3 −1 1 0
f. � �� �
4 −2 −2 −3

3. Calculate each sum or difference.

1 3 4 2
a. � �+� �
2 4 3 1
−4 −5 −2 3
b. � �+� �
−6 −7 −1 4
−5 3
c. � � + � �
4 2
3 7
d. � � − � �
−5 9
−4 −5 −2 3
e. � �−� �
−6 −7 −1 4

1
4. In video game programming, Fahad translates a car, whose coordinate is � �, 2 units up and 4 units to the right,
1
𝜋𝜋 𝜋𝜋
rotates it radians counterclockwise, reflects it about the 𝑥𝑥-axis, reflects it about the 𝑦𝑦-axis, rotates it radians
2 2
counterclockwise, and finally translates it 4 units down and 2 units to the left. What point represents the final
location of the car?

Lesson 25: Matrix Multiplication and Addition

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 26: Getting a Handle on New Transformations

Classwork
Opening Exercise
Perform the following matrix operations:
3 −2 −1
a. � �� �
1 5 3

3 −2 1 0
b. � �� �
1 5 0 1

3 −2 1 −3
c. � �� �
1 5 2 4

3 −2 𝑎𝑎 𝑏𝑏
d. � �� �
1 5 𝑐𝑐 𝑑𝑑

3 −2 1 0
e. � �+� �
1 5 0 1

3 −2 1 −3
f. � �+� �
1 5 2 4

3 −2 𝑎𝑎 𝑏𝑏
g. � �+� �
1 5 𝑐𝑐 𝑑𝑑

h. Can you add the two matrices in part (a)? Why or why not?

Lesson 26: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M1
PRECALCULUS AND ADVANCED TOPICS

Exercises
109 3
1. Perform the transformation � � on the unit square.
1 −2
a. Sketch the image. What is the shape of the image?

b. What are the coordinates of the vertices of the image?

c. What is the area of the image? Show your work.

𝑎𝑎 −𝑏𝑏
2. In the Exploratory Challenge, we drew the image of a general rotation/dilation of the unit square � �.
𝑏𝑏 𝑎𝑎
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.

b. Confirm the area using the determinant of the resulting matrix.

Lesson 26: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M1
PRECALCULUS AND ADVANCED TOPICS

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.

b. How does a dilation with a scale factor of 𝑘𝑘 change the area of the unit square? Calculate the determinant of a
matrix representing a pure dilation of 𝑘𝑘.

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.

Lesson 26: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson Summary
Definition
 The area of the image of the unit square under the linear transformation represented by a 2 × 2 matrix is
called the determinant of that matrix.

Problem Set

1. Perform the following transformation on the unit square: Sketch and state the area of the image.
3 −1 1 −3
a. � � b. � �
1 3 3 1
4 −2 2 −4
c. � � d. � �
2 4 4 2

2. Perform the following transformation on the unit square: Sketch the image, find the determinant of the given
matrix, and find the area the image.
1 3 1 2
a. � � b. � �
2 4 3 4
3 1 4 2
c. � � d. � �
2 4 3 1
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?

3. Perform the following transformation on the unit square: Sketch the image, find the determinant of the given
matrix, and find the area the image.
−1 −3 −1 −3
a. � � b. � �
−2 −4 2 4
1 3 −1 3
c. � � d. � �
−2 −4 −2 4
1 −3 −1 3
e. � � f. � �
2 −4 2 −4
1 −3
g. � �
−2 4

Lesson 26: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 27 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 27: Getting a Handle on New Transformations

Classwork
Opening Exercise
Explain the geometric effect of each matrix.
𝑎𝑎 −𝑏𝑏
a. � �
𝑏𝑏 𝑎𝑎

cos(𝜃𝜃) −sin(𝜃𝜃)
b. � �
sin(𝜃𝜃) cos(𝜃𝜃)

𝑘𝑘 0
c. � �
0 𝑘𝑘

1 0
d. � �
0 1

0 0
e. � �
0 0

𝑎𝑎 𝑐𝑐
f. � �
𝑏𝑏 𝑑𝑑

Lesson 27: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 27 M1
PRECALCULUS AND ADVANCED TOPICS

Example 1
𝑘𝑘 0
Given the transformation � � with 𝑘𝑘 > 0:
𝑘𝑘 1
a. Perform this transformation on the vertices of the unit square. Sketch the image, and label the vertices.

b. Calculate the area of the image using the dimensions of the image parallelogram.

c. Confirm the area of the image using the determinant.

𝑥𝑥
d. Perform the transformation on �𝑦𝑦�.

e. In order for two matrices to be equivalent, each of the corresponding elements must be equivalent. Given
5 𝑥𝑥
that, if the image of this transformation is � �, find �𝑦𝑦�.
4

Lesson 27: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 27 M1
PRECALCULUS AND ADVANCED TOPICS

1
f. Perform the transformation on � �. Write the image matrix.
1

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.

Exercise
𝑘𝑘 0
Perform the transformation � � with 𝑘𝑘 > 1 on the vertices of the unit square.
1 𝑘𝑘
a. What are the vertices of the image?

b. Calculate the area of the image.

𝑥𝑥 −2 𝑥𝑥
c. If the image of the transformation on �𝑦𝑦� is � �, find �𝑦𝑦� in terms of 𝑘𝑘.
−1

Example 2
2 5
Consider the matrix 𝐿𝐿 = � �. For each real number 0 ≤ 𝑡𝑡 ≤ 1, consider the point (3 + 𝑡𝑡, 10 + 2𝑡𝑡).
−1 3
a. Find point 𝐴𝐴 when 𝑡𝑡 = 0.

Lesson 27: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 27 M1
PRECALCULUS AND ADVANCED TOPICS

b. Find point 𝐵𝐵 when 𝑡𝑡 = 1.

1
c. Show that for 𝑡𝑡 = , (3 + 𝑡𝑡, 10 + 2𝑡𝑡) is the midpoint of ����
𝐴𝐴𝐴𝐴 .
2

d. Show that for each value of 𝑡𝑡, (3 + 𝑡𝑡, 10 + 2𝑡𝑡) is a point on the line through 𝐴𝐴 and 𝐵𝐵.

e. Find 𝐿𝐿𝐿𝐿 and 𝐿𝐿𝐿𝐿.

f. What is the equation of the line through 𝐿𝐿𝐿𝐿 and 𝐿𝐿𝐿𝐿?

3 + 𝑡𝑡
g. Show that 𝐿𝐿 � � lies on the line through 𝐿𝐿𝐿𝐿 and 𝐿𝐿𝐿𝐿.
10 + 2𝑡𝑡

Lesson 27: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 27 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

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.
1 0
a. � �
1 1
2 0
b. � �
2 1
3 0
c. � �
3 1
2 0
d. � �
2 2
3 0
e. � �
3 3
1 0
f. � �
1 2
1 0
g. � �
1 3
2 0
h. � �
2 3
3 0
i. � �
3 5

𝑘𝑘 0 𝑥𝑥 𝑘𝑘𝑘𝑘 𝑥𝑥
2. Given � � �𝑦𝑦� = � �. Find �𝑦𝑦� if the image of the transformation is the following:
𝑘𝑘 1 𝑘𝑘𝑘𝑘 + 𝑦𝑦
4
a. � �
5
−3
b. � �
2
5
c. � �
−6

𝑘𝑘 0 𝑥𝑥 𝑘𝑘𝑘𝑘
3. Given � �� � = � �. Find the value of 𝑘𝑘 so that:
𝑘𝑘 1 𝑦𝑦 𝑘𝑘𝑘𝑘 + 𝑦𝑦
𝑥𝑥 3 24
a. �𝑦𝑦� = � � and the image is � �.
−2 22
𝑥𝑥 27 18
b. �𝑦𝑦� = � � and the image is � �.
3 21

𝑘𝑘 0 𝑥𝑥 𝑘𝑘𝑘𝑘
4. Given � �� � = � �. Find the value of 𝑘𝑘 so that:
1 𝑘𝑘 𝑦𝑦 𝑥𝑥 + 𝑘𝑘𝑘𝑘
𝑥𝑥 −4 −12
a. �𝑦𝑦� = � � and the image is � �.
5 11
5
𝑥𝑥 −15
b. �𝑦𝑦� = �3 � and the image is � 1 �.
2 −
3
9

Lesson 27: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 27 M1
PRECALCULUS AND ADVANCED TOPICS

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.
2 0
a. � �
1 2
3 0
b. � �
1 3
2 0
c. � �
3 2
2 0
d. � �
4 2
4 0
e. � �
2 4
3 0
f. � �
5 3

1 3
6. Consider the matrix 𝐿𝐿 = � �. For each real number 0 ≤ 𝑡𝑡 ≤ 1, consider the point (3 + 2𝑡𝑡, 12 + 2𝑡𝑡).
2 4
a. Find the point 𝐴𝐴 when 𝑡𝑡 = 0.
b. Find the point 𝐵𝐵 when 𝑡𝑡 = 1.
1
c. Show that for 𝑡𝑡 = , (3 + 2𝑡𝑡, 12 + 2𝑡𝑡) is the midpoint of ����
𝐴𝐴𝐴𝐴.
2
d. Show that for each value of 𝑡𝑡, (3 + 2𝑡𝑡, 12 + 2𝑡𝑡) is a point on the line through 𝐴𝐴 and 𝐵𝐵.
e. Find 𝐿𝐿𝐿𝐿 and 𝐿𝐿𝐿𝐿.
f. What is the equation of the line through 𝐿𝐿𝐿𝐿 and 𝐿𝐿𝐿𝐿?
3 + 2𝑡𝑡
g. Show that 𝐿𝐿 � � lies on the line through 𝐿𝐿𝐿𝐿 and 𝐿𝐿𝐿𝐿.
12 + 2𝑡𝑡

Lesson 27: Getting a Handle on New Transformations

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 28 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 28: When Can We Reverse a Transformation?

Classwork
Opening Exercise
3 −2
Perform the operation � � on the unit square.
1 1
a. State the vertices of the transformation.

b. Explain the transformation in words.

c. Find the area of the transformed figure.

d. If the original square was 2 × 2 instead of a unit square, how would the transformation change?

e. What is the area of the image? Explain how you know.

Example
What transformation reverses a pure dilation from the origin with a scale factor of 𝑘𝑘?
𝑎𝑎 𝑐𝑐
a. Write the pure dilation matrix, and multiply it by � �.
𝑏𝑏 𝑑𝑑

Lesson 28: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 28 M1
PRECALCULUS AND ADVANCED TOPICS

b. What values of 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, and 𝑑𝑑 would produce the identity matrix? (Hint: Write and solve a system of
equations.)

c. Write the matrix, and confirm that it reverses the pure dilation with a scale factor of 𝑘𝑘.

Exercises
Find the inverse matrix and verify.
1 0
1. � �
1 1

3 1
2. � �
5 2

−2 −5
3. � �
1 2

Lesson 28: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 28 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1 0
1. In this lesson, we learned 𝑅𝑅𝜃𝜃 𝑅𝑅−𝜃𝜃 = � �. Chad was saying that he found an easy way to find the inverse matrix,
0 1
�1 0�
1
which is 𝑅𝑅−𝜃𝜃 = 0 1 . His argument is that if we have 2𝑥𝑥 = 1, then 𝑥𝑥 = .
𝑅𝑅𝜃𝜃 2
a. Is Chad correct? Explain your reason.
b. If Chad is not correct, what is the correct way to find the inverse matrix?

2. Find the inverse matrix and verify it.

3 2
a. � �
7 5
−2 −1
b. � �
3 1
3 −3
c. � �
−2 2
0 1
d. � �
−1 3
4 1
e. � �
2 1

𝑥𝑥
3. Find the starting point �𝑦𝑦� if:
4
a. The point � � is the image of a pure dilation with a factor of 2.
2
4 1
b. The point � � is the image of a pure dilation with a factor of .
2 2
−10
c. The point � � is the image of a pure dilation with a factor of 5.
35
4
d. 9 � is the image of a pure dilation with a factor of 2.
The point � 16
3
21

4. Find the starting point if:

a. 3 + 2𝑖𝑖 is the image of a reflection about the real axis.
b. 3 + 2𝑖𝑖 is the image of a reflection about the imaginary axis.
c. 3 + 2𝑖𝑖 is the image of a reflection about the real axis and then the imaginary axis.
d. −3 − 2𝑖𝑖 is the image of a 𝜋𝜋 radians counterclockwise rotation.

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 28 M1
PRECALCULUS AND ADVANCED TOPICS

cos (𝜃𝜃) −sin (𝜃𝜃)

5. Let’s call the pure counterclockwise rotation of the matrix � � as 𝑅𝑅𝜃𝜃 , and the “undo” of the pure
sin (𝜃𝜃) cos (𝜃𝜃)
cos (−𝜃𝜃) −sin (−𝜃𝜃)
rotation is � � as 𝑅𝑅−𝜃𝜃 .
sin (−𝜃𝜃) cos (−𝜃𝜃)
cos (−𝜃𝜃) −sin (−𝜃𝜃)
a. Simplify � �.
sin (−𝜃𝜃) cos (−𝜃𝜃)
b. What would you get if you multiply 𝑅𝑅𝜃𝜃 to 𝑅𝑅−𝜃𝜃 ?
𝜋𝜋
c. Write the matrix if you want to rotate radians counterclockwise.
2
𝜋𝜋
d. Write the matrix if you want to rotate radians clockwise.
2
𝜋𝜋
e. Write the matrix if you want to rotate radians counterclockwise.
6
𝜋𝜋
f. Write the matrix if you want to rotate radians counterclockwise.
4
0 𝜋𝜋 𝑥𝑥
g. If the point � � is the image of radians counterclockwise rotation, find the starting point �𝑦𝑦�.
2 4
1
2 𝜋𝜋 𝑥𝑥
h. If the point � � � is the image of radians counterclockwise rotation, find the starting point �𝑦𝑦�.
3 6
2

Lesson 28: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 29 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 29: When Can We Reverse a Transformation?

Classwork
Opening Exercise
−7 −2
Find the inverse of � �. Show your work. Confirm that the matrices are inverses.
4 1

Exercises
5 3
1. Find the inverse of � �. Confirm your answer.
2 4

Find the inverse matrix and verify.

3 −3
2. � �
1 4

5 −2
3. � �
4 −3

Lesson 29: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 29 M1
PRECALCULUS AND ADVANCED TOPICS

𝑎𝑎 𝑐𝑐
4. � �
𝑏𝑏 𝑑𝑑

Example
1 2
Find the determinant of � �.
2 4

Lesson 29: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 29 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

Find the inverse matrix of the following.

1 0
a. � �
0 1
0 1
b. � �
1 0
1 1
c. � �
1 1
1 0
d. � �
1 0
0 1
e. � �
0 1
−2 2
f. � �
−5 4
4 6
g. � �
5 8
6 −9
h. � �
5 −7
1 2
i. � 2 − 3�
−6 4
0.8 0.4
j. � �
−0.75 −0.5

Lesson 29: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

Lesson 30: When Can We Reverse a Transformation?

Classwork
Opening Exercise
a. What is the geometric effect of the following matrices?
𝑘𝑘 0
i. � �
0 𝑘𝑘

𝑎𝑎 −𝑏𝑏
ii. � �
𝑏𝑏 𝑎𝑎

cos(𝜃𝜃) − sin(𝜃𝜃)
iii. � �
sin(𝜃𝜃) cos(𝜃𝜃)

0 0 1 0
b. Jadavis says that the identity matrix is � �. Sophie disagrees and states that the identity matrix is � �.
0 0 0 1
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?

ii. Mr. Kuzy then asks each of them to explain the geometric effect that their matrix would have on the unit
square.

Lesson 30: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

c. Given the matrices below, answer the following:

2 3 5 2
𝐴𝐴 = � � 𝐵𝐵 = � �
1 4 10 4
i. Which matrix does not have an inverse? Explain algebraically and geometrically how you know.

ii. If a matrix has an inverse, find it.

Example

1 �3
−
Given � 2 2�
�3 1
2 2
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.

c. Find the area of the image.

d. Find the inverse of this transformation.

Lesson 30: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

e. Explain the meaning of the inverse transformation on the unit square.

Exercises
1
0
1. Given � 2 1
�
0
2
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.

c. Find the area of the image.

d. Find the inverse of this transformation.

e. Explain the meaning of the inverse transformation on the unit square.

f. If any matrix produces a dilation with a scale factor of 𝑘𝑘, what would the inverse matrix produce?

1 1
−
�2 �2
2. Given � 1 1
�
�2 �2

a. Perform this transformation on the unit square, and sketch the results on graph paper. Label the vertices.

Lesson 30: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

b. Explain the transformation that occurred to the unit square.

c. Find the area of the image.

d. Find the inverse of the transformation.

e. Explain the meaning of the inverse transformation on the unit square.

f. Rewrite the original matrix if it also included a dilation with a scale factor of 2.

g. What is the inverse of this matrix?

3. Find a transformation that would create a 90° counterclockwise rotation about the origin. Set up a system of
equations, and solve to find the matrix.

Lesson 30: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

4.
a. Find a transformation that would create a 180° counterclockwise rotation about the origin. Set up a system of
equations, and solve to find the matrix.

b. Rewrite the matrix to also include a dilation with a scale factor of 5.

3 −100
5. For which values of 𝑎𝑎 does � � have an inverse matrix?
900 𝑎𝑎

𝑎𝑎 𝑎𝑎 + 4
6. For which values of 𝑎𝑎 does � � have an inverse matrix?
2 𝑎𝑎

𝑎𝑎 + 2 𝑎𝑎 − 4
7. For which values of 𝑎𝑎 does � � have an inverse matrix?
𝑎𝑎 − 3 𝑎𝑎 + 3

Lesson 30: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

cos(𝜃𝜃°) − sin(𝜃𝜃°)
8. Chethan says that the matrix � � produces a rotation 𝜃𝜃° counterclockwise. He justifies his work
sin(𝜃𝜃°) cos(𝜃𝜃°)
1 �3
cos(60°) − sin(60°) −
by showing that when 𝜃𝜃 = 60, the rotation matrix is � �= �2 2 �. Shayla disagrees and
sin(60°) cos(60°) �3 1
2 2
1 −√3
says that the matrix � � produces a 60° rotation counterclockwise. Tyler says that he has found that the
√3 1
2 −2√3
matrix � � produces a 60° rotation counterclockwise, too.
2√3 2
a. Who is correct? Explain.

b. Which matrix has the largest scale factor? Explain.

c. Create a matrix with a scale factor less than 1 that would produce the same rotation.

Lesson 30: When Can We Reverse a Transformation?

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Find a transformation that would create a 30° counterclockwise rotation about the origin and then its inverse.

2. Find a transformation that would create a 30° counterclockwise rotation about the origin, a dilation with a scale
factor of 4, and then its inverse.

3. Find a transformation that would create a 270° counterclockwise rotation about the origin. Set up a system of
equations, and solve to find the matrix.

4. Find a transformation that would create a 270° counterclockwise rotation about the origin, a dilation with a scale
factor of 3, and its inverse.

8 𝑎𝑎
5. For which values of 𝑎𝑎 does � � have an inverse matrix?
𝑎𝑎 2

𝑎𝑎 𝑎𝑎 − 4
6. For which values of 𝑎𝑎 does � � have an inverse matrix?
𝑎𝑎 + 4 𝑎𝑎

3𝑎𝑎 2𝑎𝑎 − 6
7. For which values of 𝑎𝑎 does � � have an inverse matrix?
6𝑎𝑎 4𝑎𝑎 − 12

8. In Lesson 27, we learned the effect of a transformation on a unit square by multiplying a matrix. For example,
2 1 2 1 1 2 2 1 1 3 2 1 0 1
𝐴𝐴 = � �, � �� � = � �,� � � � = � �, and � � � � = � �.
1 2 1 2 0 1 1 2 1 3 1 2 1 2
a. Sasha says that we can multiply the inverse of 𝐴𝐴 to those resultants of the square after the transformation to
get back to the unit square. Is her conjecture correct? Justify your answer.
b. From part (a), what would you say about the inverse matrix with regard to the geometric effect of
transformations?
cos(𝜃𝜃) −sin(𝜃𝜃) 𝜋𝜋
c. A pure rotation matrix is � �. Prove the inverse matrix for a pure rotation of radians
sin(𝜃𝜃) cos(𝜃𝜃) 4
𝜋𝜋 𝜋𝜋 𝑑𝑑 −𝑐𝑐
cos �− � −sin �− �
4 4 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏
counterclockwise is � 𝜋𝜋 𝜋𝜋 �, which is the same as � −𝑏𝑏 �.
𝑑𝑑
𝑠𝑠𝑠𝑠𝑠𝑠 �− � cos �− �
4 4 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏
1
0
d. Prove that the inverse matrix of a pure dilation with a factor of 4 is � 4 1�, which is the same as
0
4
𝑑𝑑 −𝑐𝑐
�𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏�.
−𝑏𝑏 𝑑𝑑
𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏

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 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 30 M1
PRECALCULUS AND ADVANCED TOPICS

𝜋𝜋
e. Prove that the matrix used to undo a radians clockwise rotation and a dilation of a factor of 2 is
3
𝜋𝜋 𝜋𝜋
cos � � −sin � � 𝑑𝑑 −𝑐𝑐
1 3 3 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏
� 𝜋𝜋 𝜋𝜋
�, which is the same as � −𝑏𝑏 𝑑𝑑
�.
2
sin � � cos � � 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏
3 3
f. Prove that any matrix whose determinant is not 0 will have an inverse matrix to “undo” a transformation. For
𝑎𝑎 𝑐𝑐 𝑥𝑥
example, use the matrix 𝐴𝐴 = � � and the point �𝑦𝑦�.
𝑏𝑏 𝑑𝑑

2 2
9. Perform the transformation � � on the unit square.
2 2
a. Can you find the inverse matrix that will “undo” the transformation? Explain your reasons arithmetically.
b. When all four vertices of the unit square are transformed and collapsed onto a straight line, what can be said
about the inverse?
c. Find the equation of the line that all four vertices of the unit square collapsed onto.
1 3
d. Find the equation of the line that all four vertices of the unit square collapsed onto using the matrix � �.
2 6
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.

10. The determinants of the following matrices are 0. Describe what pattern you can find among them.
1 2 1 1 1 2 1 −2
a. � �, � �, � �, and � �
1 2 2 2 4 8 2 −4
0 1 1 0 0 0 1 1 0 0
b. � �, � �, � �, � �, and � �
0 1 1 0 1 1 0 0 0 0

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