LESSON

1.3 Name Class Date

Transformations of 1.3 Transformations

Function Graphs of Function Graphs
Essential Question: What are the ways you can transform the graph of the function f(x)?
Resource

Common Core Math Standards Locker

The student is expected to:

COMMON Explore 1 Investigating Translations 6
y
CORE F-BF.B.3 of Function Graphs 4
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), You can transform the graph of a function in various ways. You can translate 2
and f(x + k) for specific values of k ... find the value of k given the graphs. the graph horizontally or vertically, you can stretch or compress the graph x
… Also A-CED.A.2, F-IF.C.7b horizontally or vertically, and you can reflect the graph across the x-axis or the
-6 -4 -2 0 2 4 6
y-axis. How the graph of a given function is transformed is determined by the
Mathematical Practices way certain numbers, called parameters, are introduced in the function.
-4
COMMON
CORE MP.2 Reasoning The graph of ƒ(x) is shown. Use this graph for the exploration.
-6

Language Objective A First graph g(x) = ƒ(x) + k where k is the parameter. Let k = 4 so
that g(x) = ƒ(x) + 4. Complete the input-output table and then
Identify graphs of odd and even functions and justify reasoning with a graph g(x). In general, how is the graph of g(x) = ƒ(x) + k related
partner. to the graph of ƒ(x) when k is a positive number? 6
y

4
x f(x) f(x) + 4

ENGAGE -1
1
-2
2
2
6
2
x
-6 -4 -2 0 2 4 6
Essential Question: What are the ways 3 -2 2
6 -4
you can transform the graph of the 5 2

function y = f(x)? -6
© Houghton Mifflin Harcourt Publishing Company

Possible answer: The parameter h in y =f(x - h) For k > 0, the graph of g(x) = f(x) + k is the graph of f(x) translated up k units.
produces a horizontal translation of the graph
B Now try a negative value of k in g(x) = ƒ(x) + k.
of y = f(x). The parameter k in y = f(x) + k produces Let k = -3 so that g(x) = ƒ(x) - 3. Complete the input-
x f(x) f(x) - 3
a vertical translation of the graph of y = f(x). The output table and then graph g(x) on the same grid. -1 -2 -5
In general, how is the graph of g(x) = ƒ(x) + k related -1
parameter a in y = af(x) produces a vertical stretch/ 1 2
to the graph of ƒ(x) when k is a negative number? -2 -5
compression of the graph of y = f(x) and may also 3
5 2 -1
produce a reflection across the x-axis. The
1
(_ )
parameter b in y = f x produces a horizontal
b
For k < 0, the graph of g(x) = f(x) + k is the graph of f(x) translated down ⎜k⎟ units.
stretch/compression of the graph of y = f(x) and
may also produce a reflection across the y-axis.

Module 1 ges must

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Transforman Graphs
PERFORMANCE TASK 1.3

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F-BF.B.3 Identif of k … find the value
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.A.2, F-IF.C.7
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A2_MNLESE385894_U1M01L3 31 6/8/15 11:46 AM

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Performance Task.
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For k < 0,
Lesson 3

31

2:18 PM
6/8/15
Module 1

L3 31
4_U1M01
SE38589
A2_MNLE

31 Lesson 1.3
 C Now graph g(x) = ƒ(x - h) where h is the parameter. Let h = 2 so that g(x) = ƒ(x - 2).
Complete the mapping diagram and then graph g(x). (To complete the mapping diagram,
you need to find the inputs for g that produce the inputs for ƒ after you subtract 2. Work EXPLORE 1
backward from the inputs for ƒ to the missing inputs for g by adding 2.) In general, how is the
graph of g(x) = ƒ(x - h) related to the graph of ƒ(x) when h is a positive number?
Investigating Translations of
Input Input Output Output 6 Function Graphs
for g for f for f for g
4
-2
1 -1 -2 -2
2
3 1 2 2 x INTEGRATE TECHNOLOGY
5 3 -2 -2 -6 -4 -2 0 2 4 6 Students have the option of completing the activity
7 5 2 2
-4 either in the book or online.
-6

For h > 0, the graph of g(x) = f(x - h) is the graph of f(x) translated right h units.
AVOID COMMON ERRORS
Students may confuse directions in horizontal
translations. Emphasize that h is the number
D Make a Conjecture How would you expect the graph of g(x) = ƒ(x - h) to be related to subtracted from x. For example, in ƒ(x – 3), the value
the graph of ƒ(x) when h is a negative number?
of h is 3, a positive number, and therefore the
For h < 0, the graph of g(x) = f(x - h) is the graph of f(x) translated left ⎜h⎟ units.
translation is to the right.

Reflect
QUESTIONING STRATEGIES
1. Suppose a function ƒ(x) has a domain of ⎡⎣x 1, x 2⎤⎦ and a range of ⎡⎣y 1, y 2⎤⎦. When the graph of ƒ(x) is
translated vertically k units where k is either positive or negative, how do the domain and range change? Given the graph of a function f(x), and the
The domain remains the same, and the range changes from ⎡⎣y 1, y 2⎤⎦ to ⎡⎣y 1 + k, y 2 + k⎤⎦. graph of the image of the function after a
translation, how can you determine the rule for the
© Houghton Mifflin Harcourt Publishing Company
2. Suppose a function ƒ(x) has a domain of ⎡⎣x 1, x 2⎤⎦ and a range of ⎡⎣y 1, y 2⎤⎦. When the graph of ƒ(x) is
translated horizontally h units where h is either positive or negative, how do the domain and range change? function represented by the image? You can select a
The domain changes from ⎡⎣x 1, x 2⎤⎦ to ⎡⎣x 1 - h, x 2 - h⎤⎦, and the range remains the same.
particular point on the original graph (such as an
3. You can transform the graph of ƒ(x) to obtain the graph of g(x) = ƒ(x - h) + k by combining endpoint of a segment, or a local maximum or
transformations. Predict what will happen by completing the table.
minimum point), and see how its image was
Sign of h Sign of k Transformations of the Graph of f(x) obtained. If the image is above or below the original
+ + Translate right h units and up k units.
point, the rule will involve adding or subtracting the
+ - Translate right h units and down ⎜k⎟ units.
number of units it was translated to f(x). If the
- + Translate left ⎜h⎟ units and up k units.
image is to the left or right of the original point, the
- - Translate left ⎜h⎟ units and down ⎜k⎟ units.
rule will involve adding or subtracting the number
of units it was translated to x in f(x).

Module 1 32 Lesson 3

PROFESSIONAL DEVELOPMENT
A2_MNLESE385894_U1M01L3 32 5/14/14 3:32 PM

Math Background
Transformations change the graph of a function. When students understand the
basic transformations (translation, reflection, stretch, and compression), they are
better able to understand how to write the equation of a graph, and how to
identify the graph of a function that has been transformed.

Transformations of Function Graphs 32

 Explore 2 Investigating Stretches and Compressions
EXPLORE 2 of Function Graphs
In this activity, you will consider what happens when you multiply by a positive parameter inside or outside a
function. Throughout, you will use the same function ƒ(x) that you used in the previous activity.
Investigating Stretches and
Compressions of Function Graphs A First graph g(x) = a ⋅ ƒ(x) where a is the parameter. Let a = 2 so that g(x) = 2ƒ(x).
Complete the input-output table and then graph g(x). In general, how is the graph of
g(x) = a ⋅ ƒ(x) related to the graph of ƒ(x) when a is greater than 1?

QUESTIONING STRATEGIES x f(x) 2f(x) 6

On the coordinate plane, what is the -1 -2 -4 4

difference between a vertical stretch and a 1 2 4 2

x
3 -2 -4
vertical compression? In a vertical stretch, the -6 -4 -2 0 2 4 6
5 2 4
points of the graph are pulled away from the x-axis.
-4
In a vertical compression, they are pulled toward
-6
the x-axis.

On the coordinate plane, what is the For a > 1, the graph of g(x) = a ⋅ f(x) is the graph of f(x) stretched vertically (away from
difference between a horizontal stretch and a the x-axis) by a factor of a.
vertical stretch? In a horizontal stretch, the points of
B Now try a value of a between 0 and 1 in g(x) = a ⋅ ƒ(x). Let a = _12 so that g(x) = _12 ƒ(x).
the graph are pulled away from the y-axis. In a
Complete the input-output table and then graph g(x). In general, how is the graph of
vertical stretch, the points of the graph are pulled g(x) = a ⋅ ƒ(x) related to the graph of ƒ(x) when a is a number between 0 and 1?
away from the x-axis. __1 f(x) y
x f(x) 2 6
-1 -2 -1 4
1 2 1 2
3 -2 -1 x
© Houghton Mifflin Harcourt Publishing Company

5 2 1 -6 -4 -2 0 2 4 6

-4
-6

For 0 < a < 1, the graph of g(x) = a ⋅ f(x) is the graph of f(x) compressed vertically
(toward the x-axis) by a factor of a.

Module 1 33 Lesson 3

COLLABORATIVE LEARNING
A2_MNLESE385894_U1M01L3.indd 33 19/03/14 2:28 PM

Peer-to-Peer Activity
Have students work in pairs. Provide students with three or four functions, and
have them use a graphing calculator to explore how changes to the various
parameters in each function affect the graph of the function. Once they start to
make the connections, encourage them to try and predict each change before
graphing the transformation.

33 Lesson 1.3
 C ( )
Now graph g(x) = ƒ __b1 ⋅ x where b is the parameter. Let b = 2 so that g(x) = ƒ __12 x . ( )
Complete the mapping diagram and then graph g(x). (To complete the mapping diagram, INTEGRATE TECHNOLOGY
you need to find the inputs for g that produce the inputs for f after you multiply by _12 .
Work backward from the inputs for f to the missing inputs for g by multiplying by 2.) A graphing calculator can be used to explore
In general, how is the graph of g(x) = ƒ( __b1 x ) related to the graph of ƒ(x) when b
the effects of different values of b in the
(b )
is a number greater than 1?
1
function g(x) = ƒ _ x on the graph of ƒ(x). Choose a
Input Input Output Output y
4
for g
· 1
for f for f for g simple function for ƒ(x), graph the function, and
2 2
-2 -1 -2 -2 x have students suggest different values for b. Graph
2 1 2 2 -6 -4 -2 0 2 4 6 8 10 the transformed functions, and have students
6 3 -2 -2
compare the graphs to see how changing the
10 5 2 2 -4
parameter affects the graph.
For b > 1, the graph of g(x) = f (__b1 ⋅ x) is the graph of f(x) stretched horizontally (away
from the y-axis) by a factor of b.
INTEGRATE MATHEMATICAL
D (
Make a Conjecture How would you expect the graph of g(x) = ƒ __b1 ⋅ x to be related ) PRACTICES
to the graph of ƒ(x) when b is a number between 0 and 1?
Focus on Reasoning
For 0 < b < 1, the graph of g(x) = f (__b1 ⋅ x)is the graph of f(x) compressed horizontally MP.2 Prompt students to recognize that when the
(toward the y-axis) by a factor of b.
graph of a function passes through the origin, a
Reflect transformation involving a stretch or a compression
4. Suppose a function ƒ(x) has a domain of ⎡⎣x 1, x 2⎤⎦ and a range of ⎡⎣y 1, y 2⎤⎦. When the graph of ƒ(x) is stretched
of the function will not affect the point at the origin.
or compressed vertically by a factor of a, how do the domain and range change? Ask students to justify how this is possible, when all
The domain remains the same, and the range changes from ⎡⎣y 1, y 2⎤⎦ to ⎡⎣ay 1, ay 2⎤⎦.
points on either side of the origin are affected.
5. You can transform the graph of ƒ(x) to obtain the graph of g(x) = a ⋅ ƒ(x-h) + k by combining
transformations. Predict what will happen by completing the table.

Value of a Transformations of the Graph of f(x) © Houghton Mifflin Harcourt Publishing Company

Stretch vertically by a factor of a, and translate h units horizontally and

a>1
k units vertically.

Compress vertically by a factor of a, and translate h units

0<a<1
horizontally and k units vertically.

6. You can transform the graph of ƒ(x) to obtain the graph of g(x) = ƒ ( __1 (x - h) ) + k by combining
b
transformations. Predict what will happen by completing the table.

Value of b Transformations of the Graph of f(x)

Stretch horizontally by a factor of b, and translate h units horizontally
b>1
and k units vertically.

Compress horizontally by a factor of b, and translate h units

0<b<1
horizontally and k units vertically.

Module 1 34 Lesson 3

A2_MNLESE385894_U1M01L3 34 5/14/14 3:32 PM

Transformations of Function Graphs 34

 Explore 3 Investigating Reflections of Function Graphs
EXPLORE 3 When the parameter in a stretch or compression is negative, another transformation called a reflection is introduced.
Examining reflections will also tell you whether a function is an even ƒunction or an odd ƒunction. An even function
is one for which ƒ(-x) = ƒ(x) for all x in the domain of the function, while an odd function is one for which
Investigating Reflections of ƒ(-x) = -ƒ(x) for all x in the domain of the function. A function is not necessarily even or odd; it can be neither.

Function Graphs A First graph g(x) = a ⋅ ƒ(x) where a = -1. Complete the input- y
6
output table and then graph g(x) = -ƒ(x). In general, how is the
graph of g(x) = -ƒ(x) related to the graph of ƒ(x)? 4

INTEGRATE MATHEMATICAL 2
x f(x) -f(x)
PRACTICES x
-1 -2 2 -6 -4 -2 0 2 4 6
Focus on Technology 1 2 -2
-4
MP.5 To help students understand the symmetry of 3 -2 2
-2 -6
the graphs of even and odd functions, have students 5 2

use a graphing calculator to graph ƒ(x) = x 2 and

The graph of g(x) = -f(x) is a reflection of the graph of f(x) across the x-axis.
g(x) = ƒ(-x) = (-x) 2 to see that the two graphs
coincide. This is a consequence of the fact that Now graph g(x) = ƒ ( __1 ⋅ x ) where b = -1. Complete the input-output table and then graph
B b
replacing x with –x in a function rule causes the g(x) = ƒ(-x). In general, how is the graph of g(x) = ƒ(-x) related to the graph of ƒ(x)?
graph to be reflected across the y-axis. Since the
Input Input Output Output y
4
graph of ƒ(x) = x 2 is symmetric with respect to the for g for f for f for g
· (-1) 2
y-axis, it is unaffected when x is replaced with –x. 1 -1 -2 -2 x
-1 -6 -4 -2 0
Also have students graph ƒ(x) = x , 3 1 2 2 2 4 6
-3 3 -2 -2
g(x) = ƒ(-x) = (-x) 3, and -5 5 2 2 -4
h(x) = -ƒ(x) = -x 3, and observe that the graphs of
g(x) and h(x) coincide. The graph of g(x) is a The graph of g(x) = f(-x) is a reflection of the graph of f(x) across the y-axis.
© Houghton Mifflin Harcourt Publishing Company

reflection of the graph of ƒ(x) across the y-axis, while

the graph of h(x) is a reflection of the graph of ƒ(x) Reflect

across the x-axis. The graphs of g(x) and h(x) 7. Discussion Suppose a function ƒ(x) has a domain of ⎡⎣x 1, x 2⎤⎦ and a range of ⎡⎣y 1, y 2⎤⎦. When the graph of
coincide because reflecting the graph of ƒ(x) across ƒ(x) is reflected across the x-axis, how do the domain and range change?
The domain remains the same, and the range changes from ⎡⎣y 1, y 2⎤⎦ to ⎡⎣-y 2, -y 1⎤⎦.
both axes does not change the graph, which is
8. For a function ƒ(x), suppose the graph of ƒ(-x), the reflection of the graph of ƒ(x) across the y-axis, is
another way of saying that the graph of ƒ(x) has 180˚
identical to the graph of ƒ(x). What does this tell you about ƒ(x)? Explain.
rotational symmetry about the origin. Because f(x) = f(-x), f(x) must be an even function.

9. Is the function whose graph you reflected across the axes in Steps A and B an even function, an odd
function, or neither? Explain.
QUESTIONING STRATEGIES It is neither an even function nor an odd function, because the graph of f(-x) in Step B is

How do the domain and range of the function not identical to the graph of f(x) nor is it identical to the graph of -f(x) in Step A.

g(x) = ƒ(-x) compare to the domain and

Module 1 35 Lesson 3
range of ƒ(x)? The domain of g(x) consists of the
opposites of the elements in the domain of f(x). The
range values are the same.
A2_MNLESE385894_U1M01L3 35 6/8/15 11:47 AM

35 Lesson 1.3
 Explain 1 Transforming the Graph of the Parent
Quadratic Function EXPLAIN 1
You can use transformations of the graph of a basic function, called a y
parent function, to obtain the graph of a related function. To do so, focus on how 4
the transformations affect reference points on the graph of the parent function. Transforming the Graph of the Parent
2
For instance, the parent quadratic function is ƒ(x) = x 2. The graph of this x
Quadratic Function
function is a U-shaped curve called a parabola with a turning point, called a
-4 -2 0 2 4
vertex, at (0, 0). The vertex is a useful reference point, as are the points (-1, 1)
and (1, 1). -2
QUESTIONING STRATEGIES
-4
When drawing the graph of a transformation
of the function ƒ(x) = x 2 that involves a
Example 1 Describe how to transform the graph of f (x) = x 2 to obtain the graph of reflection across the x-axis, a horizontal translation,
the related function g(x). Then draw the graph of g(x).
and a vertical stretch, in which order should the
 g(x) = -3ƒ(x - 2) -4 transformations be applied to the graph of the parent
function? Explain. The order does not matter. The
Parameter
and Its Value Effect on the Parent Graph graph of the new function will be the same no
a = -3 vertical stretch of the graph of ƒ(x) by a factor of 3 and a reflection across the x-axis matter what the order.
b=1 Since b = 1, there is no horizontal stretch or compression.
When drawing the graph of a transformation
h=2 horizontal translation of the graph of ƒ(x) to the right 2 units
of the function ƒ(x) = x 2 that involves a
k = -4 vertical translation of the graph of ƒ(x) down 4 units
reflection, a vertical translation, and a horizontal
Applying these transformations to a point (x, y) on the parent graph results in the point (x + 2, -3y -4). compression, in which order should the
The table shows what happens to the three reference points on the graph of ƒ(x).
transformations be applied to the graph of the parent
Point on the Graph of f(x) Corresponding Point on g(x) function? Explain. The reflection and compression
(-1, 1) (-1 + 2, -3 (1) - 4) = (1, -7) need to be applied before the vertical translation.
© Houghton Mifflin Harcourt Publishing Company
(0, 0) (0 + 2, -3 (0) - 4) = (2, -4) The new function is a vertical translation of the
(1, 1) (1 + 2, -3 (1) - 4) = (3, -7) graph of f(x) = ax 2.
Use the transformed reference points to graph g(x). y x
-2 0 2 4 6
-2

-4

-6

-8

Module 1 36 Lesson 3

DIFFERENTIATE INSTRUCTION
A2_MNLESE385894_U1M01L3.indd 36 19/03/14 2:28 PM

Visual Cues
Suggest that students write the general function g(x) = af(x - h) 2 + k, (or
ƒ(x) = a(x - h) 2 + k, depending on the context) above the specific function they
are analyzing in order to correctly identify the parameters in the transformation.
Communicating Math
When analyzing transformed functions, have students list each parameter and its
value, and then write a short phrase, such as “shift 5 units to the left” or “reflect
across the x-axis” to indicate the meaning of each parameter. This may make it
easier for the student to then draw the graph of the function.

Transformations of Function Graphs 36

 B (
g(x) = ƒ _
2 )
1 (x + 5) + 2
INTEGRATE MATHEMATICAL
PRACTICES Parameter
and Its Value Effect on the Parent Graph
Focus on Math Connections a= 1 Since a = 1, there is no vertical stretch, no vertical compression, and no
MP.1 Discuss with students how the attributes of a reflection across the x-axis.

function (such as domain, range, maximum or b= 2 The parent graph is stretched/compressed horizontally by a factor
2
minimum values, and intervals over which the of . There is no reflection across the y-axis.

function is increasing or decreasing) can be h = -5 The parent graph is translated -5 units horizontally/vertically.
determined from the function written in the form
k= 2 2
g(x) = af(x- h) 2 + k, where ƒ(x) is the function ƒ(x) The parent graph is translated units horizontally/vertically.

= x 2. Applying these transformations to a point on the parent graph results in the point ( 2x - 5, y + 2 ).
The table shows what happens to the three reference points on the graph of ƒ(x).

Point on the Graph of f(x) Corresponding Point on the Graph of g(x)

(-1, 1) (
( 2 (-1) - 5, 1 + 2) = -7 , 3 )
(0, 0) ( 2 (0) - 5, 0 + 2) = ( -5 , 2 )
(1, 1) ( 2 (1) - 5, 1 + 2) = ( -3 , 3 )
Use the transformed reference points to graph g(x). y
6

2
x
-10 -8 -6 -4 -2 0
© Houghton Mifflin Harcourt Publishing Company

-2

Reflect

10. Is the function ƒ(x) = x an even function, an odd function, or neither? Explain.
2

The function f(x) = x 2 is an even function because f(-x) = f(x) for all values of x in its
domain.

11. The graph of the parent quadratic function ƒ(x) = x has the vertical line x = 0 as its axis of symmetry.
2

Identify the axis of symmetry for each of the graphs of g(x) in Parts A and B. Which transformation(s)
affect the location of the axis of symmetry?
In Part A, the axis of symmetry is x = 2. In Part B, the axis of symmetry is x = -5. Only a
horizontal translation affects the location of the axis of symmetry.

Module 1 37 Lesson 3

LANGUAGE SUPPORT
A2_MNLESE385894_U1M01L3 37 15/05/14 5:22 PM

Connect Vocabulary
Have students work individually and then with a partner on this activity. Give
each student pictures on paper or graphing calculators showing images of the
graphs of even and odd functions. Have each student identify whether the graph is
an even or odd function. Once they decide, they should work with a partner. Each
partner has to agree or disagree with the choices made by the other person, and
explain why he/she agrees or disagrees. In their explanations, encourage the use of
the terms reflection, x-axis, y-axis, coincides with graph of f.

37 Lesson 1.3
 Your Turn

12. Describe how to transform the graph of ƒ(x) = x 2 to obtain

the graph of the related function g(x) = ƒ(-4(x - 3)) + 1.
3
y EXPLAIN 2
Then draw the graph of g(x).
2 Modeling with a Quadratic Function
The graph of g(x) = f(-4(x - 3)) + 1 is a reflection
1
of the graph of f(x) across the y-axis, a horizontal x
1 _
compression by a factor of , and a translation of 0 1 2 3 4 5 6 AVOID COMMON ERRORS
4
-1
3 units to the right and 1 unit up. Students may use the wrong signs to indicate the
horizontal and vertical translations. Reinforce that
Graph of f(x) Graph of g(x)
the number indicating the horizontal translation
(-1, 1) (-_1 (-1) + 3, 1 + 1) = (3_1 , 2)
4 4 must be subtracted from x, whereas the number
(0, 0) (-_41 (0) + 3, 0 + 1) = (3, 1) indicating the vertical translation must be added to
the function.
(1, 1) (-_41 (1) + 3, 1 + 1) = (2_34 , 2)
QUESTIONING STRATEGIES
Explain 2 Modeling with a Quadratic Function When modeling a real-world object using a
You can model real-world objects that have a parabolic shape using a quadratic function. In order quadratic function, why do you need to
to fit the function’s graph to the shape of the object, you will need to determine the values of the
( )
parameters in the function g(x) = a ⋅ ƒ __1 (x - h) + k where ƒ(x) = x 2. Note that because ƒ(x) is
b
restrict the domain of the function? The graph of
simply a squaring function, it’s possible to pull the parameter b outside the function and combine it the function is part of a parabola. It has a left-most
with the parameter a. Doing so allows you to model real-objects using g(x) = a ⋅ ƒ (x - h) + k,
which has only three parameters. point and a right-most point. Thus, the domain
When modeling real-world objects, remember to restrict the domain of g(x) = a ⋅ ƒ (x - h) + k consists only of values between, and including,
to values of x that are based on the object’s dimensions.
those two points.

© Houghton Mifflin Harcourt Publishing Company

Example 2
y
An old stone bridge over a river uses a parabolic arch for x
support. In the illustration shown, the unit of measurement 0 10 20 30 40 50
for both axes is feet, and the vertex of the arch is point C. -10 C (27, -5)
Find a quadratic function that models the arch, and state the
function’s domain. -20
A (2, -20) B (52, -20)

Module 1 38 Lesson 3

A2_MNLESE385894_U1M01L3 38 6/9/15 11:22 PM

Transformations of Function Graphs 38

 Analyze Information
INTEGRATE MATHEMATICAL
PRACTICES Identify the important information.

• The shape of the arch is a parabola .

Focus on Modeling
• The vertex of the parabola is C(27, -5) .
MP.4 Have students brainstorm real-world objects A(2, -20) B(52, -20)
• Two other points on the parabola are and .
that can be modeled using quadratic functions. Make
a list of the objects mentioned, and have students Formulate a Plan
discuss (in general terms) the value of the parameter You want to find the values of the parameters a, h, and k in g(x) = a ⋅ ƒ(x - h) + k
a in ƒ(x) = a(x - h) 2 + k, for each object. where ƒ(x) = x 2. You can use the coordinates of point C to find the values of h
and k. Then you can use the coordinates of one of the other points to find the value
of a.

Solve
The vertex of the graph of g(x) is point C, and the vertex of the graph of ƒ(x) is the
origin. Point C is the result of translating the origin 27 units to the right and 5 units
down. This means that h = 27 and k = -5. Substituting these values into g(x) gives
g(x) = a ⋅ ƒ(x - 27) - 5. Now substitute the coordinates of point B into g(x) and
solve for a.
g(x) = a ⋅ ƒ (x - 27) - 5 Write the general function.

( )
g 52 = a ⋅ ƒ (52 - 27) - 5 Substitute 52 for x.

-20 = a ⋅ ƒ (52 - 27) - 5 Replace g(52) with -20, the y-value of B.

( )
-20 = a ⋅ ƒ 25 - 5 Simplify.

-20 = a (625) - 5 Evaluate ƒ (25) for ƒ(x) = x 2.

a = -___
3
125 Solve for a.

Substitute the value of a into g(x).

g(x) = -_ 3 ƒ (x - 27) - 5
© Houghton Mifflin Harcourt Publishing Company

125
The arch exists only between points A and B, so the domain of g(x) is {x⎜2 ≤ x ≤ 52.

Justify and Evaluate

To justify the answer, verify that g(2) = -20.
g(x) = -_ 3 ƒ (x - 27) - 5 Write the function.
125
( )
g 2 = -_ 3 ƒ ( 2 - 27) - 5
125
Substitute 2 for x.

= -_
125 (
3 ƒ -25 - 5
) Subtract.

= -_
125 (
3 · 625 - 5
) Evaluate ƒ (-25).

= -20 ✓ Simplify.

Module 1 39 Lesson 3

A2_MNLESE385894_U1M01L3 39 5/14/14 3:33 PM

39 Lesson 1.3
 Your Turn

13. The netting of an empty hammock hangs between

its supports along a curve that can be modeled by A (-2, 4)
y
B (8, 4)
ELABORATE
a parabola. In the illustration shown, the unit of 4
measurement for both axes is feet, and the vertex
2 C (3, 3) INTEGRATE MATHEMATICAL
of the curve is point C. Find a quadratic function
x
PRACTICES
that models the hammock’s netting, and state the
function’s domain. -2 0 2 4 6 8 Focus on Math Connections
MP.1 Remind students that the real zeros of a
The vertex is (3, 3), so h = 3 and k = 3. Substitute the values of h and k into g(x). function are the x-intercepts of the graph of the
g(x) = a ⋅ f(x - 3) + 3 where f(x) = x2
function. Have them discuss how the number of real
Substitute (8, 4) into g(x) and solve for a.
4 = a ⋅ f(8 - 3) + 3
zeros of a function can be determined by knowing
4 = a ⋅ f(5) + 3 the values of a, h, and k in f(x) = a(x - h) 2 + k.
4 = a ⋅ 25 + 3
_
1
=a
25
_
1 (
SUMMARIZE THE LESSON
So, g(x) = f x - 3) + 3. The netting exists only between points A and B, so the domain of
25 How does each of the following
g(x) is {x⎜-2 ≤ x ≤ 8}.
transformations of f(x) = x 2 affect the values
of a, h, and k in the function g(x) = af(x - h) 2 + k?
Elaborate
translation 3 units left h = -3
14. What is the general procedure to follow when graphing a function of the form g(x) = a ⋅ ƒ(x - h) + k
given the graph of ƒ(x)? translation 3 units up k = 3
The general procedure for graphing g(x) is to choose reference points on the graph of f(x)
and apply the transformations for g(x) to them. In the case of f(x) = x 2, choose three points vertical stretch by a factor of 3 a = 3
on the graph of f(x): the vertex and one point on either side of the vertex. Each point (x, y)
reflection across the x-axis a = -1
is transformed to the point (x + h, ay + k). Use the new points to draw the graph of g(x).
© Houghton Mifflin Harcourt Publishing Company
15. What are the general steps to follow when determining the values of the parameters a, h, and k in
ƒ(x) = a(x - h) + k when modeling a parabolic real-world object?
2

To model a parabolic real-world object, first define a coordinate system. Then identify
three points on the object; ideally, one of the points should be the vertex. The coordinates
of the vertex on the object, (x v, y v), are the values of h and k, respectively. Use a second
point, (x 1, y 1), on the object to solve y 1 = a(x 1 - x v) 2 + y v for a and get a _______2. You can
y1 - yv
(x 1 - x v)
use the third point as a check on your results.

16. Essential Question Check-In How can the graph of a function ƒ(x) be transformed?
The graph can be stretched or compressed horizontally or vertically, it can be reflected
across the x-axis or y-axis, and it can be translated horizontally or vertically.

Module 1 40 Lesson 3

A2_MNLESE385894_U1M01L3 40 6/8/15 11:47 AM

Transformations of Function Graphs 40

 Evaluate: Homework and Practice
EVALUATE
• Online Homework
Write g(x) in terms of ƒ(x) after performing the given transformation of the • Hints and Help
graph of ƒ(x). • Extra Practice

1. Translate the graph of ƒ(x) to the left 3 units. 2. Translate the graph of ƒ(x) up 2 units.

y y
4
4
2
2
ASSIGNMENT GUIDE -6 -4 -2 0 2 4
x
x
-2 -4 -2 0 2 4
Concepts and Skills Practice -2
Explore 1 Exercises 1–4 -4

Investigating Translations of
g(x) = f(x + 3) g(x) = f(x) + 2
Function Graphs
Explore 2 Exercises 5–8
3. Translate the graph of ƒ(x) to the right 4 units. 4. Translate the graph of ƒ(x) down 3 units.
Investigating Stretches and
Compressions of Function Graphs y y
4
Explore 3 Exercises 9–14
Investigating Reflections of Function x
Graphs x -4 -2 0 4
-4 -2 0 4 6 -2
Example 1 Exercises 15–16 -2
Transforming the Graph of the -4
Parent Quadratic Function -4

Example 2 Exercises 17–18 g(x) = f(x - 4) g(x) = f(x) - 3

Modeling with a Quadratic Function
© Houghton Mifflin Harcourt Publishing Company

5. Stretch the graph of ƒ(x) horizontally by a 6. Stretch the graph of ƒ(x) vertically by a
factor of 3. factor of 2.
AVOID COMMON ERRORS y y
4 4
Students may confuse the concepts of stretch and
2 2
compression (both vertical and horizontal). Tell them
x
that they can evaluate the function for a specific value -4 -2 0 2 4 x -4 -2 0 2 4
of x, and compare it to the value of ƒ(x). Plotting one -2 -2
or two of these points will provide a visual cue as to -4 -4
the nature of the transformation.
g(x) = f (_13 x) g(x) = 2f(x)

Module 1 41 Lesson 3

COMMON
A2_MNLESE385894_U1M01L3.indd 41
Exercise Depth of Knowledge (D.O.K.) CORE Mathematical Practices 19/03/14 2:28 PM

1–4 1 Recall of Information MP.4 Modeling

5–14 1 Recall of Information MP.2 Reasoning
15–16 1 Recall of Information MP.6 Precision
17–18 2 Skills/Concepts MP.4 Modeling
19 3 Strategic Thinking MP.2 Reasoning
20 3 Strategic Thinking MP.6 Precision

41 Lesson 1.3
 7. Compress the graph of ƒ(x) horizontally by a 8. Compress the graph of ƒ(x) vertically by a
1.
factor of _ 1.
factor of _ CRITICAL THINKING
3 2
y y Given a function ƒ(x), how do the functions
6 6
g(x) = -ƒ(x) and h(x) = ƒ(-x) differ in terms of
4 4 their effects on the ordered pairs that belong to ƒ(x)?
2 2 g(x) is the function that negates the f(x) values in
x x each ordered pair; h(x) negates the x-value in each
-4 -2 0 2 4 -4 -2 0 2 4
-2 -2
ordered pair.

g(x) = f(3x) g(x) = _1 f(x)

2
INTEGRATE MATHEMATICAL
9. Reflect the graph of ƒ(x) across the y-axis. 10. Reflect the graph of ƒ(x) across the x-axis.
PRACTICES
y y
Focus on Patterns
4
4 MP.8 Encourage students to use their graphing
calculators to explore graphs of functions of higher
x 2
x degree and to see whether they can determine any
-4 -2 0 4
-2 -4 -2 0 2 4 patterns that would indicate when a function is even
-2 and when it is odd.
-4
-4
g(x) = f(-x)

g(x) = -f(x)

11. Reflect the graph of ƒ(x) across the y-axis. 12. Reflect the graph of ƒ(x) across the x-axis.

© Houghton Mifflin Harcourt Publishing Company

y y
4 4

2 2
x x
-4 -2 0 2 4 -4 -2 0 2 4
-2 -2

-4 -4

g(x) = f(-x) g(x) = -f(x)

Module 1 42 Lesson 3

A2_MNLESE385894_U1M01L3 42 6/8/15 11:47 AM

Transformations of Function Graphs 42

 13. Determine if each function is an even function, an odd function, or neither.
INTEGRATE TECHNOLOGY
a. y b. y c. y
Students can check their work by graphing their 2
functions on a graphing calculator. This provides x 2 x
x -4 -2 0
immediate feedback, as well as the opportunity to -4 -2 0 4 2 4
-2 -4 -2 0 2 4 -2
visually observe how corrections to the various -2
parameters affect the graph of the function.

Odd function Even function Neither

14. Determine whether each quadratic function is an even function. Answer yes or no.

a. ƒ(x) = 5x
2 Yes b. ƒ(x) = (x - 2)
2 No

x
() Yes Yes
2
c. ƒ(x) = _ d. ƒ(x) = x + 6
2
3

Describe how to transform the graph of f(x) = x 2 to obtain the

graph of the related function g(x). Then draw the graph of g(x).

ƒ(x + 4)
15. g(x) = - _ 16. g(x) = ƒ (2x) + 2
3
reflection of the graph of f(x) across the horizontal compression of the graph of
x-axis, a vertical compression by a factor f(x) by a factor of _
1
2
and a translation of
of _
1
3
, and a translation of 4 units to the left 2 units up
© Houghton Mifflin Harcourt Publishing Company

y y
4 6

2 4
x
2
-6 -4 -2 0
x
-2
-4 -2 0 2 4
-4 -2

Module 1 43 Lesson 3

A2_MNLESE385894_U1M01L3.indd 43 19/03/14 2:28 PM

43 Lesson 1.3
 17. Architecture Flying buttresses were used in the construction
of cathedrals and other large stone buildings before the advent of
INTEGRATE MATHEMATICAL
more modern construction materials to prevent the walls of large, PRACTICES
high-ceilinged rooms from collapsing.
Focus on Reasoning
The design of a flying buttress includes an arch. In the illustration y
shown, the unit of measurement for both axes is feet, and the vertex
12
MP.2 Students can check their functions for
of the arch is point C. Find a quadratic function that models the
arch, and state the function’s domain.
C(2, 12) correctness by substituting the coordinates of a point
10
on the parabola into the rule and checking to see
The vertex is (2, 12), so h = 2 and k = 12. Substitute the 8
values of h and k into g(x). g(x) = a ⋅ f(x - 2) + 12 whether the resulting equation is true.
Substitute (8, 6) into g(x) and solve for a. 6 B(8, 6)

6 = a ⋅ f(8 - 2) + 12 4
6 = a ⋅ f(6) + 12 x CONNECT VOCABULARY
6 = a ⋅ 36 + 12 0 4 6 8 10
Review terms such as horizontal translation, vertical
-_
1
=a
6
translation, stretch/compression by having students
So, g(x) = -_
1 (
f x - 2) + 12. The arch exists only between points C and B, so the domain of
6 look at graphs of functions with different parameters,
g(x) is {x⎜2 ≤ x ≤ 8}.
and then show the kind of translation by using their
hands. For example, holding a hand horizontally for
18. A red velvet rope hangs between two stanchions and forms a curve y a horizontal translation, and moving it up or down
that can be modeled by a parabola. In the illustration shown, the unit A (1, 4) B (7, 4)
of measurement for both axes is feet, and the vertex of the curve is
4 for a vertical one.
point C. Find a quadratic function that models the rope, and state the C (4, 3.5)
2
function’s domain.
x
The vertex is (4, 3.5), so h = 4 and k = 3.5. Substitute the 0 2 4 6 8
values of h and k into g(x). g(x) = a ⋅ f(x - 4) + 3.5
Substitute (4, 3.5) into g(x) and solve for a.
4 = a ⋅ f(7 - 4) + 3.5
4 = a ⋅ f(3) + 3.5
4 = a ⋅ 9 + 3.5 © Houghton Mifflin Harcourt Publishing Company

__
1
=a
18

So, g(x) = __
1 (
18
f x - 4) + 3.5. The rope exists only between points A and B, so the domain of
g(x) is {x⎜1 ≤ x ≤ 7}.

Module 1 44 Lesson 3

A2_MNLESE385894_U1M01L3 44 5/14/14 3:33 PM

Transformations of Function Graphs 44

PEER-TO-PEER DISCUSSION H.O.T. Focus on Higher Order Thinking

19. Multiple Representations The graph of the function y

Ask students to discuss with a partner how a 4
(2 )
2

real-world situation that can be modeled by a vertical g(x) = __1 x + 2 is shown.

2
stretch of the parent quadratic function compares to Use the graph to identify the transformations of the graph of x
ƒ(x) = x 2 needed to produce the graph of g(x). (If a stretch or -8 -6 -4 -2 0
a real-world situation that can be modeled by a compression is involved, give it in terms of a horizontal stretch or
-2
vertical compression of the parent quadratic function. compression rather than a vertical one.) Use your list of

Have them share their conclusions with the class. transformations to write g(x) in the form g(x) = ƒ _1 (x - h) + k.
b ( ) -4
Then show why the new form of g(x) is algebraically equivalent to
A situation that can be modeled by a vertical stretch the given form.
of f(x) = x 2 will contain function values that will
The graph of g(x) shows that the graph of f(x) has been stretched horizontally by a factor
increase or decrease at a faster rate than the
function values in a situation that can be modeled
of 2 and translated to the left 4 units. So, g(x) = f _
2(
1
(
x - (-4)) . This equation is equivalent )
to the given form of g(x) because x - (-4) can be rewritten as x + 4, _
1
can be distributed to
by a vertical compression of f(x) = x 2. 2
the terms of x + 4, and f(x) can be replaced by the action (squaring) that it performs on its
input values. So, g(x) = f _
2( (
x - (-4)) = f _ ) (
(x + 4) = f _12 x + 2 = _12 x + 2 . ) ( ) ( )
2
1 1
2
JOURNAL
20. Represent Real-World Situations The graph of the ceiling
Have students explain how the parameters a, h, and k function, ƒ(x) = ⎡x⎤, is shown. This function accepts any real
in the function g(x) = af(x - h) + k affect the graph
2 number x as input and delivers the least integer greater than or
© Houghton Mifflin Harcourt Publishing Company · Image Credits: ©Michael Dwyer/Alamy
equal to x as output. For instance, ƒ(1.3) = 2 because 2 is the least
of ƒ(x) = x 2. integer greater than or equal to 1.3. The ceiling function is a type of
step function, so named because its graph looks like a set of steps.
Write a function g(x) whose graph is a transformation of the
graph of ƒ(x) based on this situation: A parking garage charges
$4 for the first hour or less and $2 for every additional hour or
fraction of an hour. Then graph g(x).

g(x) = 2f(x) + 2
y f
4

Parking fee (dollars)

8
2
x 6

-4 0 2 4 4
-2
2
-4
t
0
1 2 3 4
Time (h)

Module 1 45 Lesson 3

A2_MNLESE385894_U1M01L3 45 6/8/15 11:47 AM

45 Lesson 1.3
 Lesson Performance Task CONNECT VOCABULARY
Some students may not be familiar with the term
You are designing two versions of a chair, one without armrests and one with armrests. The diagrams show side
views of the chair. Rather than use traditional straight legs for your chair, you decide to use parabolic legs. Given the
armrest, a compound word made up of arm and rest.
function ƒ(x) = x 2, write two functions, g(x) and h(x), whose graphs represent the legs of the two chairs and involve Show students an example of a chair with armrests
transformations of the graph of ƒ(x). For the chair without armrests, the graph of g(x) must touch the bottom of the
chair’s seat. For the chair with armrests, the graph of h(x) must touch the bottom of the armrest. After writing each and another without armrests. Discuss how the
function, graph it. armrests in the given design need to have the support
32
y
32
y of the parabolic legs beneath them to provide the
30 30 strength to hold up under the weight of a person
28 28
26 26 leaning on the armrests.
Vertical dimensions (in.)

Vertical dimensions (in.)

24 24
22 22
20 20
18
16
18
16
QUESTIONING STRATEGIES
14 14
12 12
Is a stretch or a compression used to form the
10 10 legs of the chair with armrests? Is it vertical or
8 8
6 6 horizontal? vertical stretch
4 4
2
x
2
x Why is it necessary for the values of a to be
0 4 8 12 16 20 0 4 8 12 16 20 negative in the equations for the
Horizontal dimensions (in.) Horizontal dimensions (in.)
parabolas? The parabolas open downward.
Sample answer: For the chair without armrests, let (10, 16) be the vertex of the parabola, Which of the two chairs do you think is likely
and terminate the parabola at (0, 0) and (20, 0). The vertex is (10, 16), so h = 10 and k = 16.
Substitute the values of h and k into g(x). g(x) = a ⋅ f(x - 10) + 16 to be the more stable? Why? The chair with
Substitute (20, 0) into g(x) and solve for a. the armrests would be more stable because it is
0 = a ⋅ f(20 - 10) + 16 secured at two points on each pair of legs rather
0 = a ⋅ f(10) + 16 © Houghton Mifflin Harcourt Publishing Company
than just one point on each. The seat on the chair
0 = a ⋅ 100 + 16
with the vertex centered on it could tip forward
-0.16 = a
So, g(x) = -0.16f(x - 10) + 16 with domain {x⎜0 ≤ x ≤ 20}. The graph of g(x) is shown. or backward depending on where the weight is
For the chair with armrests, let (10, 24) be the vertex of the parabola, and again terminate shifted.
the parabola at (0, 0) and (20, 0). Note that this parabola is a vertical stretch of the previous
parabola by a factor of __
24
16
= 1.5. So, h(x) = 1.5 g(x) = 1.5(-0.16f(x - 10) + 16)
( )
= -0.24f x - 10 + 24, also with domain {x⎜0 ≤ x ≤ 20}. The graph of h(x) is shown.

Module 1 46 Lesson 3

EXTENSION ACTIVITY
A2_MNLESE385894_U1M01L3.indd 46 19/03/14 2:28 PM

Have students research the differences between a catenary curve and a parabolic
curve and find examples of catenaries in real life (for example, a hanging cable, or
the Gateway Arch in St. Louis). Students should find that the two types of curves
look very similar in that they are both symmetrical and have similar shapes. A
parabola in its simplest form is ƒ(x) = x 2 while a catenary is of the form
ƒ(x) = cosh(x). Parabolas are often used to model catenaries when the differences
Scoring Rubric
between them are not consequential. 2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.

Transformations of Function Graphs 46