CHAPTER 3 LESSON 16 Effects on the Graph When a Linear

LESSON 16 Effects on the Graph When a Linear Function is ChangedCHAPTER 3

Algebra I | End-of-Coursemasteryeducation.com92 Copying is prohibited.

GUIDED PRACTICE

Words to Knowparent function

parent graph

The parent function f(x) � 2x � 1 is transformed to f(3x). Explain the effect on the parent graph.

Problem 1

UNPACKING the TEKS | S A.3(E)

The basic form of a linear function is f(x) � mx � b, where m is the slope, b is the y-intercept, x is the input, and f(x) is the output.

A parent function is the simplest function of a group of functions. When you transform a parent function, the parent graph also is transformed.

Step 1 Write the transformed function.

Replace the input, x, in the parent function with 3x. Write the new function:

f (3x) � � 1

Then, simplify to f (3x) � .

Step 2 Graph f(x) � 2x � 1 and f(3x) � 6x � 1.

Step 2 Analyze the graphs.

The slope of f(x) � 2x � 1 is .

The slope of f(3x) � 6x � 1 is .

The y-intercept is in both functions.

So, the graph of f(3x) rises 3 times as fast as the parent graph of f(x). The rate of change of f (3x) is 3 times as great as the rate of change of the parent function f (x).

Step 1

Step 2

Step 3

So, the graph of the transformed function is times as steep as the graph of the parent function.

Solution

y

f(x)

f(3x)

0 x1

23

4567

89

�5

�4�3�2

�1

�6

�7

�8�9

1 2 3 4 5 6�4 �3 �2 �1�5�6●

Multiplying the output of f(x) by a transforms the function to af(x) and changes the slope and y-intercept of the graph.

Multiplying the input of f(x) by b transforms the function to f(bx) and changes the slope of the parent graph.

Subtracting c from the input of f(x) transforms the function to f(x � c) and changes the y-intercept as the graph shifts horizontally. The x-intercept changes by the value of c.

Adding d to the output of f(x) transforms the function to f(x) � d and changes the y-intercept as the graph shifts vertically. The y-intercept changes by the value of d.

9781609794408_TX_ALG_EOC_SE_Book_8-2018.indb 929781609794408_TX_ALG_EOC_SE_Book_8-2018.indb 92 8/27/2018 4:49:13 PM8/27/2018 4:49:13 PMProcess BlackProcess Black

SAMPLE

LESSON 16 • EFFECTS ON THE GRAPH WHEN A LINEAR FUNCTION IS CHANGED

masteryeducation.com 93Chapter 3 | Graphs of Linear Equations, Inequalities, and Systems of EquationsCopying is prohibited.

The parent function f(x) � 2x � 1 is transformed to 3f(x). Explain the effect on the parent graph.

Problem 2

Step 1 Write the new function.

Multiply the output, 2x � 1, by 3.

Write the new function:

3f(x) � 3(2x � 1)

Simplify:

3f (x) �

Step 1 Graph both functions.

y

f(x)

3f(x)

0 x1

23

4567

89

�5

�4�3�2

�1

�6

�7

�8�9

1 2 3 4 5 6�4 �3 �2 �1�5�6●


For f(x) � 2x � 1, the slope is and the y-intercept is �1.

For 3f(x) � 6x � 3, the slope is and the y-intercept is .

For both f(x) and 3f(x), notice the y-intercept changes, but the x-intercept does not change.

So, the graph of 3f(x) rises 3 times as fast as the parent graph of f(x) because of the change in slope. The graph of 3f(x) is stretched vertically, away from the x-axis, compared to the graph of f(x).

Step 1

Step 2

Step 3

The effect on the graph of the parent function f(x) � 2x � 1 is that the transformed graph of 3f(x) is times as steep as the graph of f(x). The graph of 3f(x) is 3 times farther from the than the graph of f(x).

Solution


SAMPLE

x LESSON 16 • EFFECTS ON THE GRAPH WHEN A LINEAR FUNCTION IS CHANGED


Step 1 Write the new functions.

For f(x) � 2, add 2 to the parent function. Remember to add 2 to both sides of the equation.

Write the new function:

f(x) � 2 � (2x � 1) � 2

Simplify:

f(x) � 2 � 2x � 1

For f(x) � 3, subtract 3 from the parent function (both sides of the equation). Write the new function:

f(x) � 3 � (2x � 1) � 3

Simplify:

f(x) � 3 � 2x � 4

Step 2 Graph both of the functions.


The graph of f (x) intercepts the y-axis at .

The graph of f (x) � 2 intercepts the y-axis at .

The graph of f (x) � 3 intercepts the y-axis at .

The slope of all three graphs is the same, and the y-intercept changes by the same value as the number being added to or subtracted from f (x).

Note that transforming f (x) to f (x) � d shifts the entire graph vertically up or down.

Step 1

Step 2 y

0 x1

23

4567

89

�5

�4�3�2

�1

�6

�7

�8�9

1 2 3 4 5 6�4 �3 �2 �1�5�6

f(x)

f(x) � 2

f(x) � 3●

●

●

Step 3

A parent function is given as f(x) � 2x � 1. Explain the difference between the parent graph and the graphs of f(x) � 2 and f(x) � 3.

Problem 3

The effect of transforming the parent function f(x) � 2x � 1 is that f(x) � 2 is translated 2 units up from the graph of f(x) and the graph of f(x) � 3 is translated 3 units from the graph of f(x).

Solution


SAMPLE


masteryeducation.com 95Chapter 3 | Graphs of Linear Equations, Inequalities, and Systems of EquationsCopying is prohibited.

Step 1 Write the new functions.

For f(x � 2), replace x in the parent function with x � 2. Write the new function:

� 2(x � 2) � 1

Simplify:

f(x � 2) �

For f(x � 3), replace x in the parent function with x � 3. Write the new function:

f(x � 3) �

Simplify:

f(x � 3) �

Step 2 Graph both of the functions.


The graph of f(x) intercepts the x-axis at 1 __ 2 .

The graph of f(x � 2) intercepts the x-axis at .

The graph of f(x � 3) intercepts the x-axis at .

The slope of all three graphs is the , and the

x-intercept changes by the same value as .

Step 1

Step 2y

0 x1

23

4567

89

�5

�4�3�2

�1

�6

�7

�8�9

1 2 3 4 5 6�4 �3 �2 �1�5�6

f(x � 2)

f(x � 3)

f(x)

● ●●

Step 3

A parent graph of f(x) � 2x � 1 is transformed to f(x � 2) and to f(x � 3). Explain the differences between the graphs.

Problem 4

The effect of transforming the parent function is that the graph of f(x � 2) is translated 2 units from the graph of f(x) and the graph of f(x � 3) is translated 3 units from the graph of f(x).

Solution

Transforming f(x) to f(x � c) shifts the entire graph horizontally left or right. Determining the direction of the shift can be counterintuitive because of the subtraction sign in f(x � c). For example, given f(x � 3), remember the subtraction sign. It actually is a positive 3 being subtracted, and the graph shifts right.


SAMPLE

x LESSON 16 • EFFECTS ON THE GRAPH WHEN A LINEAR FUNCTION IS CHANGED


SHORT RESPONSE QUESTIONS1. The function f(x) graphed below represents the height, in meters,

of a roller coaster beginning its initial ascent.

Part A The function is transformed to f( 1 __ 2 x), and the new function is graphed. How does the graph change, and how does this affect the meaning of the graph?

Part B The original function is transformed to f(x) � 2, and the new function is graphed. How does the graph change, and how does this affect the meaning of the graph?

2. Sam and Rachel both invest in separate interest-bearing savings accounts, as shown in the graph below. Explain the meaning of the y-intercept and the slope of each account. Then, use function notation to tell how Sam’s graph could be transformed so that it equals Rachel’s graph.

100

200

300

500

400

543210

y

x

Time in Years

Acc

ount

Bal

ance

in D

olla

rs

Sam

Rachel

SavingsAccount Balances

2468

101214

1820

16

1 2 3 4 9 1080

y

x6

Height ofRoller Coaster


SAMPLE

masteryeducation.com


97Chapter 3 | Graphs of Linear Equations, Inequalities, and Systems of EquationsCopying is prohibited.

PRACTICE

1 A parent function f(x) � 16x � 6 is transformed to f(2x). Which statement is true about how the graph of f(2x) changed from the graph of f(x)?

A It got steeper, and the x-intercept changed.

B It got steeper, and the y-intercept changed.

C It got flatter, and the x-intercept changed.

D It got flatter, and the y-intercept changed.

[TEKS A.3(E), DOK 2]

2 The function f(x) � 0.4x � 1,000 represents the number of bacteria present on a test slide after x hours. The function is transformed so that the y-intercept is reduced to 500. Which statement best describes how the original function changed?

F The initial number of bacteria decreased, and f(x) transformed to f(x � 500).

G The initial number of bacteria increased, and f(x) transformed to f(x � 500).

H The initial number of bacteria decreased, and f(x) transformed to f(x) � 500.

J The initial number of bacteria increased, and f(x) transformed to f(x) � 500.

[TEKS A.1(F), A.3(E), DOK 2]

3 A parent function g(x) � 1 _ 2 x � 3 is transformed to 2g(x). What is the y-intercept of 2g(x)?

A 2

B 3

C 5

D 6 [TEKS A.3(E), DOK 2]

4 A parent function f(x) � 3x � 5 is transformed to f(x � 4). Which statement about the graph of f(x � 4) is true?

F It shifted right from the parent graph, and the y-intercept increased to 7.

G It shifted left from the parent graph, and the y-intercept increased to �1.

H It shifted left from the parent graph, and the y-intercept increased to 7.

J It shifted right from the parent graph, and the y-intercept increased to �1.


5 A parent function f(x) � 5x is transformed to 2f(x � 3). What are the slope and the y-intercept of 2f(x � 3)?

A slope � 5 and y-intercept � �15

B slope � 5 and y-intercept � �30

C slope � 10 and y-intercept � �15

D slope � 10 and y-intercept � �30


DIRECTIONS Read each question and choose the best answer.Then circle the letter for the correct answer.


SAMPLE

Algebra I | End-of-Coursemasteryeducation.com98

LESSON 16 • EFFECTS ON THE GRAPH WHEN A LINEAR FUNCTION IS CHANGEDx

Copying is prohibited.

ASSESSMENT

1 Jim charges customers a flat fee of $20 plus an additional $7 per hour to mow lawns. Which would be the equation representing how much Jim charges to mow a lawn that takes x hours if his flat fee increased by 50%?

A y � 7x � 10

B y � 7x � 20

C y � 7x � 30

D y � 7x � 40

[TEKS A.1(A), A.3(E), DOK 1]

2 A parent function f(x) � 12x is transformed to f(x � 6). Which statement is true about how the graph of f(x � 6) changed from the graph of f(x)?

F The graph of f(x � 6) shifted left.

G The graph of f(x � 6) shifted right.

H The graph of f(x � 6) shifted up.

J The graph of f(x � 6) shifted down.


3 The function f(x) � 12x represents the number of drops of water dripping from a leaky faucet after x minutes. The function f(x) is transformed to show a rate of change that is triple the previous rate. What is the slope of the transformed function?

A 4

B 12

C 15

D 36


4 A parent function f(x) � �3x � 4 is transformed to 2f(x). What are the slope and the y-intercept of 2f(x)?

F slope � �6 and y-intercept � �4

G slope � �3 and y-intercept � �4

H slope � �6 and y-intercept � �8

J slope � �3 and y-intercept � �8


5 The cost of running a business is represented by f(x) � �0.46x � 1,050. The original function is transformed to show half the amount of startup costs. When the new function is graphed, what will the new y-intercept be?

Record your answer and fill in the bubbles on the grid below. Be sure to use the correct place value.


DIRECTIONS Read each question and choose the best answer.Then circle the letter for the correct answer.


SAMPLE

Documents

CHAPTER 3 LESSON 16 Effects on the Graph When a Linear