Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.

114

Transforming Linear Functions Reading Strategies: Identify Relationships

The table shows three types of transformations.

Transformation Translation Rotation Reflection

Description When b in y = mx + b increases (decreases), the line is translated up (down).

When m in y = mx + b increases (decreases), the line becomes more steep (less steep).

When m in y = mx + b is multiplied by −1, the line is reflected across the y-axis.

Graph

Equations f(x) = −3x + 2 g(x) = −3x − 3

f(x) = −3x + 2 g(x) = −1x + 2

f(x) = −3x + 2 g(x) = 3x + 2

Answer the following.

1. Look at the graph of the translation. How many vertical units is f(x) shifted down to g(x)? _________________

Look at the related functions. Subtract the y-intercept of g(x) from the y-intercept of f(x). _________________

2. Look at the graph of the rotation. Which graph is steeper? _________________

Look at the slopes of the related functions. For which function is the absolute value of the slope greater? _________________

3. Look at the graph of the reflection. Which axis is a line of symmetry? _________________

Look at the slopes of the related functions. Are they the same or are they opposite? _________________

Describe the transformation from the graph of f(x) to the graph of g(x).

4. f(x) = 4x − 3, g(x) = 2x − 3 ________________________________________________________

5. f(x) = −2x + 2, g(x) = 2x + 2 ________________________________________________________

6. f(x) = x + 2, g(x) = x + 8 ________________________________________________________

LESSON

6-4

Name ________________________________________ Date __________________ Class __________________

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78

Transforming Linear Functions Reteach

For a linear function f(x) = mx + b, changing the value of b moves the graph up or down.

Description Equation y-intercept or b

Parent function f(x) = x 0 Translate up 2 g(x) = x + 2 2 Translate down 4 h(x) = x − 4 −4

Changing the absolute value of the slope m makes the line more or less steep.

If m is positive, the line goes up from left to right.

If m is negative, the line goes down from left to right.

Predict the change in the graph from f(x) to g(x). Then graph both lines to check your prediction. 1. f(x) = x; g(x) = x + 5 2. f(x) = −3x + 1; g(x) = 3x + 1

________________________________________ ________________________________________

LESSON

6-4

Name ________________________________________ Date __________________ Class __________________

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111

Transforming Linear Functions Practice and Problem Solving: A/B

Identify the steeper line. 1. y = 3x + 4 or y = 6x + 11 2. y = −5x − 1 or y = −2x − 7

________________________________________ ________________________________________

Each transformation is performed on the line with the equation y = 2x − 1. Write the equation of the new line. 3. vertical translation down 3 units 4. slope increased by 4

________________________________________ ________________________________________

5. slope divided in half 6. shifted up 1 unit

________________________________________ ________________________________________

7. slope increased by 50% 8. shifted up 3 units and slope doubled

________________________________________ ________________________________________

A salesperson earns a base salary of $4000 per month plus 15% commission on sales. Her monthly income, f(s), is given by the function f(s) = 4000 + 0.15s, where s is monthly sales, in dollars. Use this information for Problems 9–12. 9. Find g(s) if the salesperson’s commission is lowered to 5%.

_________________________________________________________________________________________

10. Find h(s) if the salesperson’s base salary is doubled.

_________________________________________________________________________________________

11. Find k(s) if the salesperson’s base salary is cut in half and her commission is doubled.

_________________________________________________________________________________________

12. Graph f(s) and k(s) on the coordinate grid below.

LESSON

6-4

Name ________________________________________ Date __________________ Class __________________

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112

Transforming Linear Functions Practice and Problem Solving: C

Identify the steeper line. 1. y = 2x − 3 or x − 5y = 20 2. x + 10y = 1 or 3x + 20y = 1

________________________________________ ________________________________________

Each transformation is performed on the line with the equation y = 4x − 20. Write the equation of the new line. 3. slope cut in half 4. vertical translation 25 units upward

________________________________________ ________________________________________

5. shifted up 8 units and slope tripled 6. reflection across the y-axis

________________________________________ ________________________________________

Solve. 7. Compare the steepness of the lines whose equations are 8x + y = 1

and −8x + y = 2. Explain your reasoning.

_________________________________________________________________________________________

_________________________________________________________________________________________

8. f(x) is an increasing linear function that passes through the point (4, 0). Show that if written in the form ( ) ,f x mx b= + m > 0 and b < 0.

_________________________________________________________________________________________

_________________________________________________________________________________________

9. A salesperson earns a base salary of $400 per week plus 20% commission on sales. He is offered double his base salary if he’ll accept half his original commission. Graph and label the original deal and the new deal below. Next to the graph, find when the original deal is a better choice. Explain your thinking.

______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________

LESSON

6-4

Name ________________________________________ Date __________________ Class __________________

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113

Transforming Linear Functions Practice and Problem Solving: Modified

A line’s y-intercept is changed from b to b*. State whether the line shifts up or down. The first one is done for you. 1. b = 8 and b* = 2 2. b = −4 and b* = −6 3. b = −1 and b* = 0

________________________ _______________________ ________________________

A line’s slope is changed from m to m*. State whether the line becomes more steep or less steep. The first one is done for you.

4. m = 2 and m* = 3 5. m = 5 and m* = 15

6. m = −4 and m* = −9

________________________ _______________________ ________________________

A taxi charges an initial fee of $3 plus $2 for each mile driven. This is shown in each graph below. For each situation described, draw the new graph. The first one is done for you. 7. The initial fee is decreased to $1. 8. The fee per mile is decreased to $1.

9. The initial fee is eliminated. 10. Both fees are changed to $4.

LESSON

6-4

down

more steep