algebra 2 unit 5.9

UNIT 5.9 TRANFORMING UNIT 5.9 TRANFORMING POLYNOMIAL FUNCTIONSPOLYNOMIAL FUNCTIONS

Warm UpLet g be the indicated transformation of f(x) = 3x + 1. Write the rule for g.

g(x) = 3x – 2 1. horizontal translation 1 unit right

g(x) = 12x + 1

2. vertical stretch by a factor of 2

3. horizontal compression by a factor of 4

g(x) = 6x + 2

Transform polynomial functions.

Objective

You can perform the same transformations on polynomial functions that you performed on quadratic and linear functions.

For f(x) = x3 – 6, write the rule for each function and sketch its graph.

Example 1A: Translating a Polynomial Function

g(x) = f(x) – 2

g(x) = (x3 – 6) – 2

g(x) = x3 – 8

To graph g(x) = f(x) – 2, translate the graph of f(x) 2 units down.

This is a vertical translation.

For f(x) = x3 – 6, write the rule for each function and sketch its graph.

Example 1B: Translating a Polynomial Function

h(x) = f(x + 3)

h(x) = (x + 3)3 – 6

To graph h(x) = f(x + 3), translate the graph 3 units to the left.

This is a horizontal translation.

Check It Out! Example 1a

For f(x) = x3 + 4, write the rule for each function and sketch its graph.

g(x) = f(x) – 5

g(x) = (x3 + 4) – 5

g(x) = x3 – 1

To graph g(x) = f(x) – 5, translate the graph of f(x) 5 units down.

This is a vertical translation.

Check It Out! Example 1b

For f(x) = x3 + 4, write the rule for each function and sketch its graph.

g(x) = f(x + 2)

g(x) = (x + 2)3 + 4

g(x) = x3 + 6x2 + 12x + 12

To graph g(x) = f(x + 2), translate the graph 2 units left.

This is a horizontal translation.

Example 2A: Reflecting Polynomial Functions

Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation.

g(x) = –f(x)

Reflect f(x) across the x-axis.

g(x) = –(x3 + 5x2 – 8x + 1)

g(x) = –x3 – 5x2 + 8x – 1

Check Graph both functions. The graph appears to be a reflection.

Example 2B: Reflecting Polynomial Functions

Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation.

g(x) = f(–x)

Reflect f(x) across the y-axis.

g(x) = (–x)3 + 5(–x)2 – 8(–x) + 1

g(x) = –x3 + 5x2 + 8x + 1



Let f(x) = x3 – 2x2 – x + 2. Write a function g that performs each transformation.

g(x) = –f(x)

Reflect f(x) across the x-axis.

g(x) = –(x3 – 2x2 – x + 2)

g(x) = –x3 + 2x2 + x – 2



g(x) = f(–x)

Reflect f(x) across the y-axis.

g(x) = (–x)3 – 2(–x)2 – (–x) + 2

g(x) = –x3 – 2x2 + x + 2


Let f(x) = x3 – 2x2 – x + 2. Write a function g that performs each transformation.

Example 3A: Compressing and Stretching Polynomial Functions

Let f(x) = 2x4 – 6x2 + 1. Graph f and g on the same coordinate plane. Describe g as a transformation of f.

g(x) is a vertical compression of f(x).

g(x) = f(x)12

g(x) = (2x4 – 6x2 + 1) 12

g(x) = x4 – 3x2 + 12

Example 3B: Compressing and Stretching Polynomial Functions


g(x) is a horizontal stretch of f(x).

h(x) = f( x)13

h(x) = 2( x)4 – 6( x)2 + 1 13

13

h(x) = x4 – x2 + 1 281

23



g(x) = f(x)

g(x) is a vertical compression of f(x).

g(x) = (16x4 – 24x2 + 4) 14

g(x) = 4x4 – 6x2 + 1

14



h(x) = f( x)

g(x) is a horizontal stretch of f(x).

12

h(x) = 16( x)4 – 24( x)2 + 4 12

12

h(x) = x4 – 3x2 + 4

Example 4A: Combining Transformations

Write a function that transforms f(x) = 6x3 – 3 in each of the following ways. Support your solution by using a graphing calculator.

Compress vertically by a factor of , and shift 2 units right.1

3

g(x) = f(x – 2)13

g(x) = (6(x – 2)3 – 3)13

g(x) = 2(x – 2)3 – 1

Example 4B: Combining Transformations

Write a function that transforms f(x) = 6x3 – 3 in each of the following ways. Support your solution by using a graphing calculator.

Reflect across the y-axis and shift 2 units down.

g(x) = f(–x) – 2

g(x) = (6(–x)3 – 3) – 2

g(x) = –6x3 – 5

Check It Out! Example 4a Write a function that transforms f(x) = 8x3 – 2 in each of the following ways. Support your solution by using a graphing calculator.Compress vertically by a factor of , and move the x-intercept 3 units right.

12

g(x) = f(x – 3)12

g(x) = (8(x – 3)3 – 212

g(x) = 4(x – 3)3 – 1

g(x) = 4x3 – 36x2 + 108x – 1


Write a function that transforms f(x) = 6x3 – 3 in each of the following ways. Support your solution by using a graphing calculator.Reflect across the x-axis andmove the x-intercept 4 units left.

g(x) = –f(x + 4)

g(x) = –6(x + 4)3 – 3

g(x) = –8x3 – 96x2 – 384x – 510

Example 5: Consumer ApplicationThe number of skateboards sold per month can be modeled by f(x) = 0.1x3 + 0.2x2 + 0.3x + 130, where x represents the number of months since May. Let g(x) = f(x) + 20. Find the rule for g and explain the meaning of the transformation in terms of monthly skateboard sales.

Step 1 Write the new rule.

The new rule is g(x) = f(x) + 20

Step 2 Interpret the transformation.

The transformation represents a vertical shift 20 units up, which corresponds to an increase in sales of 20 skateboards per month.

g(x) = 0.1x3 + 0.2x2 + 0.3x + 130 + 20g(x) = 0.1x3 + 0.2x2 + 0.3x + 150

The number of bicycles sold per month can be modeled by f(x) = 0.01x3 + 0.7x2 + 0.4x + 120, where x represents the number of months since January. Let g(x) = f(x – 5). Find the rule for g and explain the meaning of the transformation in terms of monthly skateboard sales.

Step 1 Write the new rule.

The new rule is g(x) = f(x – 5).

Step 2 Interpret the transformation.

The transformation represents the number of sales since March.

g(x) = 0.01(x – 5)3 + 0.7(x – 5)2 + 0.4(x – 5) + 120g(x) = 0.01x3 + 0.55x2 – 5.85x + 134.25

Check It Out! Example 5

Lesson Quiz: Part I

1. For f(x) = x3 + 5, write the rule for g(x) = f(x – 1) – 2 and sketch its graph.

g(x) = (x – 1)3 + 3

2.

Lesson Quiz: Part II

h(x) = –2x3 – 4

3.

Write a function that reflects f(x) = 2x3 + 1 across the x-axis and shifts it 3 units down.

The number of videos sold per month can be modeled by f(x) = 0.02x3 + 0.6x2 + 0.2x + 125, where x represents the number of months since July. Let g(x) = f(x) – 15. Find the rule for g and explain the meaning of the transformation in terms of monthly video sales. 0.02x3 + 0.6x2 + 0.2x + 110; vertical

shift 15 units down; decrease of 15 units per month

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