1. Objective
Transform polynomial functions.

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

3. Example 1A: Translating a Polynomial Function
For f(x) = x3 – 6, write the rule for each function and sketch its graph.
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.

4. Example 1B: Translating a Polynomial Function
For f(x) = x3 – 6, write the rule for each function and sketch its graph.
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.

5. 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.

6. 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.

7. Reflect f(x) across the x-axis.
Example 2A: Reflecting Polynomial Functions
Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation.
Reflect f(x) across the x-axis.
g(x) = –f(x)
g(x) = –(x3 + 5x2 – 8x + 1)
g(x) = –x3 – 5x2 + 8x – 1
Check Graph both functions. The graph appears to be a reflection.

8. Reflect f(x) across the y-axis.
Example 2B: Reflecting Polynomial Functions
Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation.
Reflect f(x) across the y-axis.
g(x) = f(–x)
g(x) = (–x)3 + 5(–x)2 – 8(–x) + 1
g(x) = –x3 + 5x2 + 8x + 1
Check Graph both functions. The graph appears to be a reflection.

9. Reflect f(x) across the x-axis.
Check It Out! Example 2a
Let f(x) = x3 – 2x2 – x + 2. Write a function g that performs each transformation.
Reflect f(x) across the x-axis.
g(x) = –f(x)
g(x) = –(x3 – 2x2 – x + 2)
g(x) = –x3 + 2x2 + x – 2
Check Graph both functions. The graph appears to be a reflection.

10. Reflect f(x) across the y-axis.
Check It Out! Example 2b
Let f(x) = x3 – 2x2 – x + 2. Write a function g that performs each transformation.
Reflect f(x) across the y-axis.
g(x) = f(–x)
g(x) = (–x)3 – 2(–x)2 – (–x) + 2
g(x) = –x3 – 2x2 + x + 2
Check Graph both functions. The graph appears to be a reflection.

11. Example 3A: Compressing and Stretching Polynomial Functions
Let f(x) = 2x4 – 6x Graph f and g on the same coordinate plane. Describe g as a transformation of f.
g(x) = f(x) 1 2
g(x) = (2x4 – 6x2 + 1) 1 2
g(x) = x4 – 3x2 + 1 2
g(x) is a vertical compression of f(x).

12. Example 3B: Compressing and Stretching Polynomial Functions
Let f(x) = 2x4 – 6x Graph f and g on the same coordinate plane. Describe g as a transformation of f.
g(x) = f( x) 1 3
g(x) = 2( x)4 – 6( x)2 + 1 1 3
g(x) = x4 – x2 + 1 2 81 3
g(x) is a horizontal stretch of f(x).

13. g(x) is a vertical compression of f(x).
Check It Out! Example 3a
Let f(x) = 16x4 – 24x Graph f and g on the same coordinate plane. Describe g as a transformation of f. 1 4
g(x) = f(x)
g(x) = (16x4 – 24x2 + 4) 1 4
g(x) = 4x4 – 6x2 + 1
g(x) is a vertical compression of f(x).

14. g(x) is a horizontal stretch of f(x).
Check It Out! Example 3b
Let f(x) = 16x4 – 24x Graph f and g on the same coordinate plane. Describe g as a transformation of f. 1 2
g(x) = f( x)
g(x) = 16( x)4 – 24( x)2 + 4 1 2
g(x) = x4 – 3x2 + 4
g(x) is a horizontal stretch of f(x).

15. 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) 1 3
g(x) = (6(x – 2)3 – 3) 1 3
g(x) = 2(x – 2)3 – 1

16. 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

17. 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. 1 2
g(x) = f(x – 3) 1 2
g(x) = (8(x – 3)3 – 2 1 2
g(x) = 4(x – 3)3 – 1
g(x) = 4x3 – 36x x – 1

18. Check It Out! Example 4b
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 and move the x-intercept 4 units left.
g(x) = –f(x + 4)
g(x) = –6(x + 4)3 – 3
g(x) = –8x3 – 96x2 – 384x – 510

19. Example 5: Consumer Application
The number of skateboards sold per month can be modeled by f(x) = 0.1x x x + 130, where x represents the number of months since May. Let g(x) = f(x) 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
g(x) = 0.1x x x
g(x) = 0.1x x x + 150
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.

20. g(x) = 0.01(x – 5)3 + 0.7(x – 5)2 + 0.4(x – 5) + 120
Check It Out! Example 5
The number of bicycles sold per month can be modeled by f(x) = 0.01x x x + 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).
g(x) = 0.01(x – 5) (x – 5) (x – 5) + 120
g(x) = 0.01x x2 – 5.85x
Step 2 Interpret the transformation.
The transformation represents the number of sales since March.

21. 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

22. Lesson Quiz: Part II 2.
Write a function that reflects f(x) = 2x3 + 1 across the x-axis and shifts it 3 units down.
h(x) = –2x3 – 4 3.
The number of videos sold per month can be modeled by f(x) = 0.02x x x + 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.
0.02x x x + 110; vertical shift 15 units down; decrease of 15 units per month