Objective Transform polynomial functions.

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

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.

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.

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.

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.

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.

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.

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.

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) = 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).

Example 3B: 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) = 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).

g(x) is a vertical compression of f(x). Check It Out! Example 3a Let f(x) = 16x4 – 24x2 + 4. 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).

g(x) is a horizontal stretch of f(x). Check It Out! Example 3b Let f(x) = 16x4 – 24x2 + 4. 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).

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

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. 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 – 36x2 + 108x – 1

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

Example 5: Consumer Application The 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 g(x) = 0.1x3 + 0.2x2 + 0.3x + 130 + 20 g(x) = 0.1x3 + 0.2x2 + 0.3x + 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.

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) = 0.01x3 + 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). g(x) = 0.01(x – 5)3 + 0.7(x – 5)2 + 0.4(x – 5) + 120 g(x) = 0.01x3 + 0.55x2 – 5.85x + 134.25 Step 2 Interpret the transformation. The transformation represents the number of sales since March.

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

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) = 0.02x3 + 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