Chapter 3

3-8 transforming polynomial functions

SAT Problem of the day Lines l and m are perpendicular lines that

intersect at the origin. If line l passes through the point (2,-1), then line m must pass through which of the following points?

A) (0,2) B)(1,3) C)(2,1) D)(3,6) E)(4,0)

solution Right Answer: D

Objectives Transform polynomial functions.

Transforming polynomial functions You can perform the same transformations

on polynomial functions that you performed on quadratic and linear functions.

Transforming Polynomial functions

Example#1 Translating polynomial For f(x) = x3 – 6, write the rule for

each function and sketch its graph. g(x) = f(x) – 2 Solution:

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

This is a vertical translation.

Example#1 continue

Example#2 For f(x) = x3 – 6, write the rule for

each function and sketch its graph.

h(x) = f(x + 3)

Solution:

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

This is a horizontal translation.

Example#2 continue

Example#3 For f(x) = x3 + 4, write the rule for

each function and sketch its graph. g(x) = f(x) – 5 Solution:

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

This is a vertical translation.

Example#3 continue

Student guided practice Do problems 1 and 4 in your book page 207

Reflecting polynomial functions Example#4 Let f(x) = x3 + 5x2 – 8x + 1. Write a

function g that performs each transformation.

Reflect f(x) across the x-axis. Solution : g(x) = –f(x) g(x) = –(x3 + 5x2 – 8x + 1) g(x) = –x3 – 5x2 + 8x – 1

Example#5 Let f(x) = x3 + 5x2 – 8x + 1. Write a

function g that performs each transformation.

Reflect f(x) across the y-axis. Solution: g(x) = f(–x) g(x) = (–x)3 + 5(–x)2 – 8(–x) + 1 g(x) = –x3 + 5x2 + 8x + 1

Student guided practiceDo problems 5 and 6 in your book page 207

Do compressions/stretches Let f(x) = 2x4 – 6x2 + 1. Graph f and g

on the same coordinate plane. Describe g as a transformation of f.

Solution: g(x) = 1/2f(x)

g(x) = 1/2 (2x4 – 6x2 + 1)

g(x) = x4 – 3x2 + 1/2

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

Example continue

Example 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( 1/3 x) Solution: g(x) = 2( 1/3x)4 – 6(1/3x)2 + 1 g(x) = 2/81x4 – 2/3 x2 + 1 g(x) is a horizontal stretch of f(x).

Student guided practice Do problems 7-9

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 1/3 , and shift 2 units right.

Solution: g(x) = 1/3f(x – 2) g(x) = 1/3(6(x – 2)3 – 3) g(x) = 2(x – 2)3 – 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 y-axis and shift 2

units down. Solution: g(x) = f(–x) – 2 g(x) = (6(–x)3 – 3) – 2 g(x) = –6x3 – 5

Student guided practice Do problems 10-12 pg. 207

Homework!! Do problems 14-20 page 207 and 208 in your

book

Closure Today we learn about transforming polynomial Next class we are going to learn about

Exponential functions , growth, and decay