CHAPTER 6 6-4 Transforming Functions

OBJECTIVES

Transform functions.

Recognize transformations of functions.

TRANSFORMING FUNCTIONS

In previous lessons, you learned how to transform several types of functions. You can transform piecewise functions by applying transformations to each piece independently. Recall the rules for transforming functions given in the table.

TRANSFORMING FUNCTIONS

EXAMPLE 1: TRANSFORMING PIECEWISE FUNCTIONS

Given f(x) = – 1/2x if x < 0

write the rule g(x), a vertical stretch by a factor of 3.

½ x2 if x ≥ 0

SOLUTION

Each piece of f(x) must be vertically stretched by a factor of 3. Replace every y in the function by 3y, and simplify.

3(– x) if x < 0 1 2

3( x2) if x ≥ 0 1 2

g(x) = 3f(x) =

– x if x < 0 3 2

x2 if x ≥ 0 3 2

CHECK IT OUT!!

Given f(x) = write the rule

for g(x), a horizontal stretch of f(x) by a factor of 2.

x2 if x ≤ 0 x – 3 if x > 0

TRANSFORMING FUNCTIONS

When functions are transformed, the intercepts may or may not change. By identifying the transformations, you can determine the intercepts, which can help you graph a transformed function

TRANSFORMING FUNCTIONS

TRANSFORMING FUNCTIONS

EXAMPLE 2A: IDENTIFYING INTERCEPTS

Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts.

f(x) =–2x – 4 ; g(x) =

SOLUTION

Find the intercepts of the original function y-intercept x-

intercept

–2 = x The y-intercept is –4, and the x-intercept is -2. Note that g(x) is a horizontal stretch of f(x) by a factor of 2. So the y-intercept of g(x) is also –4. The x-intercept is 2(–2), or –4.

f(0) = –2(0) – 4 = – 4 0 = –2x – 4

EXAMPLE 2B: IDENTIFY INTERCEPTS

f(x) = x2 – 1; g(x) = f(–x)

CHECK IT OUT!!!

Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts.

f(x) = x + 4 and g(x) = –f(x) 2 3

The y-intercept is 4, and the x-intercept is –6. Note that g(x) is a reflection of f(x) across the x-axis. So the x-intercept of g(x) is also –6. The y-intercept is –1(4), or –4.

EXAMPLE 3: COMBINING TRANSFORMATIONS

Given f(x) = 1/3(x – 2)2 and g(x) = 2f(x) – 3 and graph g(x).

SOLUTION

Step 1 Graph f(x). The graph of f(x) has y-intercept (0, 4/3) and x-intercept (2, 0).

SOLUTION Step 2 Analyze each transformation one at a

time. The first transformation is a vertical stretch

by a factor of 2. After the vertical stretch, the x-intercept will remain 2, but the y-intercept will be .

The second transformation is a vertical translation of 3 units down. Use a table to shift each identified point down 3 units.

SOLUTION

Intercept Points ( 2, 0) Shifted (2, –3)

8 3

1 3

Step 3 Graph the final result.

CHECK IT OUT!!!

Given f(x) = 2x – 4 and g(x)= – 1/2f(x), graph g(x).

APPLICATION

A movie theater charges $5 for children under 12 and $7.50 for anyone 12 and over. The theater decides to increase its prices by 20%. It charges an additional $0.50 fee for online ticket purchases. Write an equation for the online ticket prices.

STUDENT GUIDED PRACTICE

Do problems 1-7 in your book page 436

HOMEWORK

Do problems 8-17 page 436 and 437