Warm Up Evaluate each expression for f (4) and f (-3).

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# Warm Up Evaluate each expression for f (4) and f (-3). - PowerPoint PPT Presentation

5. f ( x ) = x + 2; vertical stretch by a factor of 4. Warm Up Evaluate each expression for f (4) and f (-3). 1. f ( x ) = –| x + 1|. –5; –2. 2. f ( x ) = 2| x | – 1. 7; 5. 3. f ( x ) = | x + 1| + 2. 7; 4.

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5.f(x) = x + 2; vertical stretch by a factor of 4

Warm Up

Evaluate each expression for f(4) and f(-3).

1.f(x) = –|x + 1|

–5; –2

2.f(x) = 2|x| – 1

7; 5

3.f(x) = |x + 1| + 2

7; 4

Let g(x) be the indicated transformation of f(x). Write the rule for g(x).

4. f(x) = –2x + 5; vertical translation 6 units down

g(x) = –2x– 1

g(x) = 2x + 8

### 2-9: Absolute Value Functions

Part 1: Vertical and Horizontal Translations

Definitions

Absolute value function – an equation containing an absolute value expression. Shaped like a V.

f(x) = |x| or y = |x|

Examples of absolute value functions:

f(x) = |x| + 3

f(x) = |x| - 2

f(x) = 2x + 3

f(x) = |x – 1|

f(x) = x3 – 2x + 5

f(x) = |x + 3|

Example 1

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

5 units down

f(x) = |x|

g(x) = f(x) + k

g(x) = |x| – 5

Substitute.

The graph of g(x) = |x| – 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).

Example 1 Continued

The graph of g(x) = |x|– 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).

f(x)

g(x)

Example 2

Let g(x) be the indicated transformation of f(x) = |x|. Write the rule for g(x) and graph the function.

4 units down

f(x) = |x|

g(x) = f(x) + k

g(x) = |x| – 4

Substitute.

Example 2 Continued

The graph of g(x) = |x| – 4 is the graph of f(x) = |x| after a vertical shift of 4 units down. The vertex of g(x) is (0, –4).

f(x)

g(x)

Example 3

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

3 units up

f(x) = |x|

g(x) = f(x) + k

g(x) = |x| + 3

Substitute.

The graph of g(x) = |x| + 3is the graph of f(x) = |x| after a vertical shift of 3 units up. The vertex of g(x) is (0, 3).

Example 4

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

6 units up

f(x) = |x|

g(x) = f(x) + k

g(x) = |x| + 6

Substitute.

The graph of g(x) = |x| + 6is the graph of f(x) = |x| after a vertical shift of 6 units up. The vertex of g(x) is (0, 6).

Example 1

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

1 unit left

f(x) = |x|

g(x) = f(x– h)

g(x) = |x – (–1)| = |x + 1|

Substitute.

Example 1 Continued

The graph of g(x) = |x + 1| is the graph of f(x) = |x| after a horizontal shift of 1 unit left. The vertex of g(x) is (–1, 0).

f(x)

g(x)

Example 2

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

2 units right

f(x) = |x|

g(x) = f(x– h)

g(x) = |x – 2| = |x –2|

Substitute.

Example 2 Continued

The graph of g(x) = |x – 2| is the graph of f(x) = |x| after a horizontal shift of 2 units right. The vertex of g(x) is (2, 0).

f(x)

g(x)

Example 3

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

4 unit left

f(x) = |x|

g(x) = f(x– h)

g(x) = |x – (–4)| = |x + 4|

Substitute.

Example 4

Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

7 units right

f(x) = |x|

g(x) = f(x– h)

g(x) = |x – 7| = |x –7|

Substitute.