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Absolute-Value Functions

Warm upLessonHomework

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

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

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

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

5; 2

7; 5

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 downg(x) = 2x 1

g(x) = 2x + 85. f(x) = x + 2; vertical stretch by a factor of 4

Graph and transform absolute-value functions.

Your Goal Today is

absolute-value function

Vocabulary

An absolute-value function is a function whose rule contains an absolute-value expression. The graph of the parent absolute-value function f(x) = |x| has a V shape with a minimum point or vertex at (0, 0).

The absolute-value parent function is composed of two linear pieces, one with a slope of 1 and one with a slope of 1. In Lesson 2-6, you transformed linear functions. You can also transform absolute-value functions.

The general forms for translations are

Vertical:

g(x) = f(x) + k

Horizontal:

g(x) = f(x h)

Remember!

Example 1A: Translating Absolute-Value Functions

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

5 units down

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).

f(x) = |x|

g(x) = f(x) + k

g(x) = |x| 5

Example 1A 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 1B: Translating Absolute-Value Functions

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

1 unit left

Substitute.

f(x) = |x|

g(x) = f(x h )

g(x) = |x (1)| = |x + 1|

Example 1B Continued

f(x)

g(x)

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).

4 units down

Substitute.

f(x) = |x|

g(x) = f(x) + k

g(x) = |x| 4

Write In Your Notes! Example 1a

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

f(x)

g(x)

Check It Out! Example 1a 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).

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

2 units right

Substitute.

f(x) = |x|

g(x) = f(x h)

g(x) = |x 2| = |x 2|

Write In Your Notes! Example 1b

f(x)

g(x)

Check It Out! Example 1b 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).

Because the entire graph moves when shifted, the shift from f(x) = |x| determines the vertex of an absolute-value graph.

Example 2: Translations of an Absolute-Value Function

Translate f(x) = |x| so that the vertex is at (1, 3). Then graph.

g(x) = |x h| + k

g(x) = |x (1)| + (3) Substitute.

g(x) = |x + 1| 3

Example 2 Continued

The graph confirms that the vertex is (1, 3).

f(x)

The graph of g(x) = |x + 1| 3 is the graph of f(x) = |x| after a vertical shift down 3 units and a horizontal shift left 1 unit.

g(x)

Write In Your Notes! Example 2

Translate f(x) = |x| so that the vertex is at (4, 2). Then graph.

g(x) = |x h| + k

g(x) = |x 4| + (2) Substitute.

g(x) = |x 4| 2

The graph confirms that the vertex is (4, 2).

Check It Out! Example 2 Continued

g(x)

The graph of g(x) = |x 4| 2 is the graph of f(x) = |x| after a vertical down shift 2 units and a horizontal shift right 4 units.

f(x)

Reflection across x-axis: g(x) = f(x)

Reflection across y-axis: g(x) = f(x)

Remember!

Absolute-value functions can also be stretched, compressed, and reflected.

Vertical stretch and compression : g(x) = af(x)

Horizontal stretch and compression: g(x) = f

Remember!

Example 3A: Transforming Absolute-Value Functions

Perform the transformation. Then graph.

g(x) = f(x)

g(x) = |(x) 2| + 3

Take the opposite of the input value.

Reflect the graph. f(x) =|x 2| + 3 across the y-axis.

g f

Example 3A Continued

The vertex of the graph g(x) = |x 2| + 3 is (2, 3).

g(x) = af(x)

g(x) = 2(|x| 1) Multiply the entire function by 2.

Example 3B: Transforming Absolute-Value Functions

Stretch the graph. f(x) = |x| 1 vertically by a factor of 2.

g(x) = 2|x| 2

Example 3B Continued

The graph of g(x) = 2|x| 2 is the graph of f(x) = |x| 1 after a vertical stretch by a factor of 2. The vertex of g is at (0, 2).

f(x) g(x)

Example 3C: Transforming Absolute-Value Functions

Compress the graph of f(x) = |x + 2| 1 horizontally by a factor of .

g(x) = |2x + 2| 1 Simplify.

Substitute for b.

f

The graph of g(x) = |2x + 2| 1 is the graph of

f(x) = |x + 2| 1 after a horizontal compression by

a factor of . The vertex of g is at (1, 1).

Example 3C Continued

g

Perform the transformation. Then graph.

g(x) = f(x)

g(x) = |x 4| + 3

Take the opposite of the input value.

Reflect the graph. f(x) = |x 4| + 3 across the y-axis.

Write In Your Notes! Example 3a

g(x) = |(x) 4| + 3

The vertex of the graph g(x) = |x 4| + 3 is (4, 3).

Check It Out! Example 3a Continued

fg

Compress the graph of f(x) = |x| + 1 vertically

by a factor of .

Simplify.

Write In Your Notes! Example 3b

g(x) = a(|x| + 1)

g(x) = (|x| + 1)

g(x) = (|x| + )

Multiply the entire function by .

Check It Out! Example 3b Continued

f(x)

g(x)

The graph of g(x) = |x| + is the graph of g(x) = |x| + 1 after a vertical compression by a factor of . The vertex of g is at ( 0, ).

Substitute 2 for b.

Stretch the graph. f(x) = |4x| 3 horizontally by a factor of 2.

g(x) = |2x| 3

Write In Your Notes! Example 3c

Simplify.

g(x) = f( x)

g(x) = | (4x)| 3

Check It Out! Example 3c Continued

g

The graph of g(x) = |2x| 3 the graph of f(x) = |4x| 3 after a horizontal stretch by a factor of 2. The vertex of g is at (0, 3).

f