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 down

g(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