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 form for translations are Vertical: k Horizontal: h g(x) = |x – (h) | + k Remember!

Example 1A: Translating Absolute-Value Functions 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 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 f(x) = |x| g(x) = f(x – h ) g(x) = |x – (–1)| = |x + 1| Substitute.

Example 1B 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)

Check It Out! Example 1a 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.

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). f(x) g(x)

Check It Out! Example 1b 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.

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). f(x) g(x)

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

The graph confirms that the vertex is (–1, –3). Example 2 Continued 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. f(x) The graph confirms that the vertex is (–1, –3). g(x)

Check It Out! 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

Check It Out! Example 2 Continued 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. g(x) f(x) The graph confirms that the vertex is (4, –2).

Absolute-value functions can also be stretched, compressed, and reflected. Reflection across x-axis: g(x) = –f(x) Reflection across y-axis: g(x) = f(–x) Remember! 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. Reflect the graph. f(x) =|x – 2| + 3 across the y-axis. g(x) = f(–x) Take the opposite of the input value. g(x) = |(–x) – 2| + 3

The vertex of the graph g(x) = |–x – 2| + 3 is (–2, 3). Example 3A Continued The vertex of the graph g(x) = |–x – 2| + 3 is (–2, 3). f g

Example 3B: Transforming Absolute-Value Functions Stretch the graph. f(x) = |x| – 1 vertically by a factor of 2. g(x) = af(x) g(x) = 2(|x| – 1) Multiply the entire function by 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). g(x) f(x)

Example 3C: Transforming Absolute-Value Functions Compress the graph of f(x) = |x + 2| – 1 horizontally by a factor of . Substitute for b. g(x) = |2x + 2| – 1 Simplify.

Example 3C Continued 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). g f

Check It Out! Example 3a Perform the transformation. Then graph. Reflect the graph. f(x) = –|x – 4| + 3 across the y-axis. g(x) = f(–x) Take the opposite of the input value. g(x) = –|(–x) – 4| + 3 g(x) = –|–x – 4| + 3

Check It Out! Example 3a Continued The vertex of the graph g(x) = –|–x – 4| + 3 is (–4, 3). g f

Check It Out! Example 3b Compress the graph of f(x) = |x| + 1 vertically by a factor of . g(x) = a(|x| + 1) g(x) = (|x| + 1) Multiply the entire function by . g(x) = (|x| + ) Simplify.

Check It Out! Example 3b Continued 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, ). f(x) g(x)

Check It Out! Example 3c Stretch the graph. f(x) = |4x| – 3 horizontally by a factor of 2. g(x) = f( x) g(x) = | (4x)| – 3 Substitute 2 for b. g(x) = |2x| – 3 Simplify.

Check It Out! Example 3c Continued 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 g

Perform each transformation. Then graph. Lesson Quiz: Part I Perform each transformation. Then graph. 1. Translate f(x) = |x| 3 units right. f g g(x)=|x – 3|

Perform each transformation. Then graph. Lesson Quiz: Part II Perform each transformation. Then graph. 2. Translate f(x) = |x| so the vertex is at (2, –1). Then graph. f g g(x)=|x – 2| – 1

Lesson Quiz: Part III Perform each transformation. Then graph. 3. Stretch the graph of f(x) = |2x| – 1 vertically by a factor of 3 and reflect it across the x-axis. g(x)= –3|2x| + 3