Objective Graph and transform |Absolute-Value | functions. Identify characteristics of absolute-value functions and their graphs.

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 vertex is located at (h, k) has a negative sign in it) Remember! y = a|x – h| + k The vertex is located at (h, k) Stretches/Compresses Makes Narrow or Wide Translates left or right Translates up or down h Opposite sign k Same sign h is the x-coordinate of the vertex point (remember the formula has a negative sign in it) k is the y-coordinate of the vertex point a is the “slope” of the line

y = |x| + 1 Axis of symmetry Vertex

y = |x – 4| Axis of symmetry Vertex

f(x) = 3|x| Axis of symmetry x Vertex

Example 1: Translating Absolute-Value Functions Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 5 units down f(x) f(x) = |x| g(x) = f(x) + k g(x) 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).

You do it! Example 1 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) f(x) = |x| g(x) = f(x) + k g(x) g(x) = |x| – 4 Substitute.

Example 2: Translating Absolute-Value Functions Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 1 unit left f(x) f(x) = |x| g(x) g(x) = f(x – h ) g(x) = |x – (–1)| = |x + 1|

You do it! Example 2 Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 2 units right f(x) f(x) = |x| g(x) = f(x – h) g(x) g(x) = |x – 2| = |x – 2|

Example 3: Translations of an Absolute-Value Function Translate f(x) = |x| so that the vertex is at (–1, –3). Then graph. f(x) g(x) = |x – h| + k g(x) = |x – (–1)| + (–3) g(x) = |x + 1| – 3 g(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.

Translate f(x) = |x| so that the vertex is at (4, –2). Then graph. You Do IT! Example 3 Translate f(x) = |x| so that the vertex is at (4, –2). Then graph. g(x) f(x) g(x) = |x – h| + k g(x) = |x – 4| + (–2) g(x) = |x – 4| – 2 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.

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

Example 5: 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 g(x) f(x) 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).

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