I can graph and transform absolute-value functions. Entry task 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 5. f(x) = x + 2; vertical stretch by a factor of 4 g(x) = 2x + 8

2.7 Absolute Value Functions and Graphs I can graph and transform absolute-value functions. Entry Task 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 Success Criteria: Identify the transformation of function equations Without graphing, identify transformation from equations Be able to quick sketch a graph from an equation

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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 f(x) = |x| f(x) g(x) = f(x) + k g(x) = |x| – 5 Substitute. g(x) 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 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 ) f(x) g(x) g(x) = |x – (–1)| = |x + 1| 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).

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) f(x) = |x| g(x) = f(x) + k g(x) g(x) = |x| – 4 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).

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| f(x) g(x) = f(x – h) g(x) g(x) = |x – 2| = |x – 2| 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 f(x) 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. 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) f(x) g(x) = |x – 4| + (–2) g(x) = |x – 4| – 2 The graph confirms that the vertex is (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 3A: Transforming Absolute-Value Functions Perform the transformation. Then graph. Reflect the graph. f(x) =|x – 2| + 3 across the y-axis. Take the opposite of the input value. g(x) = f(–x) f g g(x) = |(–x) – 2| + 3 The vertex of the graph g(x) = |–x – 2| + 3 is (–2, 3).

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

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

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

Assignment #17 Pg 111 #18-27 On Your Yellow Sheet FILL IN: D: What did you DO today? L: What did you LEARN? I: What was INTERESTING? Q: What QUESTIONS do you have?

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