2-9 Absolute Value Functions

Transcription

1 2-9 Absolute Value Functions Warm Up Lesson Presentation Lesson Quiz 2

2 Warm Up 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 ; 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 5. f(x) = x + 2; vertical stretch by a factor of 4 g(x) = 2x + 8

3 Objective Graph and transform absolute-value functions.

4 Vocabulary absolute-value function

5 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).

6 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 absolutevalue functions.

7 Remember! The general forms for translations are Vertical: g(x) = f(x) + k Horizontal: g(x) = f(x h)

8 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).

9 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)

10 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.

11 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). g(x) f(x)

12 4 units down f(x) = x g(x) = f(x) + k g(x) = x 4 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. Substitute.

13 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)

14 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.

15 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)

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

17 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

18 Example 2 Continued The graph of g(x) = x is the graph of f(x) = x after a vertical shift down 3 units and a horizontal shift left 1 unit. g(x) f(x) The graph confirms that the vertex is ( 1, 3).

19 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) g(x) = x 4 2 Substitute.

20 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. f(x) g(x) The graph confirms that the vertex is (4, 2).

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

22 Example 3A: Transforming Absolute-Value Functions Perform the transformation. Then graph. Reflect the graph. f(x) = x across the y-axis. g(x) = f( x) Take the opposite of the input value. g(x) = ( x) 2 + 3

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

24 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

25 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)

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

27 Example 3C Continued The graph of g(x) = 2x is the graph of f(x) = x after a horizontal compression by a factor of. The vertex of g is at ( 1, 1). g f

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

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

30 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.

31 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)

32 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 g(x) = 2x 3 Substitute 2 for b. Simplify.

33 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). g f

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

35 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(x)= x 2 1 g

36 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