Warm Up Evaluate each expression for f (4) and f (-3).

1. 5.f(x) = x + 2; vertical stretch by a factor of 4 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 + 1| + 2 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 + 8

2. 2-9: Absolute Value Functions Part 1: Vertical and Horizontal Translations

3. Definitions Absolute value function – an equation containing an absolute value expression. Shaped like a V.

4. f(x) = |x| or y = |x| Examples of absolute value functions: f(x) = |x| + 3 f(x) = |x| - 2 f(x) = 2x + 3 f(x) = |x – 1| f(x) = x3 – 2x + 5 f(x) = |x + 3|

5. Identify the Absolute Value Functions

6. Identify the Absolute Value Functions

7. Writing transformed absolute value functions – vertical

8. Example 1 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 1 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 2 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.

11. Example 2 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)

12. Example 3 Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 3 units up f(x) = |x| g(x) = f(x) + k g(x) = |x| + 3 Substitute. The graph of g(x) = |x| + 3is the graph of f(x) = |x| after a vertical shift of 3 units up. The vertex of g(x) is (0, 3).

13. Example 4 Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 6 units up f(x) = |x| g(x) = f(x) + k g(x) = |x| + 6 Substitute. The graph of g(x) = |x| + 6is the graph of f(x) = |x| after a vertical shift of 6 units up. The vertex of g(x) is (0, 6).

14. Writing transformed absolute value functions – horizontal

15. Example 1 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.

16. Example 1 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)

17. Example 2 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.

18. Example 2 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)

19. Example 3 Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 4 unit left f(x) = |x| g(x) = f(x– h) g(x) = |x – (–4)| = |x + 4| Substitute.

20. Example 4 Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 7 units right f(x) = |x| g(x) = f(x– h) g(x) = |x – 7| = |x –7| Substitute.