1. 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).

2. 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.

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

4. 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).

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)

6. 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.

7. 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)

8. 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.

9. 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)

10. 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.

11. 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)

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

13. 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

14. 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)

15. 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

16. 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).

17. 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!

18. 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

19. 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

20. 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

21. 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)

22. 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.

23. 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

24. 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

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

26. 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.

27. 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)

28. 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.

29. 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

30. 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|

31. 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

32. 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