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

2. 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).

3. 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

5. y = |x| + 1
Axis of symmetry
Vertex

6. y = |x – 4|
Axis of symmetry
Vertex

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

8. 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).

9. 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.

10. 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|

11. 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|

12. 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.

13. 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.

14. 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!

15. 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).

16. 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).

17. 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.

18. 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

19. 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

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

21. 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.

22. 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)

23. 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.

24. 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

25. 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|

26. 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

27. 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