1. Graph and transform absolute-value functions.
Objective
Graph and transform absolute-value functions.
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. Vertical: g(x) = f(x) + k Horizontal: g(x) = f(x – h)
The general forms for translations are
Vertical:
g(x) = f(x) + k
Horizontal:
g(x) = f(x – h)
Remember!

4. Perform the transformation on f(x) = |x|
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
f(x)
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).

5. Perform the transformation on f(x) = |x|
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) = |x – (–1)| = |x + 1|
g(x)
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).

6. Let g(x) be the indicated transformation of f(x) = |x|
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|
f(x)
g(x) = f(x) + k
g(x) = |x| – 4
g(x)
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).

7. Perform the transformation on f(x) = |x|
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|
f(x)
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).
g(x)

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

9. 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.
The graph confirms that the vertex is (–1, –3).

10. 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 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.
The graph confirms that the vertex is (4, –2).

11. 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!

12. Take the opposite of the input value.
Perform the transformation. Then graph.
Reflect the graph. f(x) =|x – 2| + 3 across the y-axis. f g
g(x) = f(–x)
Take the opposite of the input value.
g(x) = |(–x) – 2| + 3
The vertex of the graph
g(x) = |–x – 2| + 3 is (–2, 3).

13. 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).

14. Stretch the graph. f(x) = |4x| – 3 horizontally by a factor of 2.
g(x) = f( x) f g
Substitute 2 for b.
g(x) = | (4x)| – 3
Simplify.
g(x) = |2x| – 3
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).

15. 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|

16. 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

17. 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