Use Absolute Value Functions and Transformations

Section 2.7

In Lesson 1-7, you learned that the absolute value of a real number x is defined as follows.

You can also define an absolute value function

, if is positive0, if 0

, if is negative

x xx x

x x

.f x x

Parent Function for Absolute Value Functions

The parent function for the family of all absolute value function is f (x) = |x|.The graph of f (x) = |x| is V-shaped and is symmetric about the y-axis. So, for every point (x, y) on the graph, the point (-x, y) is also on the graph.The highest or lowest point on the graph of an absolute value function is called the vertex.The vertex of the graph f (x) = |x| is (0, 0).

To the left of x = 0, the graph is given by the line y = -x.

To the right of x = 0, the graph is given by the line y = x.

(-2, 2) (2, 2)

vertex

(0, 0)

Transformations

A transformation changes a graph’s size, shape, position, or orientation.A translation is a transformation that shifts a graph horizontally and/or vertically, but does not change the size, shape, or orientation.The graph y = |x – h| + k is a graph of y = |x| translated h units horizontally and k units vertically.The vertex of y = |x – h| + k is (h, k).

y = |x|

(0, 0) h

k(h, k)

y = |x − h| + k

Graph y = | x – 1 | + 3. Compare the graph with the graph of y = | x |.Comparisony = | x – 1 | + 3 translate y = | x | one unit to the right and three units up.

Example 1

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

Stretches and Shrinks

When |a| ≠ 1, the graph of y = a|x| is a vertical stretch or a vertical shrink, depending on whether |a| is less than or greater than 1.

For |a| > 1 For |a| < 1• The graph is

vertically stretched or elongated

• The graph is vertically shrunk or compressed

• The graph of y = a|x| is narrower than the graph of y = |x|.

• The graph of y = a|x| is wider that the graph of y = |x|.

When a = -1, the graph of y = a|x| is a reflection in the x-axis of the graph of y = |x|.When a < 0 but a ≠ -1, the graph of y = a|x| is a vertical stretch or shrink with a reflection in the x-axis of the graph of y = |x|.

Reflections

Compare each graph with the graph y = | x |.Comparison(a) y = 1/3| x | is the graph of y = | x | vertically

shrunk by a factor of 1/3

Example 2

1Graph a and b 2 .3

y x y x

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

(b) Comparison:y = -2| x | is the graph of y = | x | vertically stretched by a factor of 2 and reflected over the x-axis.

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

A graph may be related to a parent function using multiple transformations.The graph of y = a|x – h| + k can involve a vertical stretch or shrink, a reflection, and a translation of the graph of y = |x|.

Multiple Transformations

Compare the graph with the graph of y = | x |.ComparisonThe graph of y = ¼| x + 3 | − 2 is the graph of y = | x | first vertically shrunk by a factor of ¼ then translated 3 units to the left and 2 units down.

Example 31Graph 3 2.4

y x

-8 -6 -4 -2 2 4 6 8

-6

-4

-2

2

4

6

A landscaper sketches the design for a triangular shrub protector on graph paper. Write an equation for the shrub protector.

Example 4

2 4 6 8 10 12 14 16-1

1

2

3

4

5

6

7

8

9

O

The vertex is (5, 6).A point on the graph is either (0, 0) or (10, 0). Now solve for a.

5 6y a x

0 0 5 6a

1.2a

1.2 5 6y x

Transformations of General Graphs

The graph of y = a ∙ f (x – h) + k can be obtained from the graph of any function y = f (x) by performing these steps:1. Stretch or shrink the graph of y = f (x)

vertically by a factor or |a| if |a| ≠ 1.If |a| > 1, stretch the graph.If |a| < 1, shrink the graph.

2. Reflect the resulting graph from #1in the x-axis if a < 0.

3. Translate the resulting graph from #2 horizontally h units and vertically k units.

The graph of a function y = f (x) is shown. Sketch the graph of the given function.

Example 5

(-3, -3)

(0, 0)

(3, -6)

1a3

y f x

(-3, -3)

(0, 0)

(3, -6)

(-3, 1)

(3, -2)

b 1 3y f x

(-3, -3)

(0, 0)

(3, -6)

(-2, 0)

(1, 3)

(4, -3)