2.7 Use Absolute Value Functions and Transformations

Absolute value function

An absolute value function is f(x) = | x |.

Vertex of an absolute value graph

The highest or lowest point on the graph of an absolute value function

Transformation

A transformation changes a graph's size, shape, position or orientation.

Translation

A transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape or orientation

Reflection

When a = -1, the graph y = a| x | is a reflection of the graph of y = | x | in the x-axis.

PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS

The parent function for the family of all absolute value functions is y = | x |. The graph of y = | 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 of an absolute value graph_. The vertex of the graph y = | x | is (_0_, _0_).

TRANSFORMATIONS OF GENERAL GRAPHS

For | a | > 1, the graph is vertically _stretched_ and y = a | x | is _narrower_ than the graph of y = | x |.

For | a | < 1, the graph is vertically _shrunk_ and y = a | x | is _wider_ than the graph of
y = | x |.

Example 1

Graph functions of the form y = a | x |

Graph (a) y = x and (b) y = -2 | x |. Compare each graph with the graph of y = | x | .

a. The graph of y = | x | is the graph of y = | x | vertically _shrunk_ by a factor of _{___ .} The graph has a vertex (_0_, _0_) and passes through (_3_, _1_) and (_-3_, _1_).

b. The graph of y = -2 | x | is the graph of y = | x | vertically _stretched_ by a factor of _2_ and then _reflected_ in the x-axis. The graph has a vertex of (_0_, _0_) and passes through (_1_, _-2_) and (__-1_, _-2_).

Graph the function. Compare the graph with y = | x |.

1. y = 3 | x |

vertically stretched by a factor of 3

Example 2

Graph a function of the form y = a| x - h | + k

Graph y = -3 | x + 2 | - 1. Compare the graph with the graph of y = | x |.

Solution

1. Identify and plot the vertex, (h, k) = (_-2_, _-1_).

To identify the vertex, rewrite the given function as
y = -3 | x - (-2) | + (-1). So, h = -2 and k = -1.

2. Plot another point on the graph such as (_-1_, _-4_). Use symmetry to pot a third point, (_-3_, _-4_).

3. Connect the points with a _V-shaped_ graph.

4. Compare with y = | x |. The graph of y = -3 | x + 2 | - 1 is the graph of y = | x | first stretched _vertically_ by a factor of _3_, then reflected in the _x-axis_ , and finally translated _left 2_ units and _down 1_ unit.

Example 3

Write an absolute value equation

Write an equation of the graph shown.

Solution

The vertex of the graph is (_3_, _2_). So, the equation has the form y = a | x - _3_| + _2_. Substitute the coordinates of the point (_1_, _-2_) into the equation and solve for a.

_-2_ = a | _1 - 3_ | + _2_ Substitute for x and for y.

_-2_ = a Solve for a.

An equation for the graph is y = _-2 | x - 3 | + 2_.

Complete the following exercises.

2. Graph the function y = - | x - 1 | - 2. Compare the graph with the graph of

y = | x |.

shrunk vertically by a factor of , reflected in the x-axis, translated right 1 unit, and down 2 units.

3. Write an equation of the graph shown.

y = 2 | x + 1 | + 3

TRANSFORMATIONS OF GENERAL GRAPHS

The graph of y = a · f(x - h) + k can be obtained from the graph of y = f(x) by performing these steps:

Step 1 Stretch or shrink the graph of y = f(x) by a factor of | a | if | a | ¹ 1. If | a | > 1, _stretch_ the graph. If | a | < 1, _shrink_ the graph.

Step 2 Reflect the resulting graph from Step 1 in the x-axis if _a< 0_.

Step 3 Translate the resulting graph from Step 2 _horizontally_ h units and _vertically_ k units.

Example 4

Apply transformations to a graph

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

a. y = · f(x)

b. y = -f(x - 1) + 2

Solution

a. The graph of y = · f(x) is the graph of y = f(x) shrunk _vertically_ by a factor of _{___}. To draw the graph, multiply the y-coordinate of each labeled point on the graph of
y = f(x) by _{___} and connect their images.

b. The graph of y = - f(x - 1) + 2 is the graph of y = f(x) _reflected_ in the x-axis, then translated _right 1_ unit and _up 2_ units. To draw the graph, first reflect the labeled points and connect their images. Then translate and connect these points to form the final image.