Example 3 from Precalculus, 1.6 Absolute Value Functions

Writing an Equation for an Absolute Value Function

Write an equation for the function graphed in Figure 5.

Figure 5

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 6.

Figure 6

We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure 7.

Figure 7

From this information we can write the equation

$\begin{array}{l} f (x) = 2 | x - 3 | - 2, & treating the stretch as a vertical stretch, or \\ f (x) = | 2 (x - 3) | - 2, & treating the stretch as a horizontal compression . \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.

How did it go?

Solution

Your turn