3.6 Absolute value functions

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions .

Understanding absolute value

Recall that in its basic form $f(x)=\left|x\right|,$ the absolute value function is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.

Graphing an absolute value function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin    in [link] .

[link] shows the graph of $y=2\left|x–3\right|+4.$ The graph of $y=\left|x\right|$ has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at $\left(3,4\right)$ for this transformed function.

Writing an equation for an absolute value function given a graph

Write an equation for the function graphed in [link] .

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 [link] .

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 [link] .

From this information we can write the equation

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

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