Exercise 6 Page 160

Let's graph $r(x)$ first and then we can compare it to the graph of $f(x).$

Graphing $r(x)$

To graph the function without going through the entire process of transforming the parent function, we can make a table of values. Then we only need to plot the ordered pairs that we find.

$x$	$\left|x\right|+5$	Simplify	$r(x)=|x|+5$
$\text{-}5$	$\left|\text{-}5\right|+5$	$5+5$	$10$
$\text{-}3$	$\left|\text{-}3\right|+5$	$3+5$	$8$
$\text{-}1$	$\left|\text{-}1\right|+5$	$1+5$	$6$
$0$	$\left|0\right|+5$	$0+5$	$5$
$1$	$\left|1\right|+5$	$1+5$	$6$
$3$	$\left|3\right|+5$	$3+5$	$8$
$5$	$\left|5\right|+5$	$5+5$	$10$

Now we can plot these ordered pairs on a coordinate plane and connect them to get the graph of $r(x).$ Note that $r(x)$ is a transformation of $f(x)$ and the graph of $f(x)=|x|$ is V-shaped. Therefore, $r(x)$ will also be a V-shaped graph.

To find the domain and range of an absolute value function, we need to think about where the vertex of the function is located. Because this type of function will always have the same basic V-shape, the $y\text{-}$value of the vertex is the minimum or maximum of the range. The minimum of the given function is $5$ and it will continue increasing indefinitely. The domain of an absolute value function will usually be all real numbers, unless specific restrictions have been imposed upon the function.

Comparing the Functions

To compare our graph to the graph $f(x)=|x|,$ let's draw them on one coordinate plane.

As we can see, the graph of $r(x)$ is a vertical translation $5$ units up of the graph $f(x).$