Exercise 34 Page 161

First we will go through all the transformations from the graph of $f(x)=|x|$ to the graph of given function. After that we will make a table of values to graph this function.

Transformations

The given equation is a transformation of the parent function $f(x)=|x|.$ We can define each transformation from the inside to the outside. We will start with the transformation closest to $x$ and end with the farthest from $x.$ Let's review our function. First, there is a horizontal translation $4$ units to the left, $\left|x\right|\Rightarrow \left|x+4\right|.$ Remember, typically $|x-h|$ is a horizontal translation $h$ units to the right, while $|x+h|$ is translated $h$ units to the left.

Next, we have a vertical translation $4$ units up, $\left|x+4\right|\Rightarrow \left|x+4\right|+4.$ Remember, in most cases $|x|+k$ is a vertical translation $k$ units up, while $|x|-k$ is translated $k$ units down.

Graphing $c(x)$

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

$x$	$\left|x+4\right|+4$	Simplify	$c(x)$
$\text{-}8$	$\left|\text{-}8+4\right|+4$	$|\text{-}4|+4$	$8$
$\text{-}6$	$\left|\text{-}6+4\right|+4$	$|\text{-}2|+4$	$6$
$\text{-}4$	$\left|\text{-}4+4\right|+4$	$|0|+4$	$4$
$\text{-}2$	$\left|\text{-}2+4\right|+4$	$|2|+4$	$6$
$0$	$\left|0+4\right|+4$	$|4|+4$	$8$

Now we can plot these points and connect them to create our graph of $c(x).$