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.

c (x) = ∣ x + 4 ∣ + 4

First, there is a horizontal translation

4

units to the left,

∣ x ∣ \Rightarrow ∣ x + 4 ∣ .

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, $∣ x + 4 ∣ \Rightarrow ∣ x + 4 ∣ + 4 .$ 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.

$x$	$∣ x + 4 ∣ + 4$	Simplify	$c (x)$
$- 8$	$∣ - 8 + 4 ∣ + 4$	$∣ - 4 ∣ + 4$	$8$
$- 6$	$∣ - 6 + 4 ∣ + 4$	$∣ - 2 ∣ + 4$	$6$
$- 4$	$∣ - 4 + 4 ∣ + 4$	$∣ 0 ∣ + 4$	$4$
$- 2$	$∣ - 2 + 4 ∣ + 4$	$∣ 2 ∣ + 4$	$6$
$0$	$∣ 0 + 4 ∣ + 4$	$∣ 4 ∣ + 4$	$8$

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

Transformations

Graphing c(x)

Graphing $c (x)$

Exercise