Be sure to think about the horizontal transformations before any vertical transformations. Work from the inside to the outside. Then make a table of values to find points on the graph.
Transformations:
A horizontal translation 2 units to the left.
A vertical translation 6 units down. Graph:
Practice makes perfect
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.
r(x)=|x+2|-6First, there is a horizontal translation 2 units to the left, |x|⇒|x+2|. 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 6 units down, |x+2|⇒|x+2|-6. Remember, in most cases |x|+k is a vertical translation k units up, while |x|-k is translated k units down.
Graphing r(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+2|-6
Simplify
r(x)
-6
| -6+2|-6
|-4|-6
-2
-4
| -4+2|-6
|-2|-6
-4
-2
| -2+2|-6
|0|-6
-6
0
| 0+2|-6
|2|-6
-4
2
| 2+2|-6
|4|-6
-2
4
| 4+2|-6
|6|-6
0
Now we can plot these points and connect them to create our graph of r(x).
