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.
Translations:
A horizontal translation 4 units to the left.
A vertical shrink by a factor of 12.
A vertical translation 1 unit 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.
m(x)=|1/2x+4|-1
First, there is a horizontal translation 4 units to the left, |x|⇒|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.
After that, we have a vertical shrink by a factor of 12, |x|⇒ 12|x|.
Finally, we have a vertical translation 1 unit down, 12|x+4|⇒ 12|x+4|-1.
Graphing m(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
1/2|x+4|-1
Simplify
m(x)
-8
1/2| -8+4|-1
1/2|-4|-1
1
-4
1/2| -4+4|-1
1/2|0|-1
-1
0
1/2| 0+4|-1
1/2|4|-1
1
Now we can plot these points and connect them to create our graph of m(x).
