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.
Tranformations:
A reflection in the y-axis.
A horizontal stretch by a factor of 3.
A horizontal translation 3 units to the right.
A vertical translation 2 units up. 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.
n(x)=|- 13x+1|+2
To see how each of the transformations affects the parent function, it would be helpful to rewrite it first. This way we can look at each of the transformations individually and work our way from the inside to the outside.n(x)=|-1/3x+1|+2
⇕
n(x)=|-1/3 ( x-3)|+2
First, we have a reflection in the y-axis, |x|⇒|- x|.
Note that since the graph is currently symmetric across the y-axis, this will not change how it looks.
Then, there is a horizontal translation 3 units to the right. |- 13x|⇒|- 13(x-3)|.
Finally, we have a vertical translation 2 units up, |- 13(x-3)|⇒|- 13(x-3)|+2.
Graphing n(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
|-1/3x+1|+2
Simplify
n(x)
-3
|-1/3( -3)+1|+2
|2|+2
4
0
|-1/3( 0)+1|+2
|1|+2
3
3
|-1/3( 3)+1|+2
|0|+2
2
6
|-1/3( 6)+1|+2
|-1|+2
3
Now we can plot these points and connect them to create our graph of n(x).
