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 reflection in the y-axis.
A horizontal translation 1 units to the left.
A vertical translation 5 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.
j(x)=|- x+1|-5
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.j(x)=|- x+1|-5
⇕
j(x)=|-( x-1)|-5
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 1 unit to the right, |- x|⇒|- (x-1)|. Remember, typically |x-h| is a horizontal translation h units to the right, while |x+h| is translated h units to the left.
Finally, we have a vertical translation 5 units down, |- x+1|⇒|- x+1|-5.
Graphing j(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+1|-5
Simplify
j(x)
-3
|-( -3)+1|-5
|4|-5
-1
-1
|-( -1)+1|-5
|2|-5
-3
1
|-( 1)+1|-5
|0|-5
-5
3
|-( 3)+1|-5
|-2|-5
-3
5
|-( 5)+1|-5
|-4|-5
-1
Now we can plot these points and connect them to create our graph of j(x).
