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 3 units to the right.
A reflection in the x-axis.
A vertical translation 5 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.
d(x)=-|x-3|+5
First, there is a horizontal translation 3 units to the right, |x|⇒|x-3|. 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 reflection in the x-axis, |x-3|⇒ -|x-3|.
Finally, we have a vertical translation 5 units up, -|x-3|⇒-|x-3|+5.
Graphing d(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-3|+5
Simplify
d(x)
-1
-| -1-3|+5
-|-4|+5
1
1
-| 1-3|+5
-|-2|+5
3
3
-| 3-3|+5
-|0|+5
5
5
-| 5-3|+5
-|2|+5
3
7
-| 7-3|+5
-|4|+5
1
Now we can plot these points and connect them to create our graph of d(x).
