Compare the given graph to the parent function, y=|x|. How many units in the horizontal and vertical direction has the vertex moved?
y=-|x-2|+4
Practice makes perfect
In order to write an absolute value equation for the given graph, we need to determine which translations of the parent function, y=|x|, took place. We can use the vertex form of absolute value functions to create the foundation of our desired equation.
y= a|x- h|+ k
In this form, each of the variables represents one of the three basic types of transformations.
Variable
Value of the Variable
Transformation
a
a=-1
reflection
a<-1
stretch + reflection
-1
compression + reflection
0
compression
1
stretch
h
h<0
translation to the left
h>0
translation to the right
k
k<0
translation down
k>0
translation up
Looking at the given graph, we can notice that it has not been stretched or compressed but we can see that it is reflected. When there is no stretch or compression but only a reflection, we have that a= -1. Now we only need to consider the vertical and horizontal translations. Let's compare the given graph to the function y=-|x|.
The graph of y=-|x| has been translated 2 units to the right and 4 units up. We can substitute these values into the general vertex form to find the equation of the function.
y= -1|x- 2|+ 4 ⇒ y=-|x-2|+4
