Compare the given graph to the parent function, y=|x|. How many units in the horizontal and vertical directions has the vertex moved?
y=|x+3|
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 constants represents one of the three basic types of transformations.
Constant
Value of the Constant
Transformation
a
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. When there is no stretch or compression, we have that a= 1. This also means that we only need to consider the vertical and horizontal translations. Let's compare it to the parent function.
The graph has been translated 3 units to the left and it has not been translated vertically. Therefore, h is equal to - 3 and k is equal to 0. We can substitute these values into the general vertex form to find the equation of the function.
y= 1|x-( -3)|+ 0 ⇔ y=|x+3|
