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+4|-5
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
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 4 units to the left and 5 units down. We can substitute these values into the general vertex form to find the equation of the function.
y= 1|x-( -4)|+( -5) ⇒ y=|x+4|-5
