Compare the given graph to the parent function f(x)=|x|. How many units in the horizontal and vertical directions has the vertex moved?
g(x)=|x+5|-1
Practice makes perfect
We want to write an equation for the absolute value function represented in the given graph. To do so, we need to determine what translations of the parent function f(x)=|x| took place. We can use the vertex form of absolute value function to create the foundation of our desired equation.
g(x)= a|x- h|+ k
In this form, the constants represent 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 nor compressed. When there is no stretch nor compression, we have that a= 1. This also means that we only need to consider vertical and horizontal translations. Let's compare the given graph with the graph of f(x)=|x|.
The graph of the parent function has been translated left 5 units and down 1 unit. We can substitute these values, as well as a= 1, into the general vertex form to find the equation of the function.
g(x)= 1|x-( -5)|+( -1) ⇒ g(x)=|x+5|-1
