Let's start by writing the general form of an absolute value function.
f(x)= a|x- h|+ k
In this form, each of the variables a, h, and k 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
Let's now consider the given function.
g(x)=|x+2|+2 ⇔ g(x)= 1*|x-( -2)|+ 2
Since a=1, h=-2, and k=2, the parent function f(x)=|x| will only be translated horizontally 2 units to the left and vertically 2 units up.
Looking at the graph of the transformed function, we can see that the domain, the possible values of x, contains all real numbers. The range, the possible values of y, contains all real numbers greater than or equal to 2. Let's write this in interval notation.
Domain:& (- ∞,∞ )
Range:& [2, ∞ )
