Vertex: (2,-6) Axis of Symmetry: x=2 Transformations:
A horizontal translation right by 2 units
A vertical translation down by 6 units
Practice makes perfect
To identify the key features of the transformed function, let's first consider the general form of the absolute value function.
y= a|x- h|+ k
In this general form, | a| is the stretch or compression factor, h is a horizontal translation, and k is a vertical translation. Because translations shift an entire function, the vertex of a translated absolute value function is located at ( h, k) and the axis of symmetry is the line x= h. Let's rewrite the given function in this form!
y=|x-2|-6 ⇔ y= 1|x- 2|+( -6)
In this case, we have a vertex of ( 2, -6) and an axis of symmetry of x= 2. Next, let's look at all of the possible transformations so that we can more clearly identify what is happening to our function.
Transformations of y=|x|
Vertical Translations
Translation up k units, k>0
y=|x|+ k
Translation down k units, k>0
y=|x|- k
Horizontal Translations
Translation right h units, h>0
y=|x- h|
Translation left h units, h>0
y=|x+ h|
Vertical Stretch or Compression
Vertical stretch, a>1
y= a|x|
Vertical compression, 0< a<1
y= a|x|
Horizontal Stretch or Compression
Horizontal stretch, 0< b<1
y=| bx|
Horizontal compression, b>1
y=| bx|
Reflections
In the x-axis
y=- |x|
In the y-axis
y=|- x|
Using the table, we can now describe the transformations.
