Absolute Value Functions and the V-Shaped Graph

Written by Jean Carlos Novaes

The absolute value of a real number is its distance from zero. Both negative 3 and positive 3 have absolute value 3. A negative input is multiplied by negative one, while a nonnegative input keeps its value. This makes the basic absolute value graph symmetric about the vertical axis.

|x|=x,x≥0
|x|=−x,x<0

Find where the expression inside the bars changes sign

In the function that takes the absolute value of x minus 2 and subtracts 1, the two branches meet at x equal to 2. At that input the expression inside the bars is zero and the output is negative 1. This is the vertex. The branch to the left has slope negative one and the branch to the right has slope positive one.

The V-shaped graph has vertex (2, −1). It crosses the horizontal axis at x = 1 and x = 3.
The V-shaped graph has vertex (2, −1). It crosses the horizontal axis at x = 1 and x = 3.

Solve by measuring the distance from 2

Set the shifted function equal to 2 and add 1 to both sides. This asks for inputs whose distance from 2 is 3. One lies 3 units to the left, at negative 1. The other lies 3 units to the right, at 5. Both give absolute value 3 before subtracting 1, so both solve the original equation.

|x−2|−1=2
|x−2|=3
x=2−3=−1
x=2+3=5

If an absolute value must equal zero, the expression inside the bars must be zero. If it must equal a negative number, there is no real solution. That test applies after isolating the absolute value. A shifted function itself can have negative outputs, as the vertex in this example shows.