6.9: Graphs of Absolute Value Equations
Suppose you're keeping a math journal and you want to explain how to graph the solutions to an absolute value equation. How would you describe the process? What steps would be involved? In this Concept, you'll learn how to create a table of values for an absolute value equation so that you can graph its solutions. With this knowledge, you can fill your math journal with useful information!
Guidance
Absolute value equations can be graphed in a way that is similar to graphing linear equations. By making a table of values, you can get a clear picture of what an absolute value equation will look like.
Example A
Graph the solutions to
Solution:
Make a table of values and graph the coordinate pairs.



–2 

–1 

0 

1 

2 

3 

The Shape of an Absolute Value Graph
Every absolute value graph will make a “V”shaped figure. It consists of two pieces: one with a negative slope and one with a positive slope. The point of their intersection is called the vertex. An absolute value graph is symmetrical, meaning it can be folded in half on its line of symmetry. The function
Example B
Graph the solutions to
Solution:
Make a table of values and plot the ordered pairs.



–2 

–1 

0 

1 

2 

3 

The graph of this function is seen below in green, graphed with the parent function in red.
Notice that the green function is just the parent function shifted over. The vertex is shifted over 1 unit to the right. This is because when
Graphing by Finding the Vertex
Absolute value equations can always be graphed by making a table of values. However, you can use the vertex and symmetry to help shorten the graphing process.
Step 1: Find the vertex by determining which value of
Step 2: Using this value as the center of the
Example C
Graph
Solution:
Start by considering where the vertex would be by solving for when the absolute value equals zero:
The vertex is at



–4 

–3 

–2 

–1 

0 

Guided Practice
Graph
Solution:
Determine which
Therefore, (–5, 0) is the vertex of the graph and represents the center of the table of values.
Create the table and plot the ordered pairs.



–7 

–6 

–5 

–4 

–3 

Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Absolute Value Equations (10:41)
In 1 – 11, graph the function.

y=x+3 
y=x−6 
y=4x+2 
y=∣∣x3−4∣∣ 
x−4=y 
−x−2=y  \begin{align*}y=x2\end{align*}
 \begin{align*}y=x+3\end{align*}
 \begin{align*}y=\frac{1}{2} x\end{align*}
 \begin{align*}y=4x2\end{align*}
 \begin{align*}y=\left \frac{1}{2} x\right +6\end{align*}
Mixed Review
 Graph the following inequality on a number line: \begin{align*}2 \le w<6\end{align*}.
 Is \begin{align*}n=4.175\end{align*} a solution to \begin{align*}n3>12\end{align*}?
 Graph the function \begin{align*}g(x)=\frac{7}{2} x8\end{align*}.
 Explain the pattern: 24, 19, 14, 9,....
 Simplify \begin{align*}(3)\left (\frac{(29)(2)8}{10}\right )\end{align*}.
Image Attributions
Here you'll learn how to make a table of values for an absolute value equation so that you can graph the solutions to the equation.