6.9: Graphs of Absolute Value Equations
What if you were given an absolute value function like
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CK12 Foundation: 0609S Graphs of Absolute Value Equations (H264)
Guidance
Now let’s look at how to graph absolute value functions.
Example A
Consider the function



2 

1 

0 

1 

2 

3 

4 

You can see that the graph of an absolute value function makes a big “V”. It consists of two line rays (or line segments), one with positive slope and one with negative slope, joined at the vertex or cusp.
We’ve already seen that to solve an absolute value equation we need to consider two options:
 The expression inside the absolute value is not negative.
 The expression inside the absolute value is negative.
Combining these two options gives us the two parts of the graph.
For instance, in the above example, the expression inside the absolute value sign is
On the other hand, when
These are both graphs of straight lines, as shown above. They meet at the point where
We can graph absolute value functions by breaking them down algebraically as we just did, or we can graph them using a table of values. However, when the absolute value equation is linear, the easiest way to graph it is to combine those two techniques, as follows:
 Find the vertex of the graph by setting the expression inside the absolute value equal to zero and solving for
x .  Make a table of values that includes the vertex, a value smaller than the vertex, and a value larger than the vertex. Calculate the corresponding values of
y using the equation of the function.  Plot the points and connect them with two straight lines that meet at the vertex.
Example B
Graph the absolute value function
Solution
Step 1: Find the vertex by solving
Step 2: Make a table of values:



8 

5 

2 

Step 3: Plot the points and draw two straight lines that meet at the vertex:
Example C
Graph the absolute value function:
Solution
Step 1: Find the vertex by solving
Step 2: Make a table of values:



0 

4 

8 

Step 3: Plot the points and draw two straight lines that meet at the vertex.
Watch this video for help with the Examples above.
CK12 Foundation: Graphs of Absolute Value Equations
Vocabulary
 The absolute value of a number is its distance from zero on a number line.

x=x ifx is not negative, andx=−x ifx is negative.  An equation or inequality with an absolute value in it splits into two equations, one where the expression inside the absolute value sign is positive and one where it is negative. When the expression within the absolute value is positive, then the absolute value signs do nothing and can be omitted. When the expression within the absolute value is negative, then the expression within the absolute value signs must be negated before removing the signs.
 Inequalities of the type
x<a can be rewritten as “−a<x<a .”  Inequalities of the type
x>b can be rewritten as “x<−b orx>b .”
Guided Practice
Graph the absolute value function: \begin{align*}y=3x4\end{align*}
Solution
Step 1: Find the vertex by solving \begin{align*}x4 = 0\end{align*}. The vertex is at \begin{align*}x = 4\end{align*}.
Step 2: Make a table of values:
\begin{align*}x\end{align*}  \begin{align*}y=3x4\end{align*} 

0  \begin{align*}y=304=34=3\cdot 4=12\end{align*} 
4  \begin{align*}y=344=30=3\cdot 0=0\end{align*} 
8  \begin{align*}y=384=34=3\cdot 4=12\end{align*} 
Notice this is the same table as Example C. The function \begin{align*}y=3x4\end{align*} is equivalent to the function \begin{align*}y=3x12.\end{align*}</math>. This is because positive numbers can be factored out, or distributed into the absolute value function.
Step 3: Plot the points and draw two straight lines that meet at the vertex.
Practice
Graph the absolute value functions.
 \begin{align*}y=x+3\end{align*}
 \begin{align*}y=x6\end{align*}
 \begin{align*}y=4x+2\end{align*}
 \begin{align*}y=56x\end{align*}
 \begin{align*}y=2x1\end{align*}
 \begin{align*}y=32x7\end{align*}
 \begin{align*}y=0.05x1.25\end{align*}
 \begin{align*}y=\frac{1}{2}x+10\end{align*}
 \begin{align*}y= \left  \frac{x}{3}4 \right \end{align*}
 \begin{align*}y= 2\left  \frac{x}{2}5 \right \end{align*}
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Here you'll learn how to make a table of values for absolute value functions so you can graph them.