What if you were given an absolute value function like

### Watch This

CK-12 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 y=|x+5|.*

**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: y=|3x−12|*

**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.

CK-12 Foundation: Graphs of Absolute Value Equations

### Guided Practice

*Graph the absolute value function: y=3|x−4|*

**Solution**

*Step 1*: Find the vertex by solving

*Step 2*: Make a table of values:

0 | |

4 | |

8 |

Notice this is the same table as Example C. The function

*Step 3*: Plot the points and draw two straight lines that meet at the vertex.

### Explore More

Graph the absolute value functions.

- \begin{align*}y=|x+3|\end{align*}
- \begin{align*}y=|x-6|\end{align*}
- \begin{align*}y=|4x+2|\end{align*}
- \begin{align*}y=|5-6x|\end{align*}
- \begin{align*}y=|2x-1|\end{align*}
- \begin{align*}y=3|2x-7|\end{align*}
- \begin{align*}y=0.05|x-1.25|\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*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.9.