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?

### Graphing Absolute Value Equations

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.

#### Let's graph the solutions to \begin{align*}y=|x|\end{align*}:

Make a table of values and graph the coordinate pairs.

\begin{align*}x\end{align*} | \begin{align*}y=|x|\end{align*} |
---|---|

–2 | \begin{align*}|-2|=2\end{align*} |

–1 | \begin{align*} |-1|=1\end{align*} |

0 | \begin{align*} |0|=0\end{align*} |

1 | \begin{align*} |1|=1\end{align*} |

2 | \begin{align*} |2|=2\end{align*} |

3 | \begin{align*} |3|=3\end{align*} |

#### 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 \begin{align*}y=|x|\end{align*} is the parent function, or most basic function, of absolute value functions.

#### Now, lets' graph the solutions to \begin{align*}y=|x-1|\end{align*} and it's parent function:

Make a table of values and plot the ordered pairs.

\begin{align*}x\end{align*} | \begin{align*}y=|x-1|\end{align*} |
---|---|

–2 | \begin{align*}|-2-1|=3\end{align*} |

–1 | \begin{align*} |-1-1|=2\end{align*} |

0 | \begin{align*} |0-1|=1\end{align*} |

1 | \begin{align*} |1-1|=0\end{align*} |

2 | \begin{align*} |2-1|=1\end{align*} |

3 | \begin{align*} |3-1|=2\end{align*} |

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 \begin{align*}x=1\end{align*}, \begin{align*}|x-1|=0\end{align*} since \begin{align*}|1-1|=0\end{align*}. The vertex will always be where the value inside the absolute value is zero.

#### 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 \begin{align*}x\end{align*} makes the distance zero.

**Step 2:** Using this value as the center of the \begin{align*}x-\end{align*}values, choose several values greater than this value and several values less than this value.

#### Finally, let's graph \begin{align*}y=|x+2|+3\end{align*}:

Start by considering where the vertex would be by solving for when the absolute value equals zero:

\begin{align*}0=|x+2| \Rightarrow x=-2\end{align*}

The vertex is at \begin{align*}x=-2\end{align*}. Choose some points on either side of that to make a table of values.

\begin{align*}x\end{align*} | \begin{align*}y=|x+2|+3\end{align*} |
---|---|

–4 | \begin{align*}|-4+2|+3=5\end{align*} |

–3 | \begin{align*} |-3+2|+3=4\end{align*} |

–2 | \begin{align*} |-2+2|+3=3\end{align*} |

–1 | \begin{align*} |-1+2|+3=4\end{align*} |

0 | \begin{align*} |0+2|+3=5\end{align*} |

### Examples

#### Example 1

Earlier, you were asked what the process for graphing an absolute value equation is.

There are multiple ways to graph an absolute value equation. One way is to create a table of values and the graph those values.

Another way is to first find the vertex by determining what value of the variable makes the distance 0. Then, you can choose several values greater and less than that value to use to plot points. This technique reduces the number of values you need to test to plot the whole shape of the graph.

#### Example 2

Graph \begin{align*}y=|x+5|\end{align*}.

Determine which \begin{align*}x-\end{align*}value equals a distance of zero.

\begin{align*}0& =|x+5|\\ x& =-5\end{align*}

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.

\begin{align*}x\end{align*} | \begin{align*}y=|x+5|\end{align*} |
---|---|

–7 | \begin{align*} |-7+5|=2\end{align*} |

–6 | \begin{align*} |-6+5|=1\end{align*} |

–5 | \begin{align*} |-5+5|=0\end{align*} |

–4 | \begin{align*} |-4+5|=1\end{align*} |

–3 | \begin{align*} |-3+5|=2\end{align*} |

### Review

In 1–11, graph the function.

- \begin{align*}y=|x+3|\end{align*}
- \begin{align*}y=|x-6|\end{align*}
- \begin{align*}y=|4x+2|\end{align*}
- \begin{align*}y=\left |\frac{x}{3}-4\right |\end{align*}
- \begin{align*}|x-4|=y\end{align*}
- \begin{align*}-|x-2|=y\end{align*}
- \begin{align*}y=|x|-2\end{align*}
- \begin{align*}y=|x|+3\end{align*}
- \begin{align*}y=\frac{1}{2} |x|\end{align*}
- \begin{align*}y=4|x|-2\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*}|n-3|>12\end{align*}?
- Graph the function \begin{align*}g(x)=\frac{7}{2} x-8\end{align*}.
- Explain the pattern: 24, 19, 14, 9,....
- Simplify \begin{align*}(-3)\left (\frac{(29)(2)-8}{-10}\right )\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.9.