# Graphs of Absolute Value Equations

## Graph two-variable absolute value equations

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Graphing Basic Absolute Value Functions

While on vacation, you go scuba diving. The surface of the water is at an unknown altitude. You descend to a depth of 90 feet below the surface of the water. What is the vertex of the absolute value function that represents your maximum possible distance from sea level?

### Graphing Absolute Value Functions

You have already learned how to solve and define absolute value equations. Now take this idea one step further and graph absolute value equations.

#### Graphing the Parent Graph of an Absolute Value Function

Let's follow the steps below to learn about the parent graph of absolute value functions.

Step 1: Graph \begin{align*}y = |x|\end{align*}. Draw a table for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, with the \begin{align*}x-\end{align*}values ranging from -3 to 3.

\begin{align*}x\end{align*} \begin{align*}|x|\end{align*} \begin{align*}y\end{align*}
\begin{align*}-3\end{align*} \begin{align*}|-3|\end{align*} \begin{align*}3\end{align*}
\begin{align*}-2\end{align*} \begin{align*}|-2|\end{align*} \begin{align*}2\end{align*}
\begin{align*}-1\end{align*} \begin{align*}|-1|\end{align*} \begin{align*}1\end{align*}
\begin{align*}|0|\end{align*}
\begin{align*}1\end{align*} \begin{align*}|1|\end{align*} \begin{align*}1\end{align*}
\begin{align*}2\end{align*} \begin{align*}|2|\end{align*} \begin{align*}2\end{align*}
\begin{align*}3\end{align*} \begin{align*}|3|\end{align*} \begin{align*}3\end{align*}

Step 2: Recall that the absolute value of a number is always positive. Plot each of the seven coordinate pairs and graph the function.

Step 3: Notice that this function is very similar to the linear function, \begin{align*}y = x\end{align*}. Draw this line on the graph in a different color or with a dashed line.

Step 4: Now, fold the graph on the \begin{align*}x-\end{align*}axis. What do you notice?

You should notice that when you fold your graph on the \begin{align*}x-\end{align*}axis, the line \begin{align*}y = x\end{align*} becomes the absolute value equation \begin{align*}y = |x|\end{align*}. That is because the absolute value of a number can never be below zero; therefore the range will always be positive. \begin{align*}y = |x|\end{align*} is considered the parent graph because it is the most basic of all the absolute value functions. All linear absolute value functions have this “V” shape.

In general, you can define the graph of \begin{align*}y = |x|\end{align*} as \begin{align*}y= \begin{cases} x; & x \ge 0\\-x; & x < 0\end{cases}\end{align*}. Each side is the mirror image of the other, over a vertical line through the vertex.

Now, let's use a table to graph the following functions.

1. Graph \begin{align*}y = |x-3|\end{align*}. Determine the domain and range.

In general, when you use a table to graph a function, pick some positive and negative numbers, as well as zero. Use the equation to help you determine which \begin{align*}x-\end{align*}values to pick. Setting what is inside the absolute value equal to zero yields \begin{align*}x = 3\end{align*}. Pick three values on either side of \begin{align*}x = 3\end{align*} and then graph.

\begin{align*}x\end{align*} \begin{align*}|x-3|\end{align*} \begin{align*}y\end{align*}
0 \begin{align*}|-3|\end{align*} 3
1 \begin{align*}|-2|\end{align*} 2
2 \begin{align*}|-1|\end{align*} 1
3 \begin{align*}|0|\end{align*} 0
4 \begin{align*}|1|\end{align*} 1
5 \begin{align*}|2|\end{align*} 2
6 \begin{align*}|3|\end{align*} 3

Notice that this graph shifts to the right 3 when compared to the parent graph. The domain will be all real numbers, \begin{align*}x \in \mathbb{R}\end{align*} , and the range will be all positive real numbers, including zero, \begin{align*}y \in [0, \infty).\end{align*}

1. Graph \begin{align*}y = |x|-5\end{align*}. Determine the domain and range.

Be careful! Here, the minus 5 is not inside the absolute value. So, first take the absolute value of the \begin{align*}x-\end{align*}value and then subtract 5. In cases like these, where a value is subtracted after taking the absolute value, the range may include negative numbers.

\begin{align*}x\end{align*} \begin{align*}|x|-5\end{align*} \begin{align*}y\end{align*}
\begin{align*}-3\end{align*} \begin{align*}|-3| - 5\end{align*} \begin{align*}-2\end{align*}
\begin{align*}-2\end{align*} \begin{align*}|-2| - 5\end{align*} \begin{align*}-3\end{align*}
\begin{align*}-1\end{align*} \begin{align*}|-1| - 5\end{align*} \begin{align*}-4\end{align*}
\begin{align*}|0| - 5\end{align*} \begin{align*}-5\end{align*}
\begin{align*}1\end{align*} \begin{align*}|1| - 5\end{align*} \begin{align*}-4\end{align*}
\begin{align*}2\end{align*} \begin{align*}|2| - 5\end{align*} \begin{align*}-3\end{align*}
\begin{align*}3\end{align*} \begin{align*}|3| - 5\end{align*} \begin{align*}-2\end{align*}

Here, the graph shifts down 5 when compared to the parent graph. The domain will be all real numbers, \begin{align*}x \in \mathbb{R} \end{align*}, and the range will be all real numbers greater than or equal to -5, \begin{align*}y \in [\text{-}5, \infty).\end{align*}

In these two absolute value graphs, you may have noticed that there is a minimum point. This point is called the vertex. For instance, in the previous problem, the vertex is (0, -5). The vertex can also be a maximum. See the next problem.

1. Use a table to graph \begin{align*}y = -|x-1|+2.\end{align*} Determine the vertex, domain, and range.

Determine the \begin{align*}x\end{align*}-value that makes the inside of the absolute value equation zero \begin{align*}(x =1)\end{align*}. Then make a table of values, including a couple values on either side of that \begin{align*}x\end{align*}-value.

\begin{align*}x\end{align*} \begin{align*}-|x-1|+2\end{align*} \begin{align*}y\end{align*}
-2 \begin{align*}-|-2 - 1| + 2\end{align*} -1
-1 \begin{align*}-|-1 - 1| + 2\end{align*} 0
0 \begin{align*}-|0 - 1| + 2\end{align*} 1
1 \begin{align*}-|1 - 1| + 2\end{align*} 2
2 \begin{align*}-|2 - 1| + 2\end{align*} 1
3 \begin{align*}-|3 - 1| + 2\end{align*} 0
4 \begin{align*}-|4 - 1| + 2\end{align*} -1

The vertex is (1, 2) and in this case, it is the maximum value. The domain is \begin{align*}x \in \mathbb{R} \end{align*}, and the range is \begin{align*}y \in (-\infty, 2].\end{align*}

### Examples

#### Example 1

Earlier, you were asked to identify the vertex of an absolute value function that represents your possible distance from sea level after diving.

The absolute value function that represents this situation is \begin{align*}y= \left| x-90 \right|,\end{align*}where \begin{align*}x\end{align*} is your altitude above or below sea level before diving. By graphing this function, you can see that the vertex occurs at the point (90, 0).

#### Example 2

Graph \begin{align*}y=\text{-} \left| x-5 \right|\end{align*} using a table. Determine the vertex, domain, and range.

Determine what makes the inside of the absolute value equation zero, \begin{align*}x =5\end{align*}. Then, to make a table of values, pick a couple values on either side of \begin{align*}x = 5\end{align*}.

\begin{align*}x\end{align*} \begin{align*}-|x-5|\end{align*} \begin{align*}y\end{align*}
2 \begin{align*}-|2 - 5|\end{align*} -3
3 \begin{align*}-|3 - 5|\end{align*} -2
-4 \begin{align*}-|4 - 5|\end{align*} -1
5 \begin{align*}-|5 - 5|\end{align*} 0
6 \begin{align*}-|6 - 5|\end{align*} -1
7 \begin{align*}-|7 - 5|\end{align*} -2
8 \begin{align*}-|8 - 5|\end{align*} -3

The vertex is (5, 0) and in this case, it is the maximum value. The domain is \begin{align*}x \in \mathbb{R} \end{align*}, and the range is \begin{align*}y \in (-\infty, 0].\end{align*}

#### Example 3

Use a table to graph \begin{align*}y= \left| x+4 \right| -2.\end{align*} Determine the vertex, domain, and range.

Determine what makes the inside of the absolute value equation zero, \begin{align*}x =-4\end{align*}. Then make a table of values, including a couple values on either side of \begin{align*}x = -4\end{align*}.

\begin{align*}x\end{align*} \begin{align*}|x+4|-2\end{align*} \begin{align*}y\end{align*}
\begin{align*}-1\end{align*} \begin{align*}|-1 + 4| - 2\end{align*} 1
-2 \begin{align*}|-2 + 4| - 2\end{align*} 0
-3 \begin{align*}|-3 + 4| - 2\end{align*} -1
-4 \begin{align*}|-4 + 4| - 2\end{align*} -2
-5 \begin{align*}|-5 + 4| - 2\end{align*} -1
-6 \begin{align*}|-6 + 4| - 2\end{align*} 0
-7 \begin{align*}|-7 + 4| - 2\end{align*} 1

The vertex is (-4, -2) and in this case, it is the minimum value. The domain is \begin{align*}x \in \mathbb{R} \end{align*}, and the range is \begin{align*}y \in [-2, \infty).\end{align*}

### Review

Graph the following functions using a table. Determine the vertex, domain, and range of each function.

1. \begin{align*}y = |x+6|\end{align*}
2. \begin{align*}y = |x-4|\end{align*}
3. \begin{align*}y = -|x|+3\end{align*}
4. \begin{align*}y = |x|-2\end{align*}
5. \begin{align*}y = -|x+3|+7\end{align*}
6. \begin{align*}y = |x-1|-6\end{align*}
7. \begin{align*}y = 2|x|\end{align*}
8. \begin{align*}y = -3|x|\end{align*}
9. \begin{align*}y = \frac{1}{3}|x|\end{align*}

Reference problems 1-9 as needed to fill in the blanks in problems 10-15.

1. If there is a negative sign in front of the absolute value, the graph is ________________ (when compared to the parent graph).
2. If the equation is \begin{align*}y = |x-h|+k\end{align*}, the vertex will be ___________________.
3. The domain of an absolute value function is always ____________________________.
4. For \begin{align*}y = a|x|\end{align*}, if \begin{align*}a > 1\end{align*}, then the graph will be ___________________ than the parent graph.
5. For \begin{align*}y = a|x|\end{align*}, if \begin{align*}0 < a < 1\end{align*}, then the graph will be ___________________ than the parent graph.
6. Without making a table, what is the vertex of \begin{align*}y = \left| x-9 \right| +7?\end{align*}

To see the Review answers, open this PDF file and look for section 2.12.

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### Vocabulary Language: English

TermDefinition
Absolute Value The absolute value of a number is the distance the number is from zero. Absolute values are never negative.
Maximum The maximum is the highest point of a graph. The maximum will yield the largest value of the range.
Minimum The minimum is the lowest point of a graph. The minimum will yield the smallest value of the range.
Parent Graph A parent graph is the simplest form of a particular type of graph. All other graphs of this type are usually compared to the parent graph.