### Graphing Absolute Value Equations Worksheets and Examples

If you obtain the absolute value of a number, then you at all times end up with a nonnegative number. It means that whether the input value is negative or positive, the result is always nonnegative (positive or zero). It is due to this fact, why solving and graphing absolute value equations are considered a little complicated. An absolute value equation in two variables is generally defined by the equation, $$y=a\left | x-h \right |+k$$ where $$\left(h,k\right)$$ is the vertex of the graph. The vertex of the graph of an absolute value equation is either its lowest or its highest point. It is the lowest (minimum) point of the graph, if it opens upward, and is the highest (maximum) point if it opens downward.

Sketching the graph of an absolute value equation is just like sketching the graph of a quadratic equation. First, identify the vertex of the graph. Next, determine the direction of opening of the graph. Create a table of values including the vertex of the graph. To be more accurate, you can include several values in the table yet for convenience let me illustrate how to graph absolute value equation by using two points on the left of its vertex and another two points on its right. To determine what kind of graph will absolute value equation has, let us now sketch the graph of the succeeding absolute value equations.

Examples

Sketch the graph of each absolute value equation.

1. $$y= \left | x \right |$$
2. $$y= \left | -x \right |$$
3. $$y= \left | x\right |-4$$
4. $$y= \left | x-4 \right |$$
5. $$y= \left | x-4 \right |+4$$

1. $$y= \left | x \right |$$

Since $$y=\left | x \right |$$ can also be written into $$y=\left | x-0 \right |+0$$ with $$a=1$$, the graph will open upward and has the minimum point at $$\left (0, 0\right)$$. The table of values is hereby presented.

$\begin{tabular}{ | l | l | l | l | l | l |} \hline x & -4 & -2 & 0 & 2 & 4\\ \hline y & 4 & 2 & 0 & 2 & 4\\ \hline \end{tabular}$

2. $$y= \left | -x \right |$$

Since $$y=-\left |x\right |$$ can also be written into $$y=-\left |x-0\right |+0$$ with $$a = –1$$, the graph will open downward and has the maximum point at $$left(0,0 \right). $\begin{tabular}{ | l | l | l | l | l | l |} \hline x & -4 & -2 & 0 & 2 & 4\\ \hline y & -4 & -2 & 0 & -2 & -4\\ \hline \end{tabular}$ 3. \(y=\left |x \right |-4$$

Since $$y=\left |x \right |-4$$ can also be written into $$y=\left |x-0\right |-4$$ with $$a = 1$$, the graph will open upward and has the minimum point at $$\left (0,-4\right)$$.

$\begin{tabular}{ | l | l | l | l | l | l |} \hline x & -4 & -2 & 0 & 2 & 4\\ \hline y & 0 & -2 & -4 & -2 & 0\\ \hline \end{tabular}$

Below is the graph of $$y=\left |x\right |-4$$.

4. $$y= \left | x-4 \right |$$

Since $$y= \left | x-4 \right |$$ can also be written into $$y= \left | x-4 \right |+0$$ with $$a = 1$$, the graph will open upward and has the minimum point at $$\left(4, 0\right)$$.

$\begin{tabular}{ | l | l | l | l | l | l |} \hline x & 2 & 3 & 4 & 5 & 6\\ \hline y & 2 & 1 & 0 & 1 & 2\\ \hline \end{tabular}$

5. $$y= \left | x-4 \right |+4$$

In the equation, $$y= \left | x-4 \right |+4$$, $$a = –1$$ and $$\left(h, k\right) = \left(4, 4\right)$$. The graph will open upward and has a maximum point at $$\left(4, 4\right)$$.

$\begin{tabular}{ | l | l | l | l | l | l |} \hline x & 2 & 3 & 4 & 5 & 6\\ \hline y & 2 & 3 & 4 & 3 & 2\\ \hline \end{tabular}$