### Graphs of Absolute Value Equations

Almost all absolute value equations have a "V" shape. The absolute value \begin{align*}y=|x|\end{align*} is a V shape centered at the origin with a slope of 1 become there is no coefficient.

All absolute value equations are essentially variations of this "V" shape. The location and the slope of the "V" change depending on the equation. If the absolute value equation has a y-intercept at the end of it, you must implement the y-intercept when graphing the "V" shape. All you have to do is shift the V up by whatever number the y-intercept is.

Absolute value equations will often have other numbers/terms inside the absolute value brackets that you need to consider when graphing an absolute value equation. If you have the equation \begin{align*}y=|x-1|\end{align*} , for exaple, you would have to shift the "V" over 1 to the right. *A negative sign means shift to the right and a positive sign means shift to the left; this is important to note because it can seem counterintuitive.* If you had the equation \begin{align*}y=|x+4|\end{align*} ,you would shift the "V" 4 to the left.

It is important to know how to integrate the two rules stated above. Practice integrating the rules above by graphing the following absolute value equations:

- \begin{align*}y=|x-4|+7\end{align*}
- \begin{align*}y=|x+9|-3\end{align*}
- \begin{align*}y=|x+3|-2\end{align*}

Also know that the slopes in these types of problems won't always be 1. If there is a coefficient symbolizing the slope in an absolute value equation, be sure to change the slant of the "V" accordingly. Say your slope is x. On the right side of the "V", make your slope x. However, on the left side of the "V", your slope will be *negative* x.

\begin{align*}y=|4x+2|\end{align*}

Practice integrating graphing absolute value equations with more complicated slopes.

- \begin{align*}y=4|x|-2\end{align*}
- \begin{align*}y=\left |\frac{1}{2} x\right |+6\end{align*}

Finally, note that sometimes you will have to graph an upsidedown "V" shape. This will happen if there is a negative sign right before the absolute value term. For example, in the absolute value equation\begin{align*}-|x-2|=y\end{align*}, the graph would still have its origin at (2,0), but it would be an upsidedown "V". However, if the negative sign *is* in the absolute value brackets, you *don't* need to graph the "V" upsidedown.

TIP: Explain why the following two graphs are shaped the way they are to a family member or friend. If you can successfully explain why to somebody else, you have a strong understanding yourself.

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

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

### Overall Practice

Use all the principles described above to graph the following graphs.

- \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*}

### Graphs of Inequalities in One & Two Variables

Review of graphing inequalities in two variables:

Graphing inequalities is fairly simple as long as you know how to graph lines. First graph the line that you are given changing the inequality sign to an equal sign for now. Once you have graphed your line (this applies to two variables as well as one variable), you will notice that you have divided the Cartesian plane into two pieces/halves. Inequality graphs require that you shade in one half/piece that you have created. To figure out which half you need to shade, pick an arbitrary point in one of the halfs. Plug the point into you picked into the inequality (with the equals sign changed back into the greater than/less than sign). Then see if the inequality is still true. If it is, shade in the section of the graph where the point you arbitrarily picked was in. If not, shade in the other side.

Review of graphing inequalities in one variable

Now that you know how to graph inequalities with two variables, graphing inequalities with one variable is extremely simple. If you had to graph the inequality \begin{align*}x>6\end{align*} on the Cartesian plane, for example, you would use the same process as graphing on a number line; simply shade in all the area on the graph where x is greater than 6. Note that these areas will extend indefinitely in the y direction.

If you are graphing a simple absolute value inequality, all you have to do is rewrite the inequality with 2 "caps" that will define where you start and stop shading in. For the absolute value inequality \begin{align*}|y| < 5\end{align*}, for example, you would solve for \begin{align*}y\end{align*} to find your caps. Your caps would be:

\begin{align*}y > -5 \quad \text{and} \quad y < 5\end{align*}

Now all you have to do is shade in the area between your two caps, in this case between the y values of -5 and 5.