# A calculus of the absurd

#### 4.9 The modulus function

The modulus function is a very happy function2323 Sorry.. This is because it’s always positive, and never takes on any negative values!

We can write it as $$\abs {x}$$. For example, all of these expressions are true

\begin{align*} & \abs {12} = 12 \\ & \abs {-12} = 12 \\ & \abs {12 - 6} = \abs {6 - 12} \\ & \abs {-x} = x \end{align*}

##### 4.9.1 Graphically

When graphing a value inside a modulus function, it is often helpful to first sketch the function without the modulus. For example, when graphing $$y=\abs {-\left (x-5\right )^{2}+1}$$, the graph of $$y=2x - 8$$ would look like this:

As the modulus function must never be unhappy (i.e. take a negative value), we need to turn that frown upside down!2424 Ok, I’ll stop, I promise. Using the quadratic formula to find the roots, we get that they are at $$x=4$$ and $$x=6$$. Therefore, everything to the left of $$4$$ and to the right of $$5$$ needs to be reflected it the $$x$$-axis.