A calculus of the absurd

4.8 The modulus function

The modulus function is a very happy function2424 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*}

4.8.1 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:

(-tikz- diagram)

As the modulus function must never be unhappy (i.e. take a negative value), we need to turn that frown upside down!2525 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.

(-tikz- diagram)