How to “Break” Your Graphing Calculator

A conversation with a colleague on function operations reminded me of one of my favorite “Oh Wow!” moments from math class.

You’ll need a TI-83 or 84.  This is a case where the Nspires are too nice for our use.

Enter functions as shown below.  We are interested in the rational function which comes from dividing the two previous functions:

Since the denominator is a factor of the numerator, we can simplify the expression, resulting in a linear function when we graph.  But what about x=-3?  What happens there?  Let’s explore a bit:

Zoom in even more….doesn’t seem all that interesting…..

Pretty boring.  But tell your students to keep zooming.  And then….

….wait for it……

…whoa!!!!!!….

Zoom in even more….

This is a nice reminder that, while we may simplify a function, we are still looking at a quotient and need to consider the parent functions used in the division.  And the “noise” we get here is a result of the algorithms TI calculators utilize to plot the points.  Actually, if somebody has a better explanation for the noise, I’d love to hear it.  Some “new” calculators will now show the hole in this function, like my new man-crush, the Desmos calculator:

But the noise is more fun.