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……
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.