Mathing…

I do something similar to this when I introduce limits. It gives students a practical understanding of what limits are and how to evaluate them.

Part 1 – When Limits Exist

Note: This post is the first of a 5-part series. The original post entitled “How understanding holes in rational functions help you graph piecewise defined functions on TI graphing calculators” got a little long and probably had more information on a single post than anyone would care for. So now you will see five where there was one.

So what am I talking about?

No, I’m not talking about drilling holes into TI graphing calculators. This is not an episode of Will it Blend? or a clip of Office Space (or if you prefer the Family Guy version). I’m talking about points on the graph that are undefined where open circles are used to indicate its presence (absence?).

Before we begin, let’s discuss holes in rational functions. We typically cover this topic in Pre-Calculus when discussing difference quotients and rational functions or in Calculus when working with slope of tangent lines and working with limits. In Pre-Calculus, we typically brush over the topic (at least in the textbook we use) and focus mainly on the vertical, horizontal, and slant asymptotes produced by rational functions. At best, we discuss how indiscriminately canceling out factors is dangerous (something we do in Algebra 2 without much regard), and that the canceling out of factors is allowed only on the caveat that the factor is never 0.

But let’s get back to our topic.

Suppose you were asked to graph the following function

We typically begin by doing a little factoring

which can be rewritten as

but because

for all values of x except 1 (where it will be undefined)

it is equivalent to simply

for all values of x except 1 (where there’s a hole since it’s undefined).

In other words, the graph of is the same as the graph of at all points except at .

On a TI-83 graphing calculator, if we enter the following equation

and we get the following graph (notice the hole at x=1)

(Note: If you’re not getting the graph above, use ZOOM-> 4:ZDecimal, it’ll make every pixel correspond to 0.1 on the screen and will force 1.0, where the hole is, to be part of the graph)

If we want additional holes in the function, we simply multiply our function by an additional term where a is the place we want the holes in the function.

For example, if we wanted additional holes at x=2 and x=-3 we simply multiply our function as follows

which simplifies to

Entering the function into our calculator

produces the desired results (notice the holes at x=1, x=2, and x=-3)

So a logical next step is, can we restrict the domain of our function to only positive numbers by making all negative numbers and zero disappear from the screen?

To do that we just multiply our function by

(-4.7 is the edge of our viewscreen when using ZDecimal Zoom). This way, every point on the viewscreen to the left of 0 is essentially a hole and therefore not graphed.

This works great until you change to a different zoom where none of the points (-0.1, -0.2, … , -4.7) is represented by any of the pixels in the viewscreen. To verify this, change the zoom to ZStandard and the holes that we saw in the earlier parts disappear.

Entering each factor again and again is tedious, time consuming, and prone to errors. Plus it would take the calculator a long time to evaluate every point. Who has that kind of time?

Luckily, there is a better way.

Proceed to Domain Restrictions on TI Graphing Calculators (Part 2/5)

UPDATE: Updated clip links.