A close look at end behavior

My problem with BOB0 - BOTN - EATS DC is two-fold. Like most mnemonic devices in K-12 mathematics:
  • It oversimplifies the concept.
  • It is a technique that encourages rote memorization, rather than conceptual understanding.
Here's how I would prefer students approach investigating end behavior. Recall that a rational function can be written as a ratio of polynomials. My advice: actually divide those polynomials. Here's an example:

.

I used polynomial long division there. So the in is called the quotient, and the is called the remainder (because goes into times with a remainder of ). Since , , and thus our rational function has a horizontal asymptote at .

Which type of rational function was this?

Check all that apply

What is the quotient in a BOB0 rational function? Explain why that's consistent with the BOB0 rule.

Here's an example that begins to illustrate why I think BOB0 - BOTN - EATS DC oversimplifies this concept:

.

So the quotient is . Use the graph below to plot this rational function.

What you're seeing is that this function is asymptotic not to a horizontal line, but to the line . is what's called a slant asymptote of . Explain how BOTN oversimplifies this situation.

Plot a rational function that is asymptotic to a parabola.

Plot a rational function that is (not equal but) asymptotic to the parabola . (Hint: You'll want to work backwards through the polynomial long division. Finding a common denominator will be a helpful technique.)