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.
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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?
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:
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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.)