Holes

Plot a rational function whose only hole occurs at x=-1.

The function has a hole. Find the coordinates of that hole.

It is not quite correct, however, to say that a rational function

of the form

has a hole at

. Plot a counterexample below.

Suppose

is a rational function that is undefined at

. Let's write

in the form

, where

does not factor

is a non-negative integer, and

is a positive integer. (We know

must be at least

because

is undefined, so our denominator must equal zero when

.) We're going to investigate how

behaves around

depending on

and

's relationship. You can use the graph at the bottom of this page to experiment in order to answer the questions below.

What happens if ?

What happens if ?

What happens if ?