# Holes

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

## Plot a rational function whose only hole occurs at (-1,3).

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 or , 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 ?