This is a handout that will help you systematically sketch functions on a coordinate plane. This handout also contains definitions of relevant terms needed for curve sketching. Another handout available in the Tutoring Center has 3 sample problems worked out completely.

ASYMPTOTES:

This handout will discuss three kinds of asymptotes: vertical, horizontal, and slant.

VERTICAL ASYMPTOTES

We define the line x = c as a vertical asymptote of the graph of , iff (if and only if) approaches infinity (or negative infinity) as x approaches c from the right or left.

The concept of an asymptote is best illustrated in the following example:

Take the function

Here, we can see that x cannot take the value of 1, otherwise, would be undefined. Also:

and

In this case, we call the line x = 1 a vertical asymptote of .

The fact that is undefined at x = 1 is not enough to conclude that we have a vertical asymptote. The function must also approach infinity or negative infinity as x approaches the value at which is undefined.

Consider the following problem:

The function is undefined at x = -2 but we do not have an asymptote. Notice the following:

We conclude that approaches -1 as x approaches -2. This function has a "hole", not an asymptote, at the value for which is undefined.

Once again, in order to have an asymptote at x = c, must have a discontinuity at c and must approach infinity or negative infinity, as x approaches c from the left or the right.

HORIZONTAL ASYMPTOTES

We define the line y = L as a horizontal asymptote of the graph of f(x), iff f(x) approaches L as x approaches infinity (or negative infinity).

For the function the line is the horizontal asymptote of the graph of .

The following limit shows why this is true:

and

When x approaches infinity, approaches the line, and when x approaches negative infinity, also approaches the line .

A quick way to determine the