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.
This handout will discuss three kinds of asymptotes: vertical, horizontal, and slant.
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:
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.
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:
When x approaches infinity, approaches the line, and when x approaches negative infinity, also approaches the line .
A quick way to determine the position of the horizontal asymptote of a rational function (having no common factors) is with the following method. Look at the highest degree in the numerator and the highest degree in the denominator.
- If the highest degree is in the denominator, then the horizontal asymptote is y = 0.
- If the highest degree in the numerator and the highest degree in the denominator
are equal, the horizontal asymptote is the ratio of the coefficient of the highest
degree term in the numerator to the coefficient of the highest degree term in the
(In our previous example y = , the highest degree in the numerator is 1 and the
highest degree in the denominator is 1. The ratio of highest degree term coefficients
is . So the horizontal asymptote is y = -2.)
- If the highest degree in the numerator is one degree larger than the highest degree in the numerator, then the function has a slant asymptote.
If the highest degree in the numerator of a rational function (having no common factors) is one degree larger than the highest degree in the denominator, we say that the function has a slant asymptote.
To determine the asymptote, rewrite the function in terms of a polynomial + another rational function. For example, let
Dividing the numerator by x - 1 we get
f(x) = -x + 2 +
Since the fraction approaches 0 as x approaches infinity and negative infinity, the function f(x) approaches the line y = -x + 2 as x approaches infinity and negative infinity. y = -x + 2 is called a slant asymptote.
Some remarks about functions with asymptotes:
- Vertical asymptotes are NEVER crossed by f(x). However, the graph of f(x) may sometimes cross a horizontal or slant asymptote.
- Asymptotes help determine the shape of the graph.
- Polynomial functions never have asymptotes.
INCREASING AND DECREASING:
A function is...
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