Graphing

In each of the following first determine the asymptotes and intercepts. Finally, make a graph which shows these details.

Example 1

Solution .

The function factors into The domain includes all values of x except for x = -1 and x = 2. Since the function reduces to , we see that it has x-intercept (1, 0), y-intercept (0, 1/2), vertical asymptote x = 2, and horizontal asymptote y =1.

Example 2

Solution .

Factoring we have . The domain is all real numbers except for x = -2 and x = 3. The x-intercepts are ( ) and ( ). The y-intercept is (0, 1). The vertical asymptotes are x = -2 and x = 3. The horizontal asymptote is y = 3.

Example 3

Solution .

(0, 0) is the x-intercept and the y-intercept. Since there are no real values for which the denominator is zero, there are no vertical asymptotes. y = 0 is the horizontal asymptote.

Example 4

Solution .

The domain is all real numbers except for x = 0. The x-intercept is (1, 0). There is no y-intercept since x cannot be zero. The vertical asymptote is x = 0. Since the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote.

It can be shown that for large values of x this function approaches This is called an oblique asymptote.