9. What are all the asymptotes of the graph of xy + y = (x 2)2 ?

A. x = 0 and y = 0

B. x = -1 and y = 0

C. x = -1 and y = 5 x

D. x = -1 and y = x - 5

E. x=-1, y=0, and y=x-5

 

Solution: The answer is D

First solve for y in terms of x as follows:

Since y is a rational function, y has a vertical asymptote at the zeros of the denominator of y . That is at,

Since the degree of the numerator is greater than the degree of the denominator, we can say that y has no horizontal asymptotes. However, since this difference in degrees is only by a value of one, then we can say that y has a slant asymptote. The slant asymptote is the quotient resulting from dividing the numerator by the denominator, using the method of long division, as follows:

 

Thus, all the asymptotes of the graph of the given curve are x = -1, and y = x - 5.