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.