Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { 2 x ^ { 2 } - x - 9 } { x ^ { 2 } + 3 x + 3 }$$ .
A)horizontal: y = 4;vertical: x = 0
B)horizontal: y = - 3;vertical: x = -4
C)horizontal: y = 2;vertical: none
D)horizontal: y = 1;vertical: x = - 3
E)horizontal: y = -2;vertical: x = 3 and x = -4

Question

Quizplus · Accepted Answer

The horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator when the degrees are the same, which gives $y = \frac{2}{1} = 2$. The denominator $x^2 + 3x + 3$ has no real roots (since its discriminant $b^2 - 4ac = 9 - 12 < 0$), so there are no vertical asymptotes.

Determine the Equations of Any Horizontal and Vertical Asymptotes Of f