Determine the equations of the vertical and horizontal asymptotes of the graph of the function $$f ( x ) = \frac { 2 x ^ { 2 } } { x ^ { 2 } - 9 }$$
A) horizontal: $y = - 3$; vertical: $x = 2$
B) horizontal: $x = 3$ and $x = - 3$; vertical: $y = - 2$
C) horizontal: $y = 2$; vertical: $x = 3$ and $x = - 3$
D) horizontal: $x = - 2$; vertical: $y = 3$ and $y = - 3$
E) horizontal: $y = 2$; vertical: $x = 3$

Question

Determine the equations of the vertical and horizontal asymptotes of the graph of the function $$f ( x ) = \frac { 2 x ^ { 2 } } { x ^ { 2 } - 9 }$$ 
A) horizontal: $y = - 3$; vertical: $x = 2$
B) horizontal: $x = 3$ and $x = - 3$; vertical: $y = - 2$
C) horizontal: $y = 2$; vertical: $x = 3$ and $x = - 3$
D) horizontal: $x = - 2$; vertical: $y = 3$ and $y = - 3$
E) horizontal: $y = 2$; vertical: $x = 3$

Quizplus · Accepted Answer

Horizontal asymptotes are determined by the behavior of the function as $x$ approaches infinity. For the given function, as $x$ approaches infinity, the ratio of the coefficients of the highest degree terms in the numerator and denominator ($\frac{2x^2}{x^2}$) approaches 2, indicating a horizontal asymptote at $y = 2$. Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points. Setting the denominator equal to zero gives $x^2 - 9 = 0$, or $x = \pm 3$, indicating vertical asymptotes at $x = 3$ and $x = -3$.

Determine the Equations of the Vertical and Horizontal Asymptotes of the Graph