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: $y = 2$; vertical: $x = 3$ and $x = - 3$
B) horizontal: $x = - 2$; vertical: $y = 3$ and $y = - 3$
C) horizontal: $y = - 3$; vertical: $x = 2$
D) horizontal: $y = 2$; vertical: $x = 3$
E) horizontal: $x = 3$ and $x = - 3$; vertical: $y = - 2$

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: $y = 2$; vertical: $x = 3$ and $x = - 3$
B) horizontal: $x = - 2$; vertical: $y = 3$ and $y = - 3$
C) horizontal: $y = - 3$; vertical: $x = 2$
D) horizontal: $y = 2$; vertical: $x = 3$
E) horizontal: $x = 3$ and $x = - 3$; vertical: $y = - 2$

Quizplus · Accepted Answer

To find the vertical asymptotes, we need to find the values of $x$ that make the denominator of the fraction equal to zero. We have $$x^2-9=(x-3)(x+3)=0.$$Thus, the vertical asymptotes are at $x=3$ and $x=-3$.

To find the horizontal asymptote, we look at the end behavior of the function as $x$ gets very large or very small in magnitude. For large values of $x$ (either positive or negative), the term $x^2$ dominates the denominator and  $$f(x) \approx \frac{2x^2}{x^2}=2.$$This means that the horizontal asymptote is at $y=2$.

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