Question

follow the seven step strategy to graph the following rational function.
$f(x) = -\frac{2}{x^2 - 9}$

find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the equation(s) of the vertical asymptote(s) is/are $x = -3, x = 3$.
(type an equation. use a comma to separate answers as needed.)
b. there is no vertical asymptote.

find the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the equation(s) of the horizontal asymptote(s) is/are \boxed{}.
(type an equation. use a comma to separate answers as needed.)
b. there is no horizontal asymptote.

Explanation:

Step1: Recall Horizontal Asymptote Rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator:

• If the degree of \( N(x) \) (denoted as \( n \)) is less than the degree of \( D(x) \) (denoted as \( m \)), the horizontal asymptote is \( y = 0 \).
• If \( n = m \), the horizontal asymptote is \( y=\frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).
• If \( n>m \), there is no horizontal asymptote (but there may be an oblique asymptote).

Step2: Determine Degrees of Numerator and Denominator

For \( f(x)=-\frac{2}{x^{2}-9} \), the numerator \( N(x)= - 2 \) has degree \( n = 0 \) (since it's a constant, \( x^{0} \) term). The denominator \( D(x)=x^{2}-9 \) has degree \( m = 2 \).

Step3: Apply the Rule

Since \( n = 0 \) and \( m = 2 \), and \( n

Answer:

A. The equation(s) of the horizontal asymptote(s) is/are \( y = 0 \)