Question

follow the seven step strategy to graph the following rational function
$f(x) = \frac{3x}{x^2 - 4}$

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 $x = -2, x = 2$
(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
(type an equation. use a comma to separate answers as needed.)
b. there is no horizontal asymptote.

Explanation:

Step1: Recall Horizontal Asymptote Rules

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 \( N(x) \) (let \( n \)) is less than the degree of \( D(x) \) (let \( m \)), 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 \( f(x)=\frac{3x}{x^2 - 4} \):

• The numerator \( N(x)=3x \) has degree \( n = 1 \).
• The denominator \( D(x)=x^2 - 4 \) has degree \( m = 2 \).

Since \( n = 1 < m = 2 \), by the horizontal asymptote rule, the horizontal asymptote is \( y = 0 \).

Answer:

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