Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{3 x^{2}+1}{x^{2}-9} ).
a. the ( y )-intercept(s) is/are at ( y=-\frac{1}{9} ).
(simplify your answer. use a comma to separate answers as needed.)
b. there are no ( y )-intercepts.
find any horizontal asymptotes of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one horizontal asymptote, (type an equation.)
b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is (type equations.)
c. there are no horizontal asymptotes.

Explanation:

Step1: Find the horizontal asymptote

For a rational function \(f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n = m\), the horizontal asymptote is \(y=\frac{a_n}{b_m}\).
For \(f(x)=\frac{3x^{2}+1}{x^{2}-9}\), here \(n = 2\), \(m = 2\), \(a_n=3\), \(b_m = 1\).

Step2: Calculate the value of the horizontal asymptote

Using the formula \(y=\frac{a_n}{b_m}\), we substitute \(a_n = 3\) and \(b_m=1\). So \(y=\frac{3}{1}=3\).

Answer:

A. The function has one horizontal asymptote, \(y = 3\).