Part 1 of 4
Points: 0 of 1
Use your knowledge of asymptotes and intercepts to match the equation, $f(x)=\frac{27}{x^{2}-9}$, with one of the graphs that follow. List all the asymptotes. Check your work using a graphing calculator.
Determine the vertical asymptote(s), if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function has one vertical asymplote, $\square$
(Simplify your answer. Type an equation. Use integers or fractions for any numbers in the equation.)
B. The function has two vertical asymptotes. The leftmost asymptote is $\square$, and the rightmost asymptote is $\square$.
(Simplify your answers. Type equations. Use integers or fractions for any numbers in the equations.)
C. The function has no vertical asymptote.
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Answer
\(\boxed{\text{The function has two vertical asymptotes. The leftmost asymptote is } x=-3, \text{ and the rightmost asymptote is } x=3}\)
Step 1 ：The vertical asymptotes of a function are the values of x for which the function is undefined. In this case, the function is undefined when the denominator is equal to zero.
Step 2 ：So, we need to solve the equation \(x^{2} - 9 = 0\) to find the vertical asymptotes.
Step 3 ：The solutions to the equation \(x^{2} - 9 = 0\) are \(x = -3\) and \(x = 3\). These are the values of x for which the function is undefined, so they are the vertical asymptotes of the function.
Step 4 ：\(\boxed{\text{The function has two vertical asymptotes. The leftmost asymptote is } x=-3, \text{ and the rightmost asymptote is } x=3}\)
