Question

which of the following could be the function graphed?
$f(x)=\frac{8x^2}{x^2 - 9}$
$f(x)=\frac{119x}{792x + 345}$
$f(x)=\frac{612}{816x - 272}$
$f(x)=\frac{237x}{421x - 515}$

Explanation:

Step1: Analyze the vertical asymptotes

For a rational function \(y = \frac{N(x)}{D(x)}\), vertical asymptotes occur at the values of \(x\) that make \(D(x)=0\).
For \(f(x)=\frac{8x^{2}}{x^{2}-9}=\frac{8x^{2}}{(x + 3)(x - 3)}\), \(D(x)=x^{2}-9=(x + 3)(x - 3)\). Setting \(D(x)=0\), we get \(x=-3\) and \(x = 3\).
For \(f(x)=\frac{119x}{792x+345}\), setting \(D(x)=792x + 345=0\), we get \(x=-\frac{345}{792}\).
For \(f(x)=\frac{612}{816x-272}\), setting \(D(x)=816x-272 = 0\), we get \(x=\frac{272}{816}=\frac{1}{3}\).
For \(f(x)=\frac{237x}{421x-515}\), setting \(D(x)=421x-515=0\), we get \(x=\frac{515}{421}\).

Step2: Analyze the \(x\) - intercepts

The \(x\) - intercepts of a rational function \(y=\frac{N(x)}{D(x)}\) occur at the values of \(x\) that make \(N(x)=0\) (and \(D(x)
eq0\) at those \(x\) values).
For \(f(x)=\frac{8x^{2}}{x^{2}-9}\), setting \(N(x)=8x^{2}=0\), we get \(x = 0\).
For \(f(x)=\frac{119x}{792x+345}\), setting \(N(x)=119x=0\), we get \(x = 0\).
For \(f(x)=\frac{612}{816x-272}\), \(N(x)=612
eq0\), so there is no \(x\) - intercept.
For \(f(x)=\frac{237x}{421x-515}\), setting \(N(x)=237x=0\), we get \(x = 0\).

Step3: Analyze the horizontal asymptotes

For a rational function \(y=\frac{N(x)}{D(x)}\) where \(N(x)=a_{n}x^{n}+\cdots\) and \(D(x)=b_{m}x^{m}+\cdots\)

• If \(n=m\), the horizontal asymptote is \(y=\frac{a_{n}}{b_{m}}\)
• If \(n\lt m\), the horizontal asymptote is \(y = 0\)
• If \(n\gt m\), there is no horizontal asymptote (for \(n=m + 1\) there is a slant asymptote)

For \(f(x)=\frac{8x^{2}}{x^{2}-9}\), \(n = m=2\), and \(y=\frac{8}{1}=8\)
For \(f(x)=\frac{119x}{792x+345}\), \(n=m = 1\), and \(y=\frac{119}{792}\approx0.15\)
For \(f(x)=\frac{612}{816x-272}\), \(n=0,m = 1\), and \(y = 0\)
For \(f(x)=\frac{237x}{421x-515}\), \(n=m = 1\), and \(y=\frac{237}{421}\approx0.56\)

Answer:

\(f(x)=\frac{8x^{2}}{x^{2}-9}\)