Question

identify the graph of this rational function. $y = \frac{-3x^{2}}{x^{2}-1}$

Explanation:

Step1: Find vertical asymptotes

Set denominator \(x^{2}-1 = 0\).
\(x^{2}-1=(x + 1)(x - 1)=0\), so \(x=-1\) and \(x = 1\) are vertical asymptotes.

Step2: Find horizontal asymptote

Since the degree of numerator and denominator is the same (\(n=m = 2\)).
For \(y=\frac{-3x^{2}}{x^{2}-1}\), the horizontal asymptote is \(y=\frac{-3}{1}=-3\).

Step3: Analyze the sign of the function

When \(x = 0\), \(y=\frac{-3\times0^{2}}{0^{2}-1}=0\).
As \(x
ightarrow\pm\infty\), \(y=\frac{-3x^{2}}{x^{2}-1}=\frac{-3}{1-\frac{1}{x^{2}}}
ightarrow - 3\).
When \(x\) is a small positive number close to \(0\) (e.g., \(x = 0.5\)), \(y=\frac{-3\times(0.5)^{2}}{(0.5)^{2}-1}=\frac{-3\times0.25}{0.25 - 1}=\frac{-0.75}{-0.75}=1>0\).
When \(|x|\) is large, \(y\approx - 3\).

Answer:

The second graph (the middle - one) is the correct graph of the function \(y=\frac{-3x^{2}}{x^{2}-1}\) as it has vertical asymptotes at \(x=-1\) and \(x = 1\), a horizontal asymptote at \(y=-3\) and passes through the origin \((0,0)\) and has the correct behavior for values of \(x\) near \(0\) and as \(x
ightarrow\pm\infty\).