Question

identify the graph of this rational function.
$y = \frac{5}{x^2 - 1}$

Explanation:

Step1: Analyze the vertical asymptotes

To find vertical asymptotes, set the denominator equal to zero: $x^2 - 1 = 0$. Factoring, we get $(x - 1)(x + 1) = 0$, so $x = 1$ and $x = -1$ are vertical asymptotes (dashed red lines).

Step2: Analyze the horizontal asymptote

For the rational function $y=\frac{5}{x^2 - 1}$, the degree of the numerator (0) is less than the degree of the denominator (2), so the horizontal asymptote is $y = 0$ (the x - axis).

Step3: Analyze the sign of the function

• When $x = 0$, $y=\frac{5}{0 - 1}=-5$. So the point $(0, - 5)$ is on the graph.
• For $x>1$, say $x = 2$, $y=\frac{5}{4 - 1}=\frac{5}{3}>0$.
• For $0
• For $-1
• For $x < - 1$, say $x=-2$, $y=\frac{5}{4 - 1}=\frac{5}{3}>0$.

So the graph should have parts above the x - axis when $x>1$ or $x < - 1$ and below the x - axis when $-11$ or $x < - 1$ and the bottom part below (passing through (0, - 5)) matches.

Answer:

The second graph (the one with the two upper curves above the horizontal asymptote (y = 0) for $x>1$ and $x < - 1$, and the lower curve below the horizontal asymptote passing through (0, - 5))