To analyze this problem, let's first examine the asymptotic behavior of the rational function $r(x)$ as $x \to -1$ and $x \to 1$ using the given graph.

Step 1: Identify Vertical Asymptotes

A vertical asymptote occurs where the function approaches $\pm \infty$. From the graph, we can observe:

• As $x \to -1$ from the left, $r(x) \to -\infty$, and as $x \to -1$ from the right, $r(x) \to +\infty$. This suggests a vertical asymptote at $x = -1$.
• As $x \to 1$ from the left, $r(x) \to +\infty$, and as $x \to 1$ from the right, $r(x) \to -\infty$. This suggests a vertical asymptote at $x = 1$.

Step 2: Confirm Behavior with a Table

The problem asks for a table that captures this asymptotic behavior. While the original graph provides the necessary visual information, a possible table would look like this:

The above table captures the behavior at points approaching $x = -1$ and $x = 1$ correctly.

Conclusion

The correct interpretation of the graph is that the function has vertical asymptotes at $x = -1$ and $x = 1$, and the behavior near these points matches the descriptions provided in the table.

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Related Questions

1. What is the general form of a rational function with two vertical asymptotes?
2. How do you find horizontal asymptotes of a rational function?
3. How can you confirm vertical asymptotes algebraically from the equation of a rational function?
4. What does the sign change around an asymptote tell you about the function's behavior?
5. How would you use limits to describe the asymptotic behavior more formally?

Tip

Remember, a rational function $\frac{p(x)}{q(x)}$ will have vertical asymptotes where $q(x) = 0$, provided $p(x) \neq 0$ at those points.