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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:

| \( x \)  | \( r(x) \)    |
|----------|---------------|
| \(-1.1\) | \(-\infty\)   |
| \(-0.9\) | \(+\infty\)   |
| \(0.9\)  | \(+\infty\)   |
| \(1.1\)  | \(-\infty\)   |

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.

Would you like further details or have any questions?

### 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.

The graph of the rational function r is shown. Which of the following tables could be used to describe the asymptotic behavior of r at x = -1 and at x = 1?

Explore the asymptotic behavior of a rational function from a graph where vertical asymptotes occur at x = -1 and x = 1. The problem covers key mathematical concepts of limits and rational functions. Suitable for Grades 11-12 students.

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