The graph in the image represents a rational function $r(x)$, and the question asks which of the given tables (though not shown) could describe the asymptotic behavior of $r$ at $x = -1$ and $x = 1$.

Analysis:

1. Vertical Asymptote at $x = 1$:

   • The graph shows a vertical asymptote at $x = 1$ because the function approaches positive or negative infinity as $x$ gets closer to 1 from either side.
   • On the left of $x = 1$ (as $x \to 1^-$), the function goes to negative infinity $(-\infty)$.
   • On the right of $x = 1$ (as $x \to 1^+$), the function goes to positive infinity $(+\infty)$.

2. Behavior at $x = -1$:

   • At $x = -1$, the graph shows that the function has a minimum point rather than an asymptote. There is no vertical asymptote here, and the function curves back upwards after reaching a low point.

Based on this information, a table describing the asymptotic behavior should include:

• At $x = 1$: The function tends to $-\infty$ from the left and $+\infty$ from the right.
• At $x = -1$: The function has a minimum and does not approach infinity.

Would you like to provide the tables so I can help further in comparing them, or do you have additional questions about this behavior?

Here are some follow-up questions you might explore:

1. What is the formal definition of vertical asymptotes in rational functions?
2. Can you explain how to find horizontal asymptotes in rational functions?
3. How does the behavior of a rational function near an asymptote differ from its behavior at other points?
4. How would the function's equation change if the vertical asymptote moved?
5. Can a rational function have more than one vertical asymptote? Why?

Tip: Vertical asymptotes occur where the denominator of a rational function equals zero, provided the numerator does not also zero out at that point.