To analyze the rational function $r(x) = \frac{x^2 - 5x - 14}{x^2 + 9x + 14}$, we need to identify its zeros, holes, and vertical asymptotes.

1. Finding Zeros: The zeros of the function occur when the numerator is zero: $x^2 - 5x - 14 = 0$ Factoring gives: $(x - 7)(x + 2) = 0$ So, $x = 7$ and $x = -2$ are the zeros.

2. Finding Holes and Vertical Asymptotes: We look for values that make the denominator zero: $x^2 + 9x + 14 = 0$ Factoring gives: $(x + 7)(x + 2) = 0$ So, the denominator is zero at $x = -7$ and $x = -2$.

   Since $x = -2$ is also a zero of the numerator, there is a hole at $x = -2$. The remaining factor $x + 7 = 0$ in the denominator creates a vertical asymptote at $x = -7$.

Summary:

• Zeros: $x = 7$ and $x = -2$
• Hole: $x = -2$
• Vertical Asymptote: $x = -7$

Conclusion:

The correct statement is: The graph of r has a hole at $x = -2$ and a vertical asymptote at $x = -7$.

Would you like further details or explanations on this? Here are some questions for consideration:

1. How do you find holes in rational functions?
2. What is the significance of vertical asymptotes?
3. Can you explain the difference between a zero and a hole?
4. How do you graph rational functions with holes and asymptotes?
5. What happens to the function's value as it approaches the asymptote?

Tip: Always factor both the numerator and denominator to clearly identify zeros, holes, and vertical asymptotes.