Question 2(Multiple Choice Worth 3 points)
(Rational Functions' Vertical Asymptotes and Holes MC)

The rational function r is defined as r of x equals the fraction with numerator x squared minus 5 x minus 14 and denominator x squared plus 9 x plus 14. Which of the following statements describes the behavior of r ?

The graph of r has an intercept at x = 2.
The graph of r has a hole at x = –7 and a vertical asymptote at x = –2.
The graph of r has a hole at x = –2 and a vertical asymptote at x = –7.
The graph of r has zeros at x = 2 and x = 7 because r (2) = r (7) = 0.

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.

The rational function r is defined as r of x equals the fraction with numerator x squared minus 5 x minus 14 and denominator x squared plus 9 x plus 14. Which of the following statements describes the behavior of r?

Explore how to analyze the rational function r(x) = (x^2 - 5x - 14)/(x^2 + 9x + 14) for its zeros, holes, and vertical asymptotes. This detailed solution reveals a hole at x = -2 and a vertical asymptote at x = -7, making it perfect for high school students studying rational functions.

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