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The given problem asks us to analyze the rate of change of the function \( g(x) = -x^2 - 9x - 20 \).

### Steps to analyze the rate of change:
1. **Find the derivative**: 
   To understand the rate of change of \( g(x) \), we calculate its derivative \( g'(x) \). The derivative represents the rate of change of the function.

   \[
   g'(x) = \frac{d}{dx} \left( -x^2 - 9x - 20 \right)
   \]
   Applying the power rule:
   \[
   g'(x) = -2x - 9
   \]
   
2. **Set the derivative equal to zero**: 
   To find critical points where the rate of change may change from positive to negative or vice versa, we set the derivative equal to zero.
   \[
   -2x - 9 = 0
   \]
   Solving for \( x \):
   \[
   -2x = 9
   \]
   \[
   x = -4.5
   \]

3. **Analyze the sign of the derivative**: 
   - For \( x < -4.5 \), substitute any value less than \( -4.5 \) (e.g., \( x = -5 \)) into \( g'(x) \):
     \[
     g'(-5) = -2(-5) - 9 = 10 - 9 = 1
     \]
     So, \( g'(x) > 0 \) for \( x < -4.5 \) (positive rate of change).
   
   - For \( x > -4.5 \), substitute any value greater than \( -4.5 \) (e.g., \( x = 0 \)) into \( g'(x) \):
     \[
     g'(0) = -2(0) - 9 = -9
     \]
     So, \( g'(x) < 0 \) for \( x > -4.5 \) (negative rate of change).

### Conclusion:
- The function has a positive rate of change for \( x < -4.5 \) and a negative rate of change for \( x > -4.5 \).
- The correct answer is **option 2**: 
  "For any interval \( x < -4.5 \), the function has a positive rate of change, and for any interval \( x > -4.5 \), the function has a negative rate of change."

Would you like more details on this process, or do you have any questions?

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Here are 5 related questions:
1. How do we interpret the sign of the derivative in terms of rate of change?
2. What is the significance of the critical point \( x = -4.5 \) for this function?
3. How would the graph of \( g(x) \) behave based on this derivative analysis?
4. How does the second derivative help determine concavity in such problems?
5. Can this process be applied to any quadratic function to determine its rate of change?

**Tip**: Understanding the sign of the derivative helps you determine when a function is increasing or decreasing, which is crucial in analyzing real-world applications.

Given the function g(x) = -x^2 - 9x - 20, which of the following describes g?

Understand how to analyze the rate of change for the function g(x) = -x^2 - 9x - 20. Learn how to calculate the derivative g'(x) = -2x - 9, find critical points, and determine the behavior of the function for different intervals. Suitable for students in Grades 10-12.

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