To solve this problem, we need to find the average rate of change of the function $f(x) = -x^2 + 5x + 14$ over the interval $[-2, 9]$.

The average rate of change is given by the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Where:

• $f(x)$ is the given function.
• $a = -2$ and $b = 9$ (the interval bounds).

Step 1: Find $f(-2)$

$f(-2) = -(-2)^2 + 5(-2) + 14 = -(4) - 10 + 14 = -4 - 10 + 14 = 0$

Step 2: Find $f(9)$

$f(9) = -(9)^2 + 5(9) + 14 = -(81) + 45 + 14 = -81 + 45 + 14 = -22$

Step 3: Apply the formula

$\text{Average Rate of Change} = \frac{f(9) - f(-2)}{9 - (-2)} = \frac{-22 - 0}{9 - (-2)} = \frac{-22}{9 + 2} = \frac{-22}{11} = -2$

So, the average rate of change is $-2$.

Would you like further details or have any questions?

Here are some related questions:

1. How would the average rate of change differ if the interval was smaller?
2. Can we find the instantaneous rate of change at a specific point?
3. How does the shape of the parabola $f(x) = -x^2 + 5x + 14$ affect its rate of change?
4. What is the significance of the negative sign in the average rate of change?
5. How would the result change if the function was linear instead of quadratic?

Tip: The average rate of change gives the slope of the secant line connecting two points on the curve.