To determine the average rate of change of the function $f(x) = -x^2 + 5x + 14$ over the interval $0 \leq x \leq 9$, we use the formula for the average rate of change of a function over an interval:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

where $a = 0$ and $b = 9$.

Step 1: Find $f(0)$

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

Step 2: Find $f(9)$

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

Step 3: Apply the formula for the average rate of change

$\text{Average rate of change} = \frac{f(9) - f(0)}{9 - 0} = \frac{-22 - 14}{9} = \frac{-36}{9} = -4$

Thus, the average rate of change of the function over the interval $0 \leq x \leq 9$ is $-4$.

Would you like further details or have any questions?

Here are five related questions you can explore:

1. How does the average rate of change compare to the instantaneous rate of change at $x = 4.5$?
2. What is the significance of a negative average rate of change?
3. How would the rate of change differ over a smaller interval, such as $0 \leq x \leq 5$?
4. Can you derive the equation of the tangent line to $f(x)$ at $x = 0$?
5. What is the vertex of the parabola represented by $f(x)$, and how does it relate to the rate of change?

Tip: The average rate of change of a quadratic function over an interval gives insight into the overall behavior of the curve—whether it rises or falls on average.