Given the function f, of, x, equals, minus, x, squared, plus, 5, x, plus, 14*f*(*x*)=−*x*2+5*x*+14, determine the average rate of change of the function over the interval 0, is less than or equal to, x, is less than or equal to, 90≤*x*≤9.

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.

Given the function f(x) = -x^2 + 5x + 14, determine the average rate of change of the function over the interval 0 ≤ x ≤ 9.

Learn how to calculate the average rate of change of the quadratic function f(x) = -x^2 + 5x + 14 over the interval 0 ≤ x ≤ 9. This solution demonstrates key algebraic techniques and formulas for understanding the behavior of quadratic functions. Suitable for students in Grades 8-10.

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