Given the function g, of, x, equals, minus, x, squared, plus, 8, x, plus, 24*g*(*x*)=−*x*2+8*x*+24, determine the average rate of change of the function over the interval 2, is less than or equal to, x, is less than or equal to, 102≤*x*≤10.

To find the average rate of change of the function \( g(x) = -x^2 + 8x + 24 \) over the interval \([2, 10]\), we'll use the formula:

\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]

where \( a = 2 \) and \( b = 10 \).

### Step 1: Evaluate the function at the endpoints of the interval.

1. **Find \( g(2) \):**

\[
g(2) = -(2)^2 + 8(2) + 24
\]
\[
g(2) = -4 + 16 + 24 = 36
\]

2. **Find \( g(10) \):**

\[
g(10) = -(10)^2 + 8(10) + 24
\]
\[
g(10) = -100 + 80 + 24 = 4
\]

### Step 2: Calculate the average rate of change.

\[
\text{Average Rate of Change} = \frac{g(10) - g(2)}{10 - 2}
\]
\[
\text{Average Rate of Change} = \frac{4 - 36}{10 - 2}
\]
\[
\text{Average Rate of Change} = \frac{-32}{8} = -4
\]

So, the average rate of change of the function \( g(x) \) over the interval \([2, 10]\) is \(-4\).

Would you like any further explanation on these steps? Here are some related questions:

1. How does the sign of the average rate of change affect the interpretation of the function's behavior?
2. What is the significance of the average rate of change in a real-world context?
3. How can the vertex of the parabola described by \( g(x) \) affect the average rate of change?
4. What would the average rate of change be if the interval were different, say from \(4\) to \(8\)?
5. How can you use the derivative to find the instantaneous rate of change at any point within the interval?

**Tip:** When dealing with quadratic functions, the average rate of change can help you understand the overall behavior of the function over a specified interval, including whether it is generally increasing or decreasing.

Learn how to calculate the average rate of change of the quadratic function g(x) = -x^2 + 8x + 24 over the interval [2, 10]. Follow a step-by-step solution including evaluating the function at interval endpoints and applying the Average Rate of Change formula. Suitable for students in Grades 9-12.

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