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We are tasked with finding the average rate of change of the function \( f(x) = -x^2 - 8x + 21 \) over the interval \([-10, 0]\).

The average rate of change of a function over an interval \([a, b]\) is given by the formula:

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

For this problem:
- The function is \( f(x) = -x^2 - 8x + 21 \)
- The interval is \([-10, 0]\), so \( a = -10 \) and \( b = 0 \)

### Step 1: Calculate \( f(-10) \)

\[
f(-10) = -(-10)^2 - 8(-10) + 21
= -100 + 80 + 21 = 1
\]

### Step 2: Calculate \( f(0) \)

\[
f(0) = -(0)^2 - 8(0) + 21 = 21
\]

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

\[
\text{Average rate of change} = \frac{f(0) - f(-10)}{0 - (-10)} = \frac{21 - 1}{10} = \frac{20}{10} = 2
\]

Thus, the average rate of change of the function over the interval \([-10, 0]\) is \( 2 \).

Would you like further details or clarification?

Here are 5 related questions you might find helpful:
1. How is the average rate of change different from the instantaneous rate of change?
2. How does the quadratic nature of \( f(x) = -x^2 - 8x + 21 \) influence its rate of change?
3. Can the average rate of change ever be negative?
4. What does the sign of the average rate of change tell us about the behavior of the function?
5. How can we interpret the average rate of change in real-life scenarios involving quadratic functions?

**Tip:** The average rate of change between two points gives the slope of the secant line connecting those points on the graph.

Given the function f(x) = -x^2 - 8x + 21, determine the average rate of change of the function over the interval -10 ≤ x ≤ 0.

Learn how to determine the average rate of change for the quadratic function f(x) = -x^2 - 8x + 21 over the interval [-10, 0]. This problem involves applying the rate of change formula and calculating specific function values. Suitable for students in Grades 9-11.

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