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To determine the average rate of change of the function \( h(x) = x^2 - 10x + 21 \) over the interval \( -1 \leq x \leq 10 \), we use the formula for the average rate of change of a function over an interval \([a, b]\):

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

Here, \( a = -1 \) and \( b = 10 \). First, we need to evaluate \( h(-1) \) and \( h(10) \).

1. **Find \( h(-1) \):**
\[
h(-1) = (-1)^2 - 10(-1) + 21 = 1 + 10 + 21 = 32
\]

2. **Find \( h(10) \):**
\[
h(10) = (10)^2 - 10(10) + 21 = 100 - 100 + 21 = 21
\]

Now, we can calculate the average rate of change:

\[
\text{Average rate of change} = \frac{h(10) - h(-1)}{10 - (-1)} = \frac{21 - 32}{10 + 1} = \frac{-11}{11} = -1
\]

Thus, the average rate of change of the function over the interval \( -1 \leq x \leq 10 \) is **-1**.

Would you like more details or have any questions?

Here are five related questions to explore:
1. What does the average rate of change tell us about a function's behavior?
2. How do we interpret a negative average rate of change geometrically?
3. What is the difference between average rate of change and instantaneous rate of change?
4. How do you find the instantaneous rate of change using derivatives?
5. How does the shape of a quadratic function affect its rate of change?

**Tip:** The average rate of change of a quadratic function over any interval depends on how the curve behaves over that range. For linear functions, the rate of change is constant over any interval.

Given the function h(x) = x^2 - 10x + 21, determine the average rate of change of the function over the interval -1 ≤ x ≤ 10.

This problem involves finding the average rate of change of a quadratic function h(x) = x^2 - 10x + 21 over the interval [-1, 10]. The solution includes key algebraic concepts such as the calculation of function values and applying the formula for the average rate of change.

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