Given the function h, of, x, equals, minus, x, squared, minus, 4, x, plus, 7*h*(*x*)=−*x*2−4*x*+7, determine the average rate of change of the function over the interval minus, 7, is less than or equal to, x, is less than or equal to, 0−7≤*x*≤0.

To determine the average rate of change of the function \( h(x) = -x^2 - 4x + 7 \) over the interval \(-7 \leq x \leq 0\), we can use the following formula:

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

where \(a = -7\) and \(b = 0\).

1. **Calculate \( h(-7) \):**

\[
h(-7) = -(-7)^2 - 4(-7) + 7
\]

\[
h(-7) = -49 + 28 + 7 = -14
\]

2. **Calculate \( h(0) \):**

\[
h(0) = -(0)^2 - 4(0) + 7 = 7
\]

3. **Apply the average rate of change formula:**

\[
\text{Average Rate of Change} = \frac{h(0) - h(-7)}{0 - (-7)} = \frac{7 - (-14)}{7} = \frac{7 + 14}{7} = \frac{21}{7} = 3
\]

So, the average rate of change of the function over the interval \([-7, 0]\) is **3**.

Would you like further details or have any questions?

Here are five related questions you might find helpful:

1. How is the average rate of change related to the slope of a secant line?
2. What is the significance of a positive average rate of change?
3. How can we interpret the average rate of change graphically?
4. What would the average rate of change be over a different interval, such as \([0, 2]\)?
5. How does the quadratic nature of the function affect its average rate of change over different intervals?

**Tip:** The average rate of change gives you the slope of the line connecting two points on the curve, which represents how much the function's output changes on average between those two points.

Learn how to calculate the average rate of change of the quadratic function h(x) = -x^2 - 4x + 7 over the interval [-7, 0]. This problem involves applying the Average Rate of Change formula and understanding quadratic functions. Suitable for high school students.

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