Given the function h, of, x, equals, x, squared, plus, 7, x, plus, 9h(x)=x
2
+7x+9, determine the average rate of change of the function over the interval minus, 9, is less than or equal to, x, is less than or equal to, 0−9≤x≤0.

To determine the average rate of change of the function \( h(x) = x^2 + 7x + 9 \) over the interval \([-9, 0]\), we can use the formula for the average rate of change:

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

where \( a \) and \( b \) are the endpoints of the interval. For this problem:

- \( a = -9 \)
- \( b = 0 \)

### Step 1: Calculate \( h(a) = h(-9) \)
\[
h(-9) = (-9)^2 + 7(-9) + 9
\]
\[
h(-9) = 81 - 63 + 9 = 27
\]

### Step 2: Calculate \( h(b) = h(0) \)
\[
h(0) = (0)^2 + 7(0) + 9
\]
\[
h(0) = 9
\]

### Step 3: Compute the Average Rate of Change
\[
\text{Average Rate of Change} = \frac{h(0) - h(-9)}{0 - (-9)} = \frac{9 - 27}{9} = \frac{-18}{9} = -2
\]

Thus, the average rate of change of the function over the interval \([-9, 0]\) is \(-2\).

Would you like more details, or do you have any questions? 

Here are 5 related questions for further practice:
1. What is the average rate of change of the function \( h(x) = x^2 + 7x + 9 \) over the interval \([1, 4]\)?
2. Determine the slope of the secant line between \( x = -3 \) and \( x = 2 \) for the function \( h(x) = x^2 + 7x + 9 \).
3. What is the average rate of change of the function \( h(x) = 2x^2 - 3x + 5 \) over the interval \([0, 3]\)?
4. Find the instantaneous rate of change of \( h(x) = x^2 + 7x + 9 \) at \( x = -9 \).
5. How does the average rate of change of a linear function differ from that of a quadratic function?

**Tip:** The average rate of change for a quadratic function over an interval can be thought of as the slope of the secant line connecting the endpoints on the graph.

Learn how to calculate the average rate of change of the quadratic function h(x) = x^2 + 7x + 9 over the interval [-9, 0]. Step-by-step solution provided using the average rate of change formula. Suitable for high school students studying quadratic functions and rates of change.

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