Given the function h, of, x, equals, x, squared, minus, 6, x, plus, 4h(x)=x
2
−6x+4, determine the average rate of change of the function over the interval minus, 2, is less than or equal to, x, is less than or equal to, 5−2≤x≤5.

To determine the average rate of change of the function \( h(x) = x^2 - 6x + 4 \) over the interval \([-2, 5]\), we can use the formula:

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

where \( a = -2 \) and \( b = 5 \).

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

1. **Find \( h(-2) \):**

\[
h(-2) = (-2)^2 - 6(-2) + 4
\]
\[
h(-2) = 4 + 12 + 4 = 20
\]

2. **Find \( h(5) \):**

\[
h(5) = (5)^2 - 6(5) + 4
\]
\[
h(5) = 25 - 30 + 4 = -1
\]

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

\[
\text{Average Rate of Change} = \frac{h(5) - h(-2)}{5 - (-2)}
\]
\[
\text{Average Rate of Change} = \frac{-1 - 20}{5 - (-2)}
\]
\[
\text{Average Rate of Change} = \frac{-21}{7} = -3
\]

So, the average rate of change of the function \( h(x) \) over the interval \([-2, 5]\) is \(-3\).

Would you like more details on any step? Here are some related questions:

1. What does the average rate of change represent graphically?
2. How can you find the instantaneous rate of change of the function at a point?
3. How does the rate of change relate to the slope of a secant line?
4. What if the interval was different? How would the average rate of change be affected?
5. Can the average rate of change be interpreted as the velocity in a physics problem?

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

Learn how to calculate the average rate of change of the function h(x) = x^2 - 6x + 4 over the interval [-2, 5]. Follow a step-by-step solution to find the endpoints, evaluate the function, and apply the average rate of change formula. Suitable for high school students studying functions and calculus concepts.

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