Given the function g, of, x, equals, minus, x, squared, plus, 3, x, plus, 5g(x)=−x 
2
 +3x+5, determine the average rate of change of the function over the interval minus, 4, is less than or equal to, x, is less than or equal to, 6−4≤x≤6.

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

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

In this case, the function is \( g(x) = -x^2 + 3x + 5 \) and the interval is \([-4, 6]\).

### Step 1: Calculate \( g(-4) \)
\[
g(-4) = -(-4)^2 + 3(-4) + 5 = -16 - 12 + 5 = -23
\]

### Step 2: Calculate \( g(6) \)
\[
g(6) = -(6)^2 + 3(6) + 5 = -36 + 18 + 5 = -13
\]

### Step 3: Apply the formula for the average rate of change
\[
\text{Average Rate of Change} = \frac{g(6) - g(-4)}{6 - (-4)} = \frac{-13 - (-23)}{6 + 4} = \frac{-13 + 23}{10} = \frac{10}{10} = 1
\]

So, the average rate of change of the function over the interval \([-4, 6]\) is 1.

Would you like further details on any part of this? Here are some related questions:

1. How does the average rate of change relate to the slope of a secant line?
2. How would the average rate of change differ over a smaller interval?
3. Can the rate of change be negative, and in what cases?
4. What does the sign of the average rate of change tell us about the function?
5. How would we compute instantaneous rate of change at a point?

**Tip:** The average rate of change formula is similar to the slope formula for a straight line, so it gives an approximation of how the function behaves between two points.

Given the function g(x) = -x^2 + 3x + 5, determine the average rate of change of the function over the interval −4 ≤ x ≤ 6.

Learn how to calculate the average rate of change for the quadratic function g(x) = -x^2 + 3x + 5 over the interval [-4, 6]. This problem involves key algebraic concepts and demonstrates how to apply the formula for the average rate of change. Suitable for students in Grades 9-10.

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