Questions: Find the average rate of change of g(x)=-x^2+2x+4 from x=-2 to x=3. Simplify your answer as much as possible.

Find the average rate of change of g(x)=-x^2+2x+4 from x=-2 to x=3. Simplify your answer as much as possible.
Transcript text: Find the average rate of change of $g(x)=-x^{2}+2 x+4$ from $x=-2$ to $x=3$. Simplify your answer as much as possible.
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Solution

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Solution Steps

Step 1: Calculate \( g(-2) \)

Substitute \( x = -2 \) into the function \( g(x) = -x^{2} + 2x + 4 \): \[ g(-2) = -(-2)^{2} + 2(-2) + 4 = -4 - 4 + 4 = -4 \]

Step 2: Calculate \( g(3) \)

Substitute \( x = 3 \) into the function \( g(x) = -x^{2} + 2x + 4 \): \[ g(3) = -(3)^{2} + 2(3) + 4 = -9 + 6 + 4 = 1 \]

Step 3: Compute the average rate of change

Use the formula for the average rate of change: \[ \text{Average rate of change} = \frac{g(3) - g(-2)}{3 - (-2)} = \frac{1 - (-4)}{3 + 2} = \frac{5}{5} = 1 \]

Final Answer

\(\boxed{1}\)

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