To find the average rate of change of a function $f(x)$ over a given interval $[a, b]$, you can use the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Given the function $f(x) = -2x^2 + 2$, let's apply this to each interval.

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(a) From 0 to 2:

1. Evaluate $f(0)$: $f(0) = -2(0)^2 + 2 = 2$

2. Evaluate $f(2)$: $f(2) = -2(2)^2 + 2 = -2(4) + 2 = -8 + 2 = -6$

3. Compute the average rate of change: $\text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0} = \frac{-6 - 2}{2} = \frac{-8}{2} = -4$

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(b) From 2 to 4:

1. Evaluate $f(2)$ (already done): $f(2) = -6$

2. Evaluate $f(4)$: $f(4) = -2(4)^2 + 2 = -2(16) + 2 = -32 + 2 = -30$

3. Compute the average rate of change: $\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-30 - (-6)}{2} = \frac{-30 + 6}{2} = \frac{-24}{2} = -12$

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(c) From 3 to 6:

1. Evaluate $f(3)$: $f(3) = -2(3)^2 + 2 = -2(9) + 2 = -18 + 2 = -16$

2. Evaluate $f(6)$: $f(6) = -2(6)^2 + 2 = -2(36) + 2 = -72 + 2 = -70$

3. Compute the average rate of change: $\text{Average Rate of Change} = \frac{f(6) - f(3)}{6 - 3} = \frac{-70 - (-16)}{3} = \frac{-70 + 16}{3} = \frac{-54}{3} = -18$

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Summary of results:

• (a) From 0 to 2: $-4$
• (b) From 2 to 4: $-12$
• (c) From 3 to 6: $-18$

Would you like any further details or explanations?

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Here are some related questions:

1. How does the average rate of change compare to the instantaneous rate of change?
2. What is the significance of a negative average rate of change?
3. Can you find the rate of change over a smaller interval, such as from $x = 1$ to $x = 1.5$?
4. How does the shape of the graph of $f(x) = -2x^2 + 2$ affect the rate of change?
5. What happens to the rate of change as $x$ increases?

Tip: The average rate of change of a quadratic function over an interval can give you an idea of how steeply the curve rises or falls between two points.