Find the average rate of change of ​f(x)equalsnegative 2 x squared plus 2 over each of the following intervals.
​(a) From 0 to 2
​(b) From 2 to 4
​(c) From 3 to 6

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:
   \[
   \text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0} = \frac{-6 - 2}{2} = \frac{-8}{2} = -4
   \]

---

### (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:
   \[
   \text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-30 - (-6)}{2} = \frac{-30 + 6}{2} = \frac{-24}{2} = -12
   \]

---

### (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.

Find the average rate of change of ​f(x) = -2x^2 + 2 over each of the following intervals: (a) From 0 to 2 (b) From 2 to 4 (c) From 3 to 6

Discover how to calculate the average rate of change of the quadratic function f(x) = -2x^2 + 2 over different intervals: (a) from 0 to 2, (b) from 2 to 4, and (c) from 3 to 6. This step-by-step solution includes essential algebraic concepts and formulas, ideal for high school students in Grades 9-12.

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