Find the average rate of change of ​f(x)equals3 x squared minus 2 from 2 to 6.

The average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is given by the formula:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

In this case, the function is \( f(x) = 3x^2 - 2 \), and we need to find the average rate of change from \( x = 2 \) to \( x = 6 \).

### Step 1: Calculate \( f(6) \)
\[
f(6) = 3(6)^2 - 2 = 3(36) - 2 = 108 - 2 = 106
\]

### Step 2: Calculate \( f(2) \)
\[
f(2) = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10
\]

### Step 3: Apply the formula
\[
\text{Average rate of change} = \frac{f(6) - f(2)}{6 - 2} = \frac{106 - 10}{6 - 2} = \frac{96}{4} = 24
\]

Thus, the average rate of change of \( f(x) = 3x^2 - 2 \) from \( x = 2 \) to \( x = 6 \) is 24.

Would you like further details on this, or have any questions?

### Related questions:
1. How do you interpret the average rate of change in real-world scenarios?
2. What is the significance of the average rate of change for quadratic functions?
3. How does the average rate of change differ from the instantaneous rate of change?
4. Can the average rate of change be negative for certain functions?
5. How does the average rate of change behave for linear functions?

**Tip:** The average rate of change is conceptually similar to the slope of a line connecting two points on a curve.

Find the average rate of change of f(x) = 3x^2 - 2 from 2 to 6.

Learn how to calculate the average rate of change for the quadratic function f(x) = 3x^2 - 2 between x = 2 and x = 6. This detailed solution covers key concepts in calculus, including the rate of change formula. Suitable for students in Grades 9-12.

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