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To determine the average rate of change of the function \( f(x) = 3x^2 - 1 \) over an interval \([a, b]\), we use the formula:

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

Now, let's calculate the average rate of change for each interval:

### (a) On the interval \([-3, -1]\):
1. \( f(-3) = 3(-3)^2 - 1 = 3(9) - 1 = 27 - 1 = 26 \)
2. \( f(-1) = 3(-1)^2 - 1 = 3(1) - 1 = 3 - 1 = 2 \)

Now, applying the formula:

\[
\frac{f(-1) - f(-3)}{-1 - (-3)} = \frac{2 - 26}{-1 + 3} = \frac{-24}{2} = -12
\]

The average rate of change on \([-3, -1]\) is \(-12\).

---

### (b) On the interval \([1, 4]\):
1. \( f(1) = 3(1)^2 - 1 = 3(1) - 1 = 2 \)
2. \( f(4) = 3(4)^2 - 1 = 3(16) - 1 = 48 - 1 = 47 \)

Now, applying the formula:

\[
\frac{f(4) - f(1)}{4 - 1} = \frac{47 - 2}{3} = \frac{45}{3} = 15
\]

The average rate of change on \([1, 4]\) is \(15\).

---

### (c) On the interval \([6, 8]\):
1. \( f(6) = 3(6)^2 - 1 = 3(36) - 1 = 108 - 1 = 107 \)
2. \( f(8) = 3(8)^2 - 1 = 3(64) - 1 = 192 - 1 = 191 \)

Now, applying the formula:

\[
\frac{f(8) - f(6)}{8 - 6} = \frac{191 - 107}{2} = \frac{84}{2} = 42
\]

The average rate of change on \([6, 8]\) is \(42\).

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Would you like any additional details or have any questions?

### Related questions:
1. How does the average rate of change relate to the derivative of a function?
2. What is the significance of the average rate of change being negative in part (a)?
3. Can the average rate of change of a quadratic function ever be constant?
4. How does the choice of interval affect the average rate of change?
5. What happens if the interval endpoints are the same (i.e., the length of the interval is 0)?

### Tip:
The average rate of change between two points on a function is equivalent to the slope of the secant line that passes through those points on the graph.

Determine the average rate of change of the function f(x) = 3x^2 - 1 on the intervals [-3, -1], [1, 4], and [6, 8].

Calculate the average rate of change of the quadratic function f(x) = 3x^2 - 1 on different intervals. Detailed step-by-step solutions provided for the intervals [-3, -1], [1, 4], and [6, 8]. Suitable for Grades 10-12 students learning about rates of change and quadratic functions.

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