We are given the function $f(x) = 3x - 6$, and we need to:

(a) Find the average rate of change from $x = 2$ to $x = 8$.

The formula for the average rate of change of a function between two points $x_1$ and $x_2$ is given by:

$\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Here, $x_1 = 2$ and $x_2 = 8$. So, we first compute $f(2)$ and $f(8)$.

• $f(2) = 3(2) - 6 = 6 - 6 = 0$
• $f(8) = 3(8) - 6 = 24 - 6 = 18$

Now we substitute into the average rate of change formula:

$\text{Average rate of change} = \frac{f(8) - f(2)}{8 - 2} = \frac{18 - 0}{8 - 2} = \frac{18}{6} = 3$

So, the average rate of change from $x = 2$ to $x = 8$ is 3.

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(b) Find an equation of the secant line containing the points $(2, f(2))$ and $(8, f(8))$.

The slope $m$ of the secant line is the same as the average rate of change, which we found to be 3.

The equation of a line is given by the point-slope form:

$y - y_1 = m(x - x_1)$

We can use the point $(2, f(2)) = (2, 0)$ and the slope $m = 3$ to write the equation of the secant line:

$y - 0 = 3(x - 2)$

Simplifying:

$y = 3(x - 2) = 3x - 6$

Thus, the equation of the secant line is:

$y = 3x - 6$

Summary:

• (a) The average rate of change from $x = 2$ to $x = 8$ is 3.
• (b) The equation of the secant line is $y = 3x - 6$.

Would you like more details on any of these steps or have any questions?

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Here are 5 related questions to extend this concept:

1. How is the average rate of change related to the derivative of the function?
2. What would be the slope of the tangent line at $x = 2$ for the function $f(x) = 3x - 6$?
3. How does the secant line differ from the tangent line geometrically?
4. What is the average rate of change of $f(x) = 3x - 6$ over the interval $[4, 10]$?
5. How do you find the equation of a line given two arbitrary points?

Tip: The average rate of change of a linear function is always the same, as it represents the constant slope of the line.