The problem asks for the average rate of change of the function $f(x) = -3x + 5$ over the interval $x = -1$ to $x = 1$.

The formula for the average rate of change of a function over an interval $[a, b]$ is: $\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Step-by-step solution:

1. Identify the values:

   • $a = -1$, $b = 1$
   • Function: $f(x) = -3x + 5$

2. Compute $f(a)$ and $f(b)$:

   • $f(-1) = -3(-1) + 5 = 3 + 5 = 8$
   • $f(1) = -3(1) + 5 = -3 + 5 = 2$

3. Substitute into the formula: $\text{Average rate of change} = \frac{f(1) - f(-1)}{1 - (-1)}$ $= \frac{2 - 8}{1 + 1}$ $= \frac{-6}{2} = -3$

Final Answer:

The average rate of change of $f(x)$ over the interval $x = -1$ to $x = 1$ is: $\boxed{-3}$

Let me know if you want a detailed explanation of any step!

Related Questions:

1. What is the formula for the average rate of change, and why does it resemble the slope formula?
2. How does the average rate of change of a linear function compare to its slope?
3. What is the interpretation of a negative average rate of change in real-world terms?
4. If the interval were $x = -2$ to $x = 2$, what would the average rate of change be for $f(x) = -3x + 5$?
5. How can you verify your result graphically?

Tip:

For linear functions like $f(x) = -3x + 5$, the average rate of change over any interval is always equal to the slope of the line!