To find the average rate of change of the function $f(x) = x^2 - 5x + 1$ from $x = 2$ to $x = 7$, we use the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

where $a = 2$ and $b = 7$.

Step 1: Evaluate $f(2)$ and $f(7)$

1. Calculate $f(2)$: $f(2) = (2)^2 - 5(2) + 1 = 4 - 10 + 1 = -5$

2. Calculate $f(7)$: $f(7) = (7)^2 - 5(7) + 1 = 49 - 35 + 1 = 15$

Step 2: Apply the Average Rate of Change Formula

$\text{Average Rate of Change} = \frac{f(7) - f(2)}{7 - 2} = \frac{15 - (-5)}{5} = \frac{15 + 5}{5} = \frac{20}{5} = 4$

Answer

The average rate of change of $f(x)$ from $x = 2$ to $x = 7$ is $4$.

Would you like more details on any step?

Here are 5 related questions to expand your understanding:

1. How do you interpret the average rate of change of a function geometrically?
2. What does a positive average rate of change indicate about the function?
3. How would you find the average rate of change if the function were nonlinear, such as $f(x) = x^3 - 2x^2 + 1$?
4. How is the average rate of change different from the instantaneous rate of change?
5. Can the average rate of change be zero? What would that imply?

Tip: The average rate of change of a function over an interval is equivalent to the slope of the secant line joining the endpoints of the interval on the graph.