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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?

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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.

Find the average rate of change of \( f(x) = x^2 - 5x + 1 \) from \( x = 2 \) to \( x = 7 \).

Calculate the average rate of change for the function \( f(x) = x^2 - 5x + 1 \) over the interval from \( x = 2 \) to \( x = 7 \). This problem demonstrates key algebraic concepts and the application of the rate of change formula. Suitable for students in Grades 9-11.

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