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The image explains how to find the average rate of change of a function \( y = f(x) \) over an interval \([a, b]\). Here's a breakdown:

1. **Average Rate of Change Formula**: 
   \[
   \frac{f(b) - f(a)}{b - a}
   \]
   This formula calculates the average rate of change of \( f(x) \) over the interval \([a, b]\), which is equivalent to the slope of the line connecting the points \( (a, f(a)) \) and \( (b, f(b)) \) on the graph of the function.

2. **Interpretation**: 
   - The average rate of change represents how much \( y \) changes per unit change in \( x \) over \([a, b]\).
   - This is also the slope of the *secant line* (a straight line between two points) joining \( (a, f(a)) \) and \( (b, f(b)) \) on the curve.

3. **Graphical Representation**:
   - In the provided graph, \( (a, f(a)) \) and \( (b, f(b)) \) are marked as points on the curve \( y = f(x) \).
   - The difference \( f(b) - f(a) \) represents the vertical change, while \( b - a \) represents the horizontal change.

Would you like a more detailed explanation or have any questions?

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Here are five follow-up questions for further exploration:

1. How does the average rate of change differ from the instantaneous rate of change?
2. What does the average rate of change signify when \( f(x) \) is a linear function?
3. Can the average rate of change be negative? If so, what does it indicate?
4. How would you interpret the average rate of change for non-linear functions over small intervals?
5. How does the secant line relate to the tangent line as \( b \) approaches \( a \)?

**Tip**: For linear functions, the average rate of change between any two points is constant and equals the slope of the line.

What is the average rate of change of a function y = f(x) over the interval [a, b]?

Understand the average rate of change of a function y = f(x) over an interval [a, b] using the formula (f(b) - f(a)) / (b - a). This concept is visualized as the slope of the line between points (a, f(a)) and (b, f(b)) on the graph of the function, which provides insight into how y changes relative to x.

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