What does the average rate of change mean graphically?

Graphically, the **average rate of change** represents the slope of the **secant line** that connects two points on a curve. These two points correspond to the endpoints of the interval over which the change is being measured.

Mathematically, the average rate of change of a function \( f(x) \) between two points \( x = a \) and \( x = b \) is given by:

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

### Graphical Interpretation:
- **Curve**: The graph of the function \( f(x) \).
- **Secant Line**: A straight line connecting the points \( (a, f(a)) \) and \( (b, f(b)) \) on the curve.
- **Slope of Secant Line**: This slope represents how much the function's value changes on average for each unit change in \( x \) between \( a \) and \( b \). It can be thought of as a rough estimate of how steep the function is on that interval.

In essence, the average rate of change gives an overall idea of how the function behaves between two points, smoothing out any local fluctuations in the graph.

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### Related Questions:
1. How does the average rate of change compare to the instantaneous rate of change?
2. Can the average rate of change be negative? What does that indicate graphically?
3. How is the average rate of change used in real-world applications?
4. What happens to the secant line as \( b \) approaches \( a \)?
5. How is the average rate of change interpreted in linear functions?

### Tip:
The average rate of change is closely related to the derivative. As the two points get closer together, the average rate of change approaches the instantaneous rate of change, which is the derivative at a point.

Discover the graphical meaning of the average rate of change in calculus, focusing on the slope of the secant line between two points on a curve. This concept is fundamental to understanding how a function changes over an interval. Suitable for high school students in Grades 10-12.

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