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?

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.