What does the average rate of change represent in a function?

The average rate of change in a function represents how much the function's output (or dependent variable) changes, on average, for a given change in the input (or independent variable) over a specific interval. Formally, it is calculated by taking the difference in the function's values at two points and dividing it by the difference in the input values at those points. This concept is essential for understanding how a function behaves between two specific values rather than at just one point.

When assessing the average rate of change, it provides a more holistic view of the function's behavior over that interval, as opposed to just looking at a single point or instantaneous change. It does not pertain to instantaneous rates of change, which would require calculus to analyze, nor does it imply constancy throughout the function's domain. Thus, this understanding is crucial in various mathematical applications, including interpreting the slope of secant lines on a graph of the function.