The average rate of change of a function over an interval describes the slope between two points on the graph of a function. This slope is that of the secant line passing through those two points. Given a function $f(x)$, and two points on its graph $(a,f(a))$ and $(b,f(b))$, the average rate of change is: $ \frac{f(b)-f(a)}{b-a} $ As the value of $b$ approaches $a$, the average rate of the function comes closer to the slope of the tangent at point $a$ until it functionally equals it as the difference between the two points becomes negligibly different. For instance, given a function $f(x)=x^2-2x$, and two input values $x=3$ and $x=5$, we calculate the average rate of change (or slope of the secant line between the two corresponding points) like so: $ \frac{f(5)-f(3)}{5-3} = \frac{(5^2-2\cdot5) - (3^2-2\cdot3)}{2} = \frac{15-3}{2} = \frac{12}{2} = 6 $ This method allows us to find the average slope of a graph over an interval. Interestingly, there will be one point on that interval where the instantaneous slope equals this average slope. As we make the two points closer and closer, the average slope becomes closer until equivalent to the tangent of the graph at that point.