How to Calculate the Average Rate of Change Algebraically

Miacademy & MiaPrep Learning Channel

This video provides a clear, step-by-step tutorial on calculating the average rate of change of a polynomial function algebraically. The instructor, Randy, begins by briefly reviewing the graphical method—identifying points on a curve and finding the slope of the secant line between them—before transitioning to the algebraic approach. He emphasizes that the goal is to obtain the same result using only the function formula and the specified interval, without relying on a visual graph. The video explores key mathematical themes including function notation, the slope formula (change in y over change in x), and evaluating polynomial expressions. It explicitly connects the abstract variables in the slope formula ($x_1, y_1, x_2, y_2$) to concrete values derived from the function $f(x) = \frac{1}{8}x^3 - x^2$. The instructor demonstrates how to substitute the interval boundaries into the function to find corresponding output values and then use those coordinates to compute the rate of change. For educators, this resource is highly valuable for bridging the gap between Algebra 1 concepts of slope and Pre-Calculus concepts of secant lines and difference quotients. It is particularly useful for visual learners who need to see the connection between the geometry of a graph and the arithmetic of function evaluation. Teachers can use this video to introduce the concept of average rate of change or to reinforce skills in evaluating functions with exponents and fractions.