Functions: Rates of Change
Lessons in this chapter
About Functions: Rates of Change
Functions: Rates of Change builds the vocabulary and tools for describing how a function behaves and changes.
Key Features of the Graph of a Function identifies a function’s intercepts, intervals of increase and decrease, and extrema directly from its graph.
Rates of Change introduces average rate of change as a general tool for measuring how fast a function’s output changes over an interval.
Rates of Change of Linear Functions and Rates of Change of Quadratic Functions apply that general tool to the two function families studied so far, showing a linear function’s constant rate versus a quadratic function’s changing rate.
Each of the four lessons, from Key Features of the Graph of a Function to Rates of Change of Quadratic Functions, opens as a practice worksheet in the page. Problems are scored as they are answered, which is how the average rate of change formula gets used often enough to stop needing to be looked up. Free to practice online.
Frequently asked questions
Key Features of the Graph of a Function covers domain, range, intervals where a function rises or falls, and extrema, all read straight off a curve. Rates of Change introduces average and instantaneous rate of change, Rates of Change of Linear Functions shows why a linear function's rate is constant, and Rates of Change of Quadratic Functions shows why a quadratic's rate of change itself changes at a constant rate. This sits in Functions (F) in the high school Common Core, and it opens Unit 1 of the AP Precalculus course framework, Polynomial and Rational Functions.
Intercepts, intervals of increase and decrease, and extrema (maximum and minimum points) describe a function’s behavior directly from its graph.
Average rate of change over an interval is the change in output divided by the change in input: the same idea as slope, generalized to any function.
A linear function’s average rate of change is always the same constant value (its slope), no matter which interval you pick.
A quadratic function’s average rate of change is different over every interval, and it changes at a constant rate itself as the interval shifts.
Assuming a rate of change is constant just because it's a single number: only a linear function's rate of change actually stays the same across every interval.