Overview
This lecture explains average rates of change in linear and quadratic functions, how to calculate them over intervals, and how the rate of change itself changes, including the concept of concavity.
Average Rate of Change & Secant Lines
- The average rate of change over an interval [a, b] is the slope of the secant line connecting (a, f(a)) and (b, f(b)).
- The formula is (f(b) - f(a)) / (b - a).
Linear Functions & Rates of Change
- Linear functions have a constant average rate of change regardless of the interval selected.
- The slope (rate of change) of a linear function does not change and is the same for any two points.
- The rate of change of the slope for linear functions is zero.
Quadratic Functions & Changing Rates
- Quadratic functions have different average rates of change depending on the interval used.
- The average rate of change increases by a constant amount as you move across equal intervals on a quadratic.
- The rate of change of the average rate of change for a quadratic function is constant (but not zero).
Calculating with Quadratic Functions
- For a quadratic y = x² - 2x, plug in the endpoints to find f(a) and f(b), then use the average rate of change formula.
- As intervals increase (e.g., [1,2], [2,3], etc.), the average rate of change for quadratics increases by a constant amount.
- The difference between consecutive average rates of change for quadratics is constant.
Concavity & Rates of Change
- If the average rate of change over equal intervals is decreasing, the function is concave down.
- If the average rate of change is increasing, the function is concave up.
- Determine concavity by checking how the average rate of change moves across intervals.
Key Terms & Definitions
- Secant Line — A line connecting two points on a graph.
- Average Rate of Change — The change in output divided by the change in input over an interval.
- Linear Function — A function whose graph is a straight line; constant rate of change.
- Quadratic Function — A function of the form ax²+bx+c with a variable rate of change.
- Concave Up — When slope increases over intervals; shaped like a "U."
- Concave Down — When slope decreases over intervals; shaped like an "n."
Action Items / Next Steps
- Practice finding average rates of change for both linear and quadratic functions over different intervals.
- For homework, use the formulas provided to calculate and compare average rates of change.
- Recognize concavity by analyzing changes in average rates.