Summary
This lesson 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 and 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 and Rates of Change
- Linear functions have a constant average rate of change regardless of the selected interval.
- 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 and 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 through equal intervals in a quadratic.
- The rate of change of the average rate of change for a quadratic function is constant (but not zero).
Calculations with Quadratic Functions
- For a quadratic y = x² - 2x, substitute 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 and 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 and 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 the slope increases over intervals; shaped like a "U".
- Concave Down — When the slope decreases over intervals; shaped like an "n".
Actions / Next Steps
- Practice finding average rates of change for linear and quadratic functions over different intervals.
- For homework, use the provided formulas to calculate and compare average rates of change.
- Recognize concavity by analyzing changes in average rates of change.