Overview
This lecture covers average rates of change in linear and quadratic functions, methods of calculation, and how these concepts relate to function characteristics such as slope and concavity.
Average Rate of Change and Secant Lines
- The average rate of change is the slope of the secant line through two points on a function.
- The formula: (f(b) - f(a)) / (b - a), where (a, f(a)) and (b, f(b)) are points on the graph.
Linear Functions
- Linear functions are straight lines, and their slope (rate of change) is constant.
- No matter which two points you choose on a linear function, the average rate of change remains the same.
- The difference in consecutive average rates of change (change in slope) for a linear function is zero.
Quadratic Functions
- Quadratic functions have the form y = x² - 2x or similar and their graphs are parabolas.
- The average rate of change between equal intervals increases or decreases at a constant rate.
- To find the average rate of change, calculate the outputs for each endpoint and use the formula.
- For each consecutive interval of length 1, the average rate of change increases or decreases by a fixed amount (constant second difference).
Rate of Change of the Average Rate of Change
- For linear functions, the change in the average rate of change between intervals is always zero.
- For quadratic functions, this second level rate is constant but not zero (e.g., always 2, -4, 6 depending on the function).
Concavity and Function Behavior
- A function is concave up if the average rate of change increases over equal intervals.
- A function is concave down if the average rate of change decreases over equal intervals.
- Concavity can be determined from a table by examining changes in the average rates of change.
Key Terms & Definitions
- Secant Line — a line passing through two points on a curve.
- Average Rate of Change — the change in the function output divided by the change in input; same as slope between two points.
- Linear Function — a function whose graph is a straight line with constant slope.
- Quadratic Function — a function defined by an equation of the form y = ax² + bx + c.
- Concave Up — a curve opening upwards; average rate of change increases.
- Concave Down — a curve opening downwards; average rate of change decreases.
Action Items / Next Steps
- Complete homework problems on average rate of change for both linear and quadratic functions.
- Practice using the formula for secant slopes with provided and new intervals.
- Analyze a table of function values to determine concavity.