Overview
This lesson introduces the concept of average rate of change for a function, explains its calculation, and demonstrates examples using the slope formula between two points.
Introduction to Average Rate of Change
- The average rate of change describes how a function changes between two points.
- Graphically, it is represented by the slope of the secant line connecting two points on the function.
Secant Lines and Slope
- A secant line is a straight line connecting two points on a function’s graph.
- The slope of the secant line gives the average rate of change between those two points.
- Slope formula: (y₂ − y₁) / (x₂ − x₁).
Real-Life Example: Travel Rate
- The average rate, like average speed, is calculated by dividing total distance by total time.
- On a distance vs. time graph, the secant line’s slope between starting and ending points shows average speed.
- If you select different intervals, the secant line’s slope shows the average speed for that specific time range.
Example Problems
- To find the average rate of change from x = 0 to x = 3 for f(x) = x² + 1:
- f(0) = 1, f(3) = 10; slope = (10 − 1)/(3 − 0) = 9/3 = 3.
- Interpretation: For every 1 increase in x, y increases by 3 on this interval.
- To find the average rate of change from x = −2 to x = 1:
- f(−2) = 5, f(1) = 2; slope = (2 − 5)/(1 − (−2)) = (−3)/3 = −1.
- Interpretation: For every 1 increase in x, y decreases by 1 on this interval.
Key Terms & Definitions
- Average Rate of Change — The change in function values divided by the change in input values over an interval.
- Secant Line — A straight line connecting two points on a function’s graph.
- Slope — The ratio (rise over run) that describes the steepness of a line.
Action Items / Next Steps
- Practice finding the average rate of change for other functions and intervals.
- Remember to simplify the final answer for the average rate of change.