Overview
This lecture explores how the average rate of change in quadratic functions behaves over consecutive equal-length intervals, and how these changes can reveal key properties of the function.
Consecutive Equal-Length Intervals in Quadratics
- Consecutive equal-length intervals are intervals that follow one another with no gaps and have the same length.
- The specific length or starting point of the intervals does not matter, as long as both conditions are met.
Average Rate of Change in Quadratic Functions
- The average rate of change over each interval is found by subtracting the y-values and dividing by the difference in x-values.
- For quadratic functions, when intervals are consecutive and equal in length, the average rate of change itself changes at a constant rate.
Graphical Example: Concave Up Parabola
- Example intervals: from -2 to 0, 0 to 2, 2 to 4, 4 to 6, 6 to 8 (length of 2 each).
- Calculated rates of change: -1.6, -0.8, 0, 0.8, 1.6 (each increases by 0.8).
- The rates of change are increasing, showing the graph is concave up.
- The sign of the rate of change switches from negative to positive across the interval containing the vertex (extrema).
- The difference between each consecutive rate of change is constant.
Tabular Example: Concave Down Parabola
- Example function: f(x) = -3x² - x + 4.
- Intervals: -7 to -4, -4 to -1, -1 to 2, 2 to 5 (length of 3 each).
- Calculated rates of change: 32, 14, -4, -22 (each decreases by 18).
- Rates of change are decreasing, so the graph is concave down.
- The difference between consecutive rates of change is a constant negative value.
Key Observations and General Properties
- In any quadratic function, the rate of change in consecutive equal-length intervals changes by a constant value.
- This constant change in rates of change means that the sequence of rates forms a linear pattern.
- For quadratics, the sign of change (increasing or decreasing) indicates whether the parabola opens up (concave up) or down (concave down).
- For linear functions, the rate of change does not change (difference is zero).
Key Terms & Definitions
- Quadratic Function — A function in the form ax² + bx + c, where a ≠ 0.
- Average Rate of Change — (Change in y) ÷ (Change in x) over an interval.
- Consecutive Equal-Length Intervals — Back-to-back intervals with equal widths and no gaps.
- Concave Up — Parabola opens upward; rates of change increase.
- Concave Down — Parabola opens downward; rates of change decrease.
- Extrema — Maximum or minimum point of a function; here, the switch from negative to positive rate of change.
Action Items / Next Steps
- Practice finding average rates of change for quadratic functions over equal-length intervals.
- Identify if rates of change are increasing or decreasing and connect this to the function's concavity.
- Watch the lecture video again if concepts need further clarification.