Overview
This lecture explains how to identify intervals of increasing, decreasing, and concavity (concave up/down) in functions by analyzing their graphs, focusing on concepts key for AP pre-calculus.
Increasing and Decreasing Functions
- A function is increasing if its y-values rise as you move from left to right on the graph.
- A function is decreasing if its y-values fall as you move from left to right on the graph.
- If a function is increasing, its rate of change (slope) is positive.
- If a function is decreasing, its rate of change (slope) is negative.
Rate of Change (Slope)
- Rate of change and slope are interchangeable terms when describing graphs.
- Positive slope means the function rises as you move right; negative slope means the function falls.
Concavity
- Concave up: The rate of change (slope) is increasing, even if the function is decreasing.
- Concave down: The rate of change (slope) is decreasing.
- Visual cues: Concave up curves look like a cup (opens upward); concave down curves look like a cap (opens downward).
- For concave up, example slopes: -10, -1, -1/10 (increasing toward zero).
- For concave down, example slopes: 10, 1, 1/10 (decreasing toward zero or negative).
Special Cases
- Linear functions have constant slope, so they are neither concave up nor concave down.
Key Terms & Definitions
- Increasing Function — y-values rise as x increases.
- Decreasing Function — y-values fall as x increases.
- Rate of Change (Slope) — Change in y over change in x; indicates how rapidly the function rises or falls.
- Concave Up — The function's slope increases as x increases.
- Concave Down — The function's slope decreases as x increases.
Action Items / Next Steps
- Make flashcards summarizing increasing/decreasing and concavity shapes.
- Practice estimating slopes and concavity on various graphs.
- Memorize the key relationships between function behavior and rate of change.