Overview
This lecture covers absolute and relative extrema (maximum and minimum points) on a graph, focusing on identifying their positions and values.
Absolute Maximum and Minimum
- The highest point on a graph is called the absolute maximum.
- The absolute maximum occurs at the point (2, 8) in this example.
- The absolute maximum value is the y-coordinate, y = 8.
- Questions may ask for the location (the point) or the value (the y-value) of the absolute maximum.
- There is no absolute minimum on the graph because it decreases without bound (goes to negative infinity).
Relative Maximum and Minimum
- Relative maximum is a high point compared to its immediate surroundings (but not the highest overall).
- The relative maximum in this example occurs at the point (–6, 5.5) or has value y = 5.5.
- Relative minimum is a low point compared to its immediate surroundings (but not the lowest overall).
- The relative minimum occurs at the point (–2, 0).
Key Terms & Definitions
- Absolute Maximum — The highest point on the entire graph; the greatest y-value.
- Absolute Minimum — The lowest point on the entire graph; the smallest y-value.
- Relative Maximum — A point higher than nearby points, but not necessarily the highest overall.
- Relative Minimum — A point lower than nearby points, but not necessarily the lowest overall.
Action Items / Next Steps
- Review definitions and example points for absolute and relative extrema.
- Practice identifying extrema on different graphs.
- Be prepared to interpret questions asking for the location or value of maxima/minima.