Overview
This lecture explains how to determine intervals where a graph is increasing, decreasing, or constant, emphasizing the use of x-values and careful notation.
Understanding Increasing, Decreasing, and Constant Intervals
- Analyze the graph from left to right, just like reading a book.
- Focus on telling the “story” of what the graph does as x increases.
- Use x-values (not y-values) to describe intervals of increasing, decreasing, or constant behavior.
- Begin your interval at negative infinity (−∞), moving toward positive infinity (+∞).
- For each interval, identify whether the graph is increasing (y-values go up), decreasing (y-values go down), or constant (y-values stay the same).
Notation & Common Mistakes
- Use parentheses for interval notation, e.g., (−∞, −4) or [−4, −1], depending on whether endpoints are included.
- Do not use union symbols or y-value ranges as you would with domain and range.
- Only x-values are used for these intervals to describe where the behavior occurs.
Example Breakdown
- From x = −∞ to x = −4, the graph is increasing.
- From x = −4 to x = −1, the graph is decreasing.
- From x = −1 to x = 3, the graph is increasing.
- From x = 3 to x = ∞, the graph is decreasing.
Key Terms & Definitions
- Increasing Interval — Range of x-values where the graph rises as x increases.
- Decreasing Interval — Range of x-values where the graph falls as x increases.
- Constant Interval — Range of x-values where the graph neither rises nor falls.
- Interval Notation — A way to represent sets of numbers using parentheses (excluded) or brackets (included).
Action Items / Next Steps
- Practice identifying increasing, decreasing, and constant intervals on different graphs.
- Use only x-values for specifying intervals.
- Review notes on interval notation to avoid confusion with domain and range.