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Understanding Periodic Phenomena and Graphs

Feb 27, 2025

Periodic Phenomena Lecture Notes

Introduction to the Topic

  • Unit 3: Focus on Periodic Phenomena.
  • Definition of Periodic Relationship:
    • As input values increase, if output values (y-values) show a repeating pattern over equal intervals, it's periodic.
    • Cyclical: This relationship is also termed cyclical.

Real-Life Examples of Periodic Phenomena

  • Phases of the Moon: Cycles of illumination.
  • Tides: High and low tides in the ocean.

Moon Animation Overview

  • Animation by Phil Hart on YouTube:
    • Illustrates the moon's phases from Earth's perspective.
    • Starts with no light (new moon), then illumination increases and decreases, repeating the cycle.
  • Graphing the Moon's Illumination:
    • Percent illumination goes from 0% (new moon) to 100% and back to 0%, forming a cyclical graph.

Constructing Periodic Graphs

  • Example 1: Sketching a graph based on one cycle.

    • Identify key points:
      • Starts at 0, ends at 0, with a 0 in the middle.
      • Full cycle takes 8 units (Period = 8).
    • Sketching tips:
      • Use half measures to find minimum (at 4) and maximum points.
  • Example 2: Sketching a weird graph.

    • Identify period (4 units for one cycle).
    • Use the copy-and-paste technique to replicate the cycle.

Defining Period of a Function

  • Period is the smallest positive value (k) such that:
    • If you add k to x, you return to the same output value, i.e., f(x + k) = f(x).
  • Example:
    • First function: k = 8 (one complete cycle).
    • Second function: k = 4.

Identifying the Period

  • Find common points in the graph to determine the period.
    • Example: From 1/4 to 3/4 is half a unit.
  • Using Minimum Values: Easier to identify repeating patterns by checking visible minimum and maximum values.

Analyzing Graphs

  • Periodic functions exhibit consistent characteristics across cycles (increasing/decreasing, concavity).
  • Example Questions:
    1. Increasing or Decreasing: Identify behavior within given intervals (e.g., from 18 to 20, it increases).
    2. Concavity: Determine if concave up or down (e.g., from 31 to 33, it is concave down).
    3. Relative Max/Min: Check for max/min values at specific points (e.g., at x=82).

Conclusion

  • Periodic phenomena are straightforward to analyze and graph.
  • Remember the Muppets' "Phenomena" for a fun reference to periodicity.
  • Mr. Kelly's Reminder: It's nice to be important, but more important to be nice.