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.