Periodic Phenomena (College Board AP® Precalculus): Revision Note

Graphs of periodic relationships

What is a periodic relationship?

• A periodic relationship exists between two quantities when

  • as the input values increase

  • the output values repeat in a pattern over successive equal-length intervals

• For example, the height of a point on a spinning wheel varies periodically with time

  • the height goes up, comes back down, and then repeats this same pattern over and over

• Other examples of periodic phenomena include

  • ocean tides

  • daily temperature cycles

  • the motion of a vibrating guitar string

  • the voltage in an alternating current (AC) electrical circuit

• The key feature that makes a relationship periodic is that

  • the same pattern of output values

  • occurs again and again

  • across equal-length intervals of the input

How can a graph of a periodic relationship be constructed?

• The graph of a periodic relationship can be built from the graph of a single cycle of the relationship

  • One cycle captures the complete pattern of output values before the pattern starts repeating

• To construct the full graph, this single cycle is copied and placed end to end, extending in both directions along the input axis

• In real-world contexts, the graph can be constructed from a verbal description of how a quantity changes

  • Identify the repeating pattern in the description

  • Sketch one complete cycle based on the described behavior

  • Repeat the cycle to extend the graph over the desired input interval

Key characteristics of periodic relationships

What is the period of a periodic function?

• The period of a periodic function is the smallest positive value such that for all in the domain

  • I.e. if you shift any input value by exactly one period, the output value stays the same

• The period represents the length of one complete cycle of the function

• Because the function repeats every units

  • the behavior of a periodic function is completely determined by any interval of width

  • If you know what the function does on one full period, you know what it does everywhere

How can the period be estimated?

• The period can be estimated by looking at successive output values at equally spaced input values and finding where the pattern begins to repeat

  • For example, given a table of values

    • look for the point where the output values start cycling through the same sequence again

• The distance along the input axis between the start of one cycle and the start of the next cycle is the period

• On a graph, the period can be estimated by identifying two consecutive points where the same behavior begins

  • e.g. two consecutive peaks, two consecutive troughs, or two consecutive points where the graph crosses the midline in the same direction

What other characteristics do periodic functions have?

• Periodic functions share many of the same characteristics as other types of functions, including:

  • Intervals of increase and decrease

  • Different concavities (concave up and concave down)

  • Various rates of change

• The important difference with periodic functions is that all characteristics found in one period are repeated in every period of the function

  • For example, if the function is increasing on a certain sub-interval within one period, it will be increasing on the corresponding sub-interval in every other period

  • Similarly, any maximum value, minimum value, or rate of change pattern within one period will appear in exactly the same way in every period