Periodic Functions

Defining Periodic Functions

a function returning to the same value at regular intervals.

Graph Behavior

A periodic function has a wave like graph. Normally this wave like pattern repeats, but not in all cases such as a

heart beat, tides or seasonal temperatures.

f(x+P)=f(x) is the formP= period, nonzero constantFor all x in the domain of f

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Meaning that the y values will repeat over some p value called the fundamental period of the function. 

Amplitude of Periodic Functions

If a periodic graph has a maximum value M and a minimum value m, then the amplitude

A of the function is:

A = (M - m)/2

A=AmplitudeM=Maximumm= minimum

Period of Periodic Functions

The period of a periodic function maps one whole phase of the graph.

To find the period you count how many units it takes for one phase to pass by. That is your

period.

*example on the next slide.

Finding the Period

Start at any phase, -3, count the units until the next phase, 3, count the units it takes you to get from -3 to 3…. Which is 6, so the period is 6.

Domain

Domain is defined as the x values in a graphFor periodic functions the domain is

different for each graph. Why? Because periodic functions do not

always have arrows at the end making it infinite. Each graph has different x minimum

and maximum values. In order to find the domain you find the x

minimum value and the x maximum value and that is the domain.

Range

Range is defined as the y values on a graph.The range of a periodic graph varies.

Why? Because periodic functions do not always have arrows at the end making it

infinite. Each graph has different y minimum and maximum values.

In order to find the range you find the y minimum value and the y maximum value and

that is the range.

Asymptotes

An asymptote is a vertical or horizontal line on a graph which a function approaches.

Periodic functions may have vertical and horizontal asymptotes.

Periodicity

Some periodic functions have periodicity. If the periodic functions has a repeating

pattern, then the function is said to have periodicity.

If the function does not repeat or have a pattern then it is said to not have periodicity.

Roots

Periodic Functions can have many roots.There are normally roots along the midline,

where the midline and a point on the graph hit.

Real World Periodic Functions

Periodic functions are popular in mapping tide patterns, seasonal patterns and also heat

beats. Many real world problems relate and use

periodic graphs.

Real World Example

• Suppose that you are on Drakes Beach in Point Reyes.  At 2:00 P.M. on October 2, the tide is (the water is at its deepest).  At that time you find that the water at the end of a pier is 1.5 meters.  At 8:00 P.M. the same day, when the tide is out, the water is at 1.1 meters.  Assume that the depth of the water varies sinusoidally with time. What is the amplitude?

• To find the amplitude you will use 1.5 meters and 1.1 meters. • You will then apply the formula--A = (M - m)/2• A=(1.5-1.1)/2 Which equals…. 0.2• So the amplitude is 0.2