Some real life examples of periodic functions are the length of a day, voltage coming out of a wall socket and finding the depth of water at high or low tide. A periodic function is defined as a function that repeats its values in regular periods. The period is the length of time it takes for the cycle to repeat itself.
Trigonometric functions are the most important examples of periodic functions, and these repeat over lengths of 2(pi). These functions are used to describe waves, oscillations and other types of patterns that exhibit periodicity.
There are four main properties of periodic functions:
- The functions cosine and sine have a period of 2(pi).
- The functions cotangent and tangent have a period of pi.
- Periodic functions cannot be monotonic, or never decreasing or increasing, on the entire domain.
- There are specific trigonometric functions for any real number "x" and any real number "k," such as sin(x+2(pi)k) is equal to sinx.
When graphed, a function is said to be periodic when it exhibits translational symmetry. This is a symmetry that results in moving a figure in a certain direction without changing shape or orientation. A visual example of this type of symmetry is sliding a stamp across the top of the wall, along the ceiling.