### 2 Periodic functions

Any function that has a definite pattern repeated at regular intervals is said to be periodic. The interval over which the repetition takes place is called the
**
period
**
of the function, and is usually given the symbol
$T$
. The period of a periodic function is usually obvious from its graph.

Figure 20 figure shows a graph of a periodic function with period $T=3$ . This function has discontinuities at values of $x$ which are divisible by $3$ .

**
Figure 20
**

Figure 21 shows a graph of a periodic function with period $T=6$ . This function has no discontinuities.

**
Figure 21
**

If a function is a periodic function with period
$T$
then, for any value of the independent variable
$x$
, the value of
$f\left(x+T\right)$
is the
**
same
**
as the value of
$f\left(x\right)$
.

##### Key Point 3

A function
$f\left(x\right)$
is
**
periodic
**
if we can find a number
$T$
such that

$\phantom{\rule{2em}{0ex}}f\left(x+T\right)=f\left(x\right)\phantom{\rule{2em}{0ex}}\text{forallvaluesof}\phantom{\rule{1em}{0ex}}x$

Often a periodic function will be defined by simply specifying the period of the function and by stating the rule for the function within one period. This information alone is sufficient to draw the graph for all values of the independent variable.

Figure 22 shows a graph of the periodic function defined by

$\phantom{\rule{2em}{0ex}}f\left(x\right)=x,\phantom{\rule{1em}{0ex}}-\pi <x<\pi ,\phantom{\rule{2em}{0ex}}\text{period}T=2\pi $

**
Figure 22
**

##### Exercises

- Explain what is meant by a periodic function.
- Sketch a graph of a periodic function which has no discontinuities.
- Sketch a graph of a periodic function which has discontinuities.
- A periodic function has period 0.01 seconds. How many times will the pattern in the graph repeat over an interval of 10 seconds ?

- 1000.