Periodic Functions

Theory

A function $f(x)$ is called periodic if there exists a positive number $p$ such that

f(x+p)=f(x) \quad \text{for all } x.

The smallest such $p$ is called the fundamental (primitive) period.

Exercises

Exercise 1

Question

Draw graphs of the following functions:

$f(x)=\sin x$
$g(x)=\cos(nx)$

Show solution ↓

Solution

1. Graph of $f(x)=\sin x$

The fundamental period of $\sin x$ is $2\pi$ .
Key points in one period:

(0,0),\quad \left(\frac{\pi}{2},1\right),\quad (\pi,0),\quad \left(\frac{3\pi}{2},-1\right),\quad (2\pi,0).

The graph repeats this pattern every $2\pi$ units.

How to draw:

Mark the key points over $[0,2\pi]$ .
Draw a smooth wave passing through them.
Repeat the same shape to the left and right.

2. Graph of $g(x)=\cos(nx)$

The fundamental period of $\cos(nx)$ is

\frac{2\pi}{n}.

Increasing $n$ makes the graph oscillate faster.
The amplitude remains $1$ .

How to draw:

Compute the period $\frac{2\pi}{n}$ .
Plot one cosine curve over $\left[0,\frac{2\pi}{n}\right]$ .
Repeat this pattern periodically.

Important observation:

Larger $n$ $\Rightarrow$ shorter period
Same maximum and minimum values ( $\pm 1$ )
More waves in the same interval