Period of a Function

The period $ t $ of a periodic function $ f(x) $ is the value $ t $ such that $$ f (x+t) = f(x) $$

Graphically, its curve is repeated each period, by translation. The function is equal to itself all the lengths $ t $ (it presents a pattern which is repeated by translation).

The value of the period $ t $ is also called the periodicity of the function.

To find the period $ t $ of a function $ f(x) $, demonstrate that $$ f(x+t)=f(x) $$

Example: The trigonometric function $ \sin(x + 2\pi) = \sin(x) $ so $ \sin(x) $ is periodic of period $ 2\pi $

Trigonometric functions are usually periodic period, to guess the period, try multiples of pi for value $ t $.

If the period is equal to 0, then the function is not periodic.

Any periodic function of period $ t $ repeats every $ t $ values. To predict the value of a periodic function, for a value $ x $ calculate $ x_t = x \mod t $ (modulo t) and find the known value of $ f(x_t) = f(x) $

Example: The function $ f(x) = \cos (x) $ has a period of $ 2\pi $, the value for $ x = 9 \pi $ is the same as for $ x \equiv 9 \pi \mod 2\pi \equiv \pi \mod 2\pi $ and therefore $ \cos(9\pi) = \cos(\pi) = -1 $

The amplitude is the absolute value of the non-periodic part of the function.

Example: $ a \sin(x) $ has for amplitude $ | a | $

If $ f $ is periodic, then it exists a real not null such as $$ f(x+t)=f(x) $$

Demonstration consists in proving that it is impossible. For example with a reductio ad absurdum or performing a calculation that leads to a contradiction.

The most common periodic functions are trigonometric functions based on sine and cosine functions (which have a period of 2 Pi).