Search for a tool
Period of a Function

Tool to compute the period of a function. The period of a function is the lowest value t such that the function repeats itself: f(x+t)=f(x-t)=f(x), that is the case for trigo functions (cos, sin, etc.)

Results

Period of a Function -

Tag(s) : Functions

Share dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Period of a Function' tool, so feel free to write! Thank you!

# Period of a Function

## Period of a Function Calculator

### What is a period of a function? (Definition)

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.

### How to find the period of a 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.

### How to find the value f(x) of a periodic function?

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 $\cos(9\pi) = \cos(\pi) = -1$

### How to find the amplitude of a periodic function?

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

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

### How to prove that a function is not periodic?

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.

### What are usual periodic functions?

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

FunctionPeriod
Sine $\sin(x) $$2\pi Cosine \cos(x)$$ 2\pi$
Tangent $\tan(x)$$\pi$

## Source code

dCode retains ownership of the online "Period of a Function" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Period of a Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Period of a Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, copy-paste, or API access for "Period of a Function" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!