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.)
Period of a Function - dCode
Tag(s) : Functions
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!
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).
Function | Period |
---|---|
Sine $ \sin(x) $ | $ 2\pi $ |
Cosine $ \cos(x) $ | $ 2\pi $ |
Tangent $ \tan(x) $ | $ \pi $ |
dCode retains ownership of the "Period of a Function" source code. Except explicit open source licence (indicated 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, or API access for "Period of a Function" are not public, same for offline use on PC, tablet, iPhone or Android !
The copy-paste of the page "Period of a Function" or any of its results, is allowed as long as you cite the online source
Reminder : dCode is free to use.