Period of a Function

The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function.Or you can say, Periodic function is a function that repeats its values after every particular interval.

The period of the function is this particular interval mentioned above.

A function f will be periodic with period m, so if we have

f (a + m) = f (a), For every m > 0.

It Shows that the function f(a) possesses the same values after an interval of “m”. One can say that after every interval of “m” the function f repeats all its values.

For example – The sine function i.e. sin a has a period 2 π because 2 π is the smallest number for which sin (a + 2π) = sin a, for all a.

We may also calculate the period using the formula derived from the basic sine and cosine equations. The period for function y = A sin( B a – c ) and y = A cos( B a – c ) is 2πB radians.

The reciprocal of the period of a function = its frequency.

Frequency is defined as the number of cycles completed in one second. If the period of a function is denoted by P and f be its frequency, then –f =1/ P.

Fundamental Period of a Function

The fundamental period of a function is the period of the function which are of the form,

f(x+k)=f(x)

f(x+k)=f(x), then k is called the period of the function and the function f is called a periodic function.

Now, Let us define the function h(t) on the interval [0,2] as follows:

Fundamental period of a function

If we extend the function h to all of R by the equation,

h(t+2)=h(t)

=> h is periodic with period 2.

The graph of the function is shown below.

Fundamental Period of a Function

Finding the Period of a Function

  • If a function repeats over at a constant period we say that is a periodic function.
  • It is represented like f(x) = f(x + p), p is the real number and this is the period of the function.
  • Period means the time interval between the two occurrences of the wave.
  • To find the period of the periodic function we have to use the following formula, Where

Period = 2pb, here b is the co – efficient of x

Sine and cosine functions have the forms of a periodic wave:

Finding the period of a function

  • Period: It is represented as “T”, Period is the distance among two repeating points on the graph function.
  • Amplitude: It is represented as “A” It is the distance between the middle point to highest or lowest point on the graph function.

sin(aθ) = 2πa and cos(aθ) = 2πa

Solved Examples

Question 1: Find the period of the given periodic function. Where f(x) = 9sin(6px7 + 5)

Solution:Given periodic function is f(x) = 9sin(6px7+ 5)

period = 2pb, here period of the periodic function = 2p(6p7) = 146 = 73

Period of a Sine Function

If we have a function f(x) = sin (xs), where s > 0, then the graph of the function makes a complete cycles between 0 and 2π and each of the function have the period, p = 2πs

Now, let’s discuss about some examples based on sin function:

Let us discuss the graph of y = sin 2x

Period of a Sine Function

Period = π

Axis: y = 0 [x-axis ]

Amplitude: 1

Maximum value = 1

Minimum value = -1

Domain: { x : x ∈ R }

Range = [ -1, 1]

Period of a Trig Function

The distance between the repetition of any function is called the period of function. For a trigonometric function, the length of one complete cycle is called a period. For any trigonometry graph function, we can take x = 0 as the starting point.

In general we have three basic trigonometric functions like sin, cos and tan functions, having 2π, 2π

and π period respectively.

Solved Examples

Question 1: Graph of y = 4 sin(a/2)

Solution:

sine function

  • Period = 4π
  • Axis: y = 0 [x-axis ]
  • Amplitude: 4
  • Maximum value = 4
  • Minimum value = -4
  • Domain: { x : x ∈ R }
  • Range = [ -4, 4]

Period of a Tangent Function

If we have a function f(a) = tan (as), where s > 0, then the graph of the function makes a complete cycles between −π/2, 0 and π/2 and each of the function have the period of p = πs

Tangent Function

Period of a Function Examples

Let’s learn some of the examples on periodic functions.

Solved Examples

Question 1: What is the period of the following periodic function?

Where f(a) = 6 sin 5a

Solution: The given periodic function is f(a) = 6 sin 5a. We have the formula for the period of function.

period = 2π/b, Where period of the periodic function = 2π/5