Period of a function

The period of the function corresponds to the above-mentioned unique timeframe.

If we have a function f that is periodic with period m, we can say that we have

For every m > 0, the function f (a + m) equals the function f (a).

It demonstrates that the function f(a) has the same values when an interval of “m” is applied to it. It is possible to state that the function f repeats all of its values after every interval of “m.”

For example, the sine function, denoted by the symbol sin a, has a period of 2 because 2 is the smallest number for which sin (a + 2) = sin a for every an is equal to the sine function.

Another method of calculating the period is to use the formula that is derived from the fundamental sine and cosine equations. The period of the functions y = A sin(Bx + C) and y = A cos(Bx + C) is 2/|B| radians for the functions y = A sin(Bx + C).

The frequency of a function is equal to the reciprocal of its period.

When it comes to frequency, it is defined as the number of cycles that are completed in one second. If the period of a function is denoted by P and the frequency of the function is denoted by f, then –f =1/P.

Fundamental Period of a Function

Consider the fundamental period

f(x+k)=f(x)

If the function f(x+k)=f(x), then k is referred to as the period of the function, and the function f is referred to as a periodical function.

Now, consider the following definition of the function h(t) on the interval [0, 2]:

The fundamental period of a function is the period during which the function is defined.

If we use the equation to extend the function h to include all of R, we get

h(t+2)=h(t)

=> h has a period of 2 and is periodic.

A Function’s Fundamental Period is defined as follows:

What is the best way to find the period of a function?

When a function repeats again and over again at a steady rate, we refer to this as a periodic function.

When written as f(x), it is represented as f(x + p), where p is a real number and this is the period of the function.

The time gap between the two occurrences of the wave is referred to as its period.

Learn how to estimate the period and amplitude of a given trigonometric function, such as the sine, cosine, tangent, and so on, using graphs and examples in this section.

Period of a Trigonometric Function

The period of a trigonometric function is a measure of how long the function lasts.

The period of a function is defined as the amount of time that elapses between each repeat of the function. The length of one complete cycle of a trigonometric function is referred to as the period of the function. We can use x = 0 as the starting point for any trigonometry graph function we want to create.

In general, we have three fundamental trigonometric functions, which are the sin, cos, and tan functions, which have periods of -2, 2, and 0 respectively.

The sine and cosine functions have the appearance of a periodic wave in the following ways:

The period is represented by the letter “T.” A period is defined as the distance between two consecutive points on a graph function.

Amplitude is symbolized by the letter “A.” A graph function’s middle point is the point at which the graph function’s highest or lowest point is the distance between them.

The sine of an is 2a, while the cosine of an is 2a.

A Sine Function’s period 

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

Now, let’s look at some examples of the sin function in practice:

Let’s have a look at the graph of y = sin 2x.

A Sine Function’s period is measured in seconds.

The period of a Tangent Function 

Suppose we have a function f(a) = tan (as), where s > 0, and the graph of the function goes through entire cycles between /2, 0, and /2, and each of the functions has a period equal to the square root of p = /s.

Conclusion

Periodic Functions Have Certain Characteristics

The following characteristics of a periodic function are useful in developing a more in-depth understanding of the ideas of periodic functions.

• The graph of a periodic function is symmetric and repeats itself along the horizontal axis, indicating that the function is periodic.

• The periodic function’s domain comprises all of the real number values, and the periodic function’s range is determined for a fixed interval of time.

• Across the entire range of a periodic function’s range, the period against which the period repeats itself equals the constant of the function.

• Assuming that f(x) is a periodic function with a period of P, it follows that 1/f(x) is also a periodic function with the same fundamental period of P.

• Assuming that f(x) has a periodic period of P and that f(ax + b) has a period of P/|a|, we can conclude that f(ax + b) has a period of P/|a|.

• Then if f(x) is a periodic function with a period of P, then (a f(x)+b) is a periodic function with a period of P, and so on.