WHAT IS THE PERIOD OF THE FUNCTION Y = 3SEN (4X)? - MATHS - 2025

The period of the function y = 3sen (4x) is 2π / 4 = π / 2. To clearly understand the reason for this statement, one must know the definition of the period of a function and the period of the function sin (x); a little bit about graphing of functions will also help.

Trigonometric functions, such as sine and cosine (sin (x) and cos (x)), are very useful in both mathematics and engineering.

The word period refers to the repetition of an event, so to say that a function is periodic is equivalent to saying "its graph is the repetition of a piece of curve." As can be seen in the previous image, the function sin (x) is periodic.

Periodic functions

A function f (x) is said to be periodic if there exists a real value p ≠ 0 such that f (x + p) = f (x) for all x in the domain of the function. In this case, the period of the function is p.

The smallest positive real number p that satisfies the definition is generally called the period of the function.

As can be seen in the previous graph, the sin (x) function is periodic and its period is 2π (the cosine function is also periodic, with a period equal to 2π).

Alterations in the graph of a function

Let f (x) be a function whose graph is known, and let c be a positive constant. What happens to the graph of f (x) if f (x) is multiplied by c? In other words, what is the graph of c * f (x) and f (cx) like?

Graph of c * f (x)

When multiplying a function, externally, by a positive constant, the graph of f (x) undergoes a change in the output values; that is, the change is vertical and there are two cases:

- If c> 1, then the graph undergoes a vertical stretch with a factor of c.

- Yes 0

Graph of f (cx)

When the argument of a function is multiplied by a constant, the graph of f (x) undergoes a change in the input values; that is, the change is horizontal and, as before, there can be two cases:

- If c> 1, then the graph undergoes horizontal compression with a factor of 1 / c.

- Yes 0

Period of the function y = 3sen (4x)

It should be noted that in the function f (x) = 3sen (4x) there are two constants that alter the graph of the sine function: one multiplying externally and the other internally.

The 3 that is outside the sine function, what it does is lengthen the function vertically by a factor of 3. This implies that the graph of function 3 sin (x) will be between the values ​​-3 and 3.

The 4 inside the sine function causes the graph of the function to undergo horizontal compression by a factor of 1/4.

On the other hand, the period of a function is measured horizontally. Since the period of the function sin (x) is 2π, considering sin (4x) the size of the period will change.

To find out what the period of y = 3sin (4x) is, just multiply the period of the function sin (x) by 1/4 (the compression factor).

In other words, the period of the function y = 3sin (4x) is 2π / 4 = π / 2, as can be seen in the last graph.

References

1. Fleming, W., & Varberg, DE (1989). Precalculus Mathematics. Prentice Hall PTR.
2. Fleming, W., & Varberg, DE (1989). Precalculus mathematics: a problem-solving approach (2, Illustrated ed.). Michigan: Prentice Hall.
3. Larson, R. (2010). Precalculus (8 ed.). Cengage Learning.
4. Pérez, CD (2006). Precalculation. Pearson Education.
5. Purcell, EJ, Varberg, D., & Rigdon, SE (2007). Calculus (Ninth ed.). Prentice Hall.
6. Saenz, J. (2005). Differential Calculus with early transcendent functions for Science and Engineering (Second Edition ed.). Hypotenuse.
7. Sullivan, M. (1997). Precalculation. Pearson Education.