Derivation of eight basic formulas of periodic function

The eight basic formulas of periodic function are derived as follows:

First, the definition of cycle

Generally speaking, if there is a non-zero constant t, there exists f(x+T) = f (x) for any x and x+t in the definition domain of the function f (x). Then, the function f(x) is called a periodic function, and the nonzero constant t is called a period of this function.

Note that in general, if a periodic function has a minimum positive period, "period" usually refers to the "minimum positive period" of this periodic function.

Second, the formula of periodic function commonly used in middle school mathematics

1, let the period (minimum positive period) of the periodic function y=f(x) be t, then f(x+nT)=f(x), and f(x-nT)=f(x). Where n can be any integer.

2. let the period (minimum positive period) of the periodic function y=f(x) be t, then y=f(x)+b, y=Af(x)+b (note: a is not equal to 0) are all periodic functions with minimum positive period of t. ..

3. Let the period (minimum positive period) of the periodic function y=f(x) be t, then y=f(wx)+b and y=Af(wx)+b are all periodic functions, and the minimum positive period is "T/|w|". (Note: A and W are not 0)

Third, the cycle of common periodic functions in high school mathematics

1, (1)y=sinx, and the minimum positive period t = 2π.

(2)y=|sinx|, and the minimum positive period T= π.

(1) y = cosx, and the minimum positive period t = 2π.

(2)y=|cosx|, and the minimum positive period T= π.

(1) y = tanx, and the minimum positive period t = π;

2)y=cotx, and the minimum positive period T=π.

4.y=Asin(wx+φ)+b, and the minimum positive period T=2π/|w|.

(Note: "A" and "W" are non-zero constants, the same below. )

5.y=Acos(wx+φ)+b, and the minimum positive period T=2π/|w|.

6.y=Atan(wx+φ)+b, and the minimum positive period T=π/|w|.

7. The constant function "y=c(c is a constant)" is a periodic function with any non-zero constant as its period.