The key word of the function is "periodic repetition".
When the independent variable increases to any real number (the independent variable is meaningful), the function value appears regularly and repeatedly.
If the function f(x)=f(x+T) (or f(x+a)=f(x-b) where a+b=T), then t is a period of the function. An integer multiple of t is also the period of a function.
1. Put forward the concept: compare the periodical appearance of "week" with the date change and the periodical appearance of sine function value with the angle change in the calendar, and find out the essence of both: when the independent variable increases a certain value, the "function value" appears regularly and repeatedly.
Show the definition of function periodicity: for the function y=f(x), if there is a non-zero constant t, so that f(x+T)=f(x) holds when x takes every value in the defined domain, then the function y=f(x) is called a periodic function, and the non-zero constant t is called the period of the function.
When the independent variable increases a certain value, the function value appears regularly.
2. Definition: For the function y=f(x), if there is a non-zero constant t, when x takes every value in the definition domain, f(x+T)=f(x).
The concretization of concepts:
When f(x)=sinx or cosx in the definition, the value of t is considered.
T=2kπ(k∈Z and k≠0)
So sine function and cosine function are both periodic functions, and the period is T=2kπ(k∈Z, k≠0).
Show pictures of sine and cosine functions.
The image shape of the periodic function changes periodically with the change of X. )
Emphasize the definition of "when x takes every value in the domain"
Let (x+T)2=x2, then x2+2xT+T2=x2.
So 2xT+T2=0, which means T(2x+T)=0.
So T=0 or T=-2x.
Emphasize "non-zero" and "constant" in the definition.
For example: trigonometric function sin(x+T)=sinx.
Cos (x+t) = t in cosx takes 2π.
3. The concept of minimum positive period:
For a function f(x), if all its periods have a minimum positive number, then this minimum positive number is called the minimum positive period of f(x).
For sine function y=sinx, as long as the independent variable x is at least increased to x+2π, the function value can be obtained repeatedly. So the minimum positive period of sine function and cosine function is 2π. (Note: Unless otherwise specified, period refers to the minimum positive period. )
On the function image, the minimum positive period is the shortest distance required for the function image to appear repeatedly.
4. Example: Find the period of the following function:
( 1)y=3cosx
Analysis: As long as the independent variable in cosx increases to at least x+2π, the value of function cosx will be repeated, so the value of function 3cosx will be repeated, so the period of y=3cosx is 2π. (Explain that the coefficient before cosx has nothing to do with the period. )
(2)y=sin(x+π/4)
The analysis shows that the angle behind x does not affect the period.
(3)y=sin2x
Analysis: Because sin2(x+π)=sin(2x+2π)=sin2x, as long as the independent variable x is at least increased to x+π, the function value will appear repeatedly. So the period of the original function is π. (Explain that the coefficient of x has an influence on the period of the function. )
(4) y=cos(x/2+π/4) (analysis omitted)
(5)y=sin(ωx+φ) (analysis omitted)
Conclusion: the shape is y=Asin(ωx+φ) or y=Acos(ωx+φ) (A, ω, φ are constants, a? 0，x？ The period of the function of r) is T=2π/ω.
Properties of periodic functions:
(1) If T(≠0) is the period of f(X), then -T is also the period of f(X).
(2) If T(≠0) is the period of f(X), then nT(n is an arbitrary non-zero integer) is also the period of f(X).
(3) If T 1 and T2 are both periods of f(X), then T 1 T2 is also a period of f(X).
(4) If f(X) has a minimum positive period T*, then any positive period t of f(X) must be a positive integer multiple of T*.
(5)T* is the minimum positive period of f(X), and T 1 and T2 are two periods of f(X) respectively, then (q is a rational number set).
(6) If T 1 and T2 are two periods of f(X) and are irrational numbers, then f(X) does not have a minimum positive period.
(7) The domain m of the periodic function f(X) must be a set with unbounded sides.