Concept demonstration:

By comparing the periodic appearance of "week" with the change of date and the periodic appearance of sine function value with the change of angle in calendar, it is found that the essence of the two is that when the independent variable increases a certain value, the function value appears regularly and repeatedly.

Show the definition of function periodicity: for 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 function y=f(x) is called a periodic function, and non-zero constant t is called the period of this 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 with a period of.

T=2kπ(k∈Z and 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,

That is, 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 increases to at least 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 reappear, so the value of function 3cosx will reappear, so the period of y=3cosx is 2π. (It means 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,

So as long as the independent variable x increases to at least 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 all constants, a? 0,

x？ r)

The period of the function is T=(2π-φ)/ω.

References:

Author: Lu Dongbao

SETTING: No.3 Middle School of Yanqing County, Beijing

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