"Function y=Asin(ωx+φ)" Trigonometric Function PPT (Properties and Applications of Function y=Asin(ωx+φ) in Lesson 2)

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"Function y=Asin(ωx+φ)" Trigonometric Function PPT (Properties and Applications of Function y=Asin(ωx+φ) in Lesson 2)

Part One: Lecture, Practice and Interaction

Find analytical expressions of trigonometric functions from images

Part of the image of function f(x)=Asin(ωx+φ)A>0, ω>0, |φ|<π2 is as shown in the figure, then the analytical formula of f(x) is ______________.

regular method

A method of finding analytical expressions based on partial graphs of functions

(1) Determine the amplitude and period directly from the image, then you can determine the parameters A and ω in the functional formula y=Asin(ωx+φ), then select the data of the maximum point and substitute it into ωx+φ=2kπ+π2, k∈Z, combined with the range of φ Find φ.

(2) Substituting several special points into the functional formula, and solving the system of equations to find the relevant undetermined coefficients A, ω, and φ.

(3) Using the method of reverse thinking, first determine the basic functional formula of the function y = Asin ωx, and then determine the relevant parameters according to the image translation law.

Track training

1. Part of the image of the known function y=Asin(ωx+φ)+B is shown in the figure. If A>0, ω>0, |φ|<π2, then ()

A． A=4B. ω=1

C． φ=π6 D． B＝4

2. It is known that the minimum value of the function y=Asin(ωx+φ)A>0, ω>0, 0<φ <π2 is -5, the abscissa difference between the two adjacent highest points and the lowest point on the image is π4, and the image After passing through points 0 and 52, find the analytical formula of this function.

Symmetry of graphs of trigonometric functions

It is known that the minimum positive period of function f(x)=sinωx+π3 (ω>0) is π, find the symmetry axis equation of this function.

regular method

How to find the symmetry axis and center of symmetry of trigonometric functions

axis of symmetry center of symmetry

y＝Asin(ωx＋φ) Let ωx＋φ＝kπ＋π2(k∈Z) Let ωx＋φ＝kπ(k∈Z) and find the abscissa of the symmetry center

y＝Acos(ωx＋φ) Let ωx＋φ＝kπ(k∈Z) Let ωx＋φ＝kπ＋π2(k∈Z) and find the abscissa of the symmetry center

y＝Atan(ωx＋φ) None Let ωx＋φ＝kπ2(k∈Z) and find the abscissa of the symmetry center

Comprehensive application of properties of trigonometric functions

(2019•Shenyang Quality Inspection (1)) It is known that the function f(x)=sin2x+π3, the following proposition is false ()

A． The graph of function f(x) is symmetric about the straight line x=π12

B. x=-π6 is a zero point of function f(x)

C． The graph of function f(x) can be obtained by shifting the graph of g(x)=sin 2x to the left by π3 unit length.

D. The function f(x) is an increasing function on 0, π12

regular method

(1) How to judge the parity of sine and cosine functions

The sine function y=Asin(ωx+φ) and the cosine function y=Acos(ωx+φ) do not necessarily have parity. For the function y=Asin(ωx+φ), it is an odd function when φ=kπ(k∈Z), and it is an even function when φ=kπ±π2(k∈Z); for the function y=Acos(ωx+φ), when φ It is an even function when =kπ(k∈Z), and an odd function when φ=kπ±π2(k∈Z).

(2) Method to determine the monotonic interval of function y = Asin (ωx + φ) (A>0, ω>0)

Using the "substitution" method for overall substitution, considering ωx + φ as a whole, we can set "z = ωx + φ", that is, by finding the monotonic interval of y = Asin z, we can find the monotonic interval of the function y = Asin (ωx + φ). If ω<0, the induction formula can be used to first convert the coefficient of x into a positive number, and then find the monotonic interval.

Function y=Asin(ωx+φ)PPT, Part 2: Feedback on compliance with standards

1. (2019•Beijing Haidian University of Science and Technology Affiliated Middle School) Translate the image of the function y=sin2x+π4 to the right by π8 unit length, and the function corresponding to the resulting image is ()

A． Non-odd and non-even functions B. both odd and even functions

C． Odd function D. even function

2. A symmetry axis of the graph of function f(x)=sin(2x+φ)(-π＜φ＜0) is the straight line x=π6, then the value of φ is ________.

3. The graph of function f(x)=Asin(ωx+φ)ω>0, A>0, |φ|<π2 is as shown in the figure.

(1) Find the analytical formula of function f(x);

(2) Find the value range of function y=f(x) on -π4, π6.

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Update Time: 2024-10-04

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