The image of the function y = sin (x + (5 / 6) π) is shifted by vector a = (m, 0) (m), and the image is symmetric about the y-axis After the image of function y = sin (x + (5 / 6) π) is translated by vector a = (m, 0) (M 〉 0), the obtained image is symmetric about y axis, and the minimum value of M is obtained

First of all, the image of y = sin (x + (5 / 6) π) is translated by vector a = (m, 0) (M 〉 0), and then y = sin (x-m + (5 / 6) π) = sin (- x-m + (5 / 6) π), namely sin (x-m - (1 / 6) π) = sin (- x-m - (1 / 6) π) = sin (- x-m - (1 / 6) π) = - sin (x + m + (1 / 6) π), then x-m

The image of function y = sin (x + (5 / 6) π) is translated by vector a = (- m, 0), and the image is symmetric about y-axis Then the minimum positive value of M is?

a=(-m,0)
That is, move the image to the left by M units,
From the image of y = sin (x + (5 / 6) π), we only need to move the original function to the left π / 2 - (5 / 6) π + K π
You can get images that are symmetrical about the y-axis
Let m = π / 2 - (5 / 6) π + K π where (k belongs to integer z)
Then, if k = 1, there is a minimum positive value of m such that the meaning of the question is satisfied, and then M = (2 / 3) π

If the function y = sin (- 1 / 2x + л / 3), the image is translated by vector a If the image of the function y = sin (- 1 / 2x + л / 3) is translated by vector a and the image of y = sin1 / 2x is obtained, then the vector a = () A.(-Л/3,0) B.(Л/3,0) C.(4Л/3,0) D.(-4Л/3,0)

C
Y = sin (- 1 / 2x + л / 3) take the point (Ψ / 3,1 / 2) a = (h, K) and the coordinates of the point after translation are (x ', y')
x'=Л/3+h y'=1/2+k
1 / 2 + k = sin1 / 2 (Ψ / 3 + H) let k = 0 obtain an H which satisfies the condition of 4 л / 3

The image of a function overlaps with the image of y = sin (2x - π / 3) + 1 after the translation vector a = (π / 6,1), what is the function

Let the coordinate of a point (x, y) on the original function image after vector a translation is (x ', y')
Then x '= x + π / 6, y' = y + 2
∵y=sin（2x-π/3）+1
∴y+2=sin[2(x+π/6)-π/3]+1
That is, y = sin2x-1,
This function is y = sin2x-1

Press the image of a function a=(π 4,2) the analytic expression of the image obtained by translation is y = sin (x + π) 4) + 2, then the original function analytic formula is () A. y=sinx B. y=cosx C. y=sinx+2 D. y=cosx+4

A kind of
a=(π
4，2)∴-
a=（-π
4，-2）
Let y = sin (x + π)
4) + 2 by vector-
After a translation, y = sin (x + π)
4+π
4）+2-2=cosx
Therefore, B

Translation of trigonometric functions For example, change y = SiNx to y = sin (2x - π) First motion period or translation? Why?

y=sin(2x-π)=sin2（x-π/2)
The abscissa of y = SiNx is reduced to half of its original value and becomes y = sin2x
Then shift π / 2 to the right to obtain sin2 (x - π / 2)
Or first move π to the right to form y = sin (x - π), and then reduce the abscissa to half of the original

Translation of trigonometric functions First, the period of the function y = 4sin (π / 6-3x) is expanded to twice the original period, and then the image of the new function is shifted to the right by π / 3. Then the analytic expression of the image obtained is () A.y=5sin(2π/3-3x/2) B.y=cos3x/2 C.y=5sin(7π/10-3x/2) D.y=5sin(π/6-2x) When I did it, I raised a 2 for both π / 6-3x / 2, but didn't I just care about x? Why did π / 6 raise a 2?

You don't understand the concept. Moving right means x might as well do it
Y = 5sin (π / 6-3x) period expansion 2 is shifted to the right by y = 5sin (π / 6-3x / 2). Note that the right shift here refers to x, so it becomes y = 5sin [π / 6-3 (x - π / 3)] and then expanded to get the answer a

How to translate trigonometric functions? For example, y = sin (Wx + G) is the first vertical axis transformation or the first horizontal axis transformation. Is there anything to pay attention to?

Fine
It doesn't matter if you transform the horizontal and vertical axes
The horizontal axis is the period
The vertical axis is the amplitude
y=sin(wx+g)
You can translate g units first
Then the horizontal axis is stretched and contracted periodically
You can also stretch the horizontal axis first and then translate g / W units
all one to
It's the order

On the translation of trigonometric functions If the ordinate of every point in the image of the function y = f (x) remains unchanged and the abscissa is extended to twice the original value, then the image is shifted to the left along the X axis by π / 2 units and down by 1 unit along the Y axis. The curve is the same as that of y = (1 / 2) SiNx image? I think there are two ways to translate trigonometric functions, that is, the one that stretches and then translates. For example, why doesn't it go back to y = (1 / 2) sin (2x + φ) + 1, and then the phase transformation is not for X. then, it can be changed into 2 (x + φ) in brackets, because π / 2 units are shifted to the left, so φ = - π / 2, Then, it is simplified to y = (1 / 2) sin (2x - π) + 1, but the answer is y = (1 / 2) sin (2x - π / 2) + 1. Why I use the method of stretching first and then translating, so please consider this aspect. I should be able to use other methods,

In order to change the shape of a trigonometric function, the method of translation or expansion of coordinate units can be used. As for the change process, the sequence of steps generally does not affect the result. The following points should be noted:
(1) In electronics, f (T) = asin (ω T + φ), G (T) = ACOS (ω T + φ), denotes a single frequency electrical signal, a is called signal amplitude, ω is called angular frequency (radian / sec), ω = 2 π F, f is called signal frequency (Hertz), f = 1 / T, t is signal period (seconds), t is time, (ω T + φ) is signal phase, and φ is initial phase (radian)
(2) For the horizontal translation of the function f (x) = asin (ω x + φ), only the horizontal coordinate X is used. The result is that the initial phase of the function will be affected, that is, the frequency of the function will not change, and the initial phase will change
For example, the function f (x) = 1 / 2Sin (x) is shifted to the right π / 2, = = > F (x) = 1 / 2Sin (x - π / 2)
(3) For the function f (x) = asin (ω x + φ), the up and down translation is only carried out for the ordinate y, and the result will affect the value of the function
For example, the function f (x) = 1 / 2Sin (x) is moved up one coordinate unit, = = > F (x) = 1 / 2Sin (x) + 1
(4) The expansion and contraction of the horizontal coordinate unit only affects the frequency of the function, but does not affect the initial phase of the function
For example, if the X coordinate unit is compressed twice, the function f (x) = 1 / 2Sin (x) = = > F (x) = 1 / 2Sin (2x); the function f (x) = 1 / 2Sin (x + φ) = > F (x) = 1 / 2Sin (2x + φ)
If the unit of X coordinate is extended by one time, the function f (x) = 1 / 2Sin (x) = = > F (x) = 1 / 2Sin (1 / 2x); the function f (x) = 1 / 2Sin (x + φ) = > F (x) = 1 / 2Sin (1 / 2x + φ)
(5) The expansion and contraction of the horizontal coordinate unit not only affects the frequency of the function, but also affects the horizontal translation of the function
For example, if the X coordinate unit is compressed twice, the function f (x) = 1 / 2Sin (x + φ) = > F (x) = 1 / 2Sin (2 (x + φ / 2))
If the unit of X coordinate is extended by one time, the function f (x) = 1 / 2Sin (x + φ) = > F (x) = 1 / 2Sin (1 / 2 (x + 2 φ))
Understand the above points, to explain your problem, in the process of change, the sequence of steps, generally will not affect the results
Let's look at compression before translation
Function f (x) = 1 / 2Sin (x)
1) Compress the horizontal coordinate to the original 1 / 2 = = > F (x) = 1 / 2Sin (2x)
2) Then shift f (x) = 1 / 2Sin (2x) horizontally to the right (π / 2) / 2 = = > F (x) = 1 / 2Sin (2 (x - π / 4)) = 1 / 2Sin (2x - π / 2)
3) Then move f (x) = 1 / 2Sin (2 (x - π / 4)) = 1 / 2Sin (2x - π / 2) up one unit = = > F (x) = 1 / 2Sin (2x - π / 2) + 1
The original function is f (x) = 1 / 2Sin (2x - π / 2) + 1

What is the monotone decreasing interval of the function f (x) = 3x + radical 1 + 2x

If the title is correct, the function has no monotonic decreasing interval