The periodicity and symmetry of functions, and several important conclusions on the symmetry and per

The function y=f(x+4) is an odd function.

y=f(x+4) image symmetry with respect to the origin.

Translate the y=f(x) image 4 units to the left.

The image of y=f(x+4) is obtained.

Pan the y=f(x+4) image to the right by 4 units.

The y=f(x) image is obtained.

The image of y=f(x) is about o'(4,0) Symmetry.

f(x) in the interval [4,+ analytically is f(x)=4 x-x+3 take x<4, then 8-x>4

The image of f(8-x)=4 (8-x)-(8-x)+3=4 (8-x)+x-5y=f(x) is about o'(4,0) Symmetry.

f(x)=-f(8-x)=4 (x-8)-x+5f(x) is parsed on r.

4/x-x+3 , x≥4)

f(x)={4/(x-8)-x+5 ,(x<4)

Since the function y=f(x+4) is odd, f(x) is symmetric with respect to the point (4,0). Let the point (x,y) be on the image of f(x), and because of the symmetry about the point (4,0), then substitute (8-x,-y) into the function f(x), and get y=4 (x-8)-x+5.

So f(x)=4 (x-8)-x+5(x<4), f(x)=4 x-x+3(x>=4).

Because f(x+4) is an odd function, so.

f(-x+4)=-f(x+4)

So this function is about point (4,0) point symmetry.

When x<4, -x> -4, 8-x>4

f(8-x)=4/(8-x)-(8-x)+3=4/(8-x)+(x-5)

Because f(x) is symmetrical about (4,0) points.

f(x)= - f(-x+8)=-4/(8-x)-x+5f(x)={ -4/(8-x)-x+5 (x<4)4/x-x+3 (x≥4)

Senior 3 Mathematics] Functional Symmetry and Periodicity.

The mantra for periodicity and symmetry of a function is the period of symmetry difference.

If f(x+a)=-f(x+b), there is one more minus sign. (x+a)-(x+b)=a-b, period x2. Periodicity, t=2|a-b|。

If f(x+a)=-f(x+b), there is one more minus sign. (x+a)+(x+b)=a+b, the axis becomes the center. Symmetry, center of symmetry ((a+b) 2,0).

Properties: 1. If the function f(x)(x d) has two axes of symmetry x=a and x=b in the defined domain, then the function f(x) is a periodic function, and the period t=2|b-a|(Not necessarily a minimum positive period).

2. If the function f(x)(x d) has two symmetry centers a(a,0) and b(b,0) in the defined domain, then the function f(x) is a periodic function, and the period t=2|b-a|(Not necessarily a minimum positive period).

3. If the function f(x)(x d) has an axis of symmetry x=a and a center of symmetry b(b, 0)(a≠b) in the defined domain, then the function f(x) is a periodic function, and the period t=4|b-a|(Not necessarily a minimum positive period).

Conclusion of functional symmetry: y=f(|x|) is an even function.

It is symmetrical with respect to the y-axis.

y=|f（x）|It is to symmetricate the image below the x-axis to the top of the x-axis, but it is impossible to determine whether it is symmetrical or not. For example, y=|lnx|There is no symmetry, and y=|sinx|But there is symmetry.

1、f（x＋a）＝－f（x）

Then f(x 2a) f (x a) is wide and pure a f(x a) f(x) f(x).

So f(x) is a periodic function with a period of 2a.

2. Difference f(x a) 1 f(x).

Then f(x 2) f(x ) a) 1 f(x a) 1 (1 f(x)) be careful with f(x).

So f(x) is a periodic function with a period of 2a.

1: Symmetry: A function: f(a+x)=f(b-x) holds, f(x) is symmetrical with respect to the line x=(a+b) 2.

f(a+x)+f(b-x)=c, f(x) is symmetrical with respect to the point ((a+b) 2, c 2.

Two functions: y=f(a+x) and y=f(b-x) are symmetrical with respect to the straight line x=(b-a) 2.

Proof: Take a point (m,n) on a function to prove that the point that has undergone a symmetrical transformation is still on the function.

For example, the central symmetry formula proves that the point (m,n) is taken on the function, and the symmetry point is (a+b-m,c-n).

f(a+(b-m))+f(b-(b-m)=c then f(a+(b-m))+n=c, that is to say, f(a+(b-m))=c-n is also a function.

2.Periodicity: f(x+a) = f(x) period 2a

f(x+a) = or 1 f(x) period 2a

Proof: Let the period be na, f(x+na)=.f(x)

3. Periodicity and symmetry appear at the same time, find the period (defined as a function on r), and then you can get an intuitive answer by drawing.

With respect to x=a, x=b symmetry period 2 (a-b).

Symmetry with respect to (a,0) and x=b Weekly Bump Period 4 (a-b).

As shown about (a,0) and x=b symmetry in period 4(a-b): f(x) = f(2a-x).

f(x)=f(2b-x)

f(2a-x) =f(2b-x)

f(2a+x) =f(2b+x)

f(x+4(a-b))=f(x+2a-2b)=f(x)

Example: y=f(x) satisfies f(x+1)=f(1-x) and f(x+3)=f(3-x) with a period of 4

Proof f(x+1)=f(1-x)=f(3+(-2-x))=f(3-(-2-x))=f(x+5).

1) If the image of a function has two pairs of fingers, then the laughing plexus function is a periodic function.

2) If the image of a function has two axes of symmetry.

Then this function is a periodic function.

3) If the image of a function has a symmetry point and an axis of symmetry, then the function is a periodic function.

1.Symmetry f(x+a)=f(b x) Remember that this equation is a general form of symmetry. As long as x has a positive and a negative. There is symmetry. As for the axis of symmetry, you can find x=a+b 2 by eating the formula

For example, f(x+3)=f(5 x).

x=3+5, 2=4, etc. This formula is common to those who do not know the equation but know the relationship between the two equations. You can apply it, but I won't give you an example here.

For known equations that require the axis of symmetry, first of all, you have to keep in mind some common symmetry equations for the axis of symmetry. For example, a primitive quadratic equation f(x) = ax2 + bx + c axis of symmetry x b 2a

The axis of symmetry of the original function and the inverse function is y x

And for some functions, it is difficult to say that their axes of symmetry are not only x 90 but also 2n if they are not restricted, such as trigonometric functions, and its axes of symmetry are not only x 90 but also 2n! 90 degrees and so on because his definition is r

f(x) x and his axis of symmetry is x 0, and it should also be noted that some axes of symmetry required after translation by simple functions can be reversed to the original and so on, and then the number of translations can be added

If f(x 3) x 3 makes t x 3, then f(t) t shows that the original equation is shifted by 3 units to the right by the elementary function, and similarly, the axis of symmetry is also shifted to the right by 3 units x 3 (remember that translation is in the form of left addition and right subtraction, as x 3 in this problem illustrates the shift by direction).

2. As for periodicity, let's first start with the general form f(x) f(x t).

Note that the x in this formula is the same sign, not like the symmetry equation, which is positive and negative, and this difference is also the key to determining whether symmetry or periodicity

Also keep in mind some common periodic functions such as trigonometric functions, what sine functions, cosine functions, tangent functions, etc., of course, their minimum periods are 2 , 2 , of course.

Their period is more than that, as long as it is a positive multiple of their minimum period, it can be the period of the problem, e.g. f(x) sinx t 2 (t 2 w).

But if it is f(x) sinx, its period is t, because after adding the absolute value, the graph below the y-axis is all turned to the top, and it is not difficult to see from the graph that the minimum symmetry week t

y1=(sinx)^2=(1-cos2x)/2

y2=(cosx)^2=(1+cos2x)/2

The above 2 equations t (t 2 w).

And for the addition and subtraction composite equation of the equation of the 2 periodic functions, if their periods are the same, then its period is still the same period e.g. y=sin2x+cos2x because they have a common period t, so its period is t

For periods that are not identical, then its period is the least common multiple of their respective periods, such as.

y=sin3 x+cos2 x t1 2 3 t2 1 then t 2 3