How to find the center point of symmetry, how to find the coordinates of the center of symmetry

Let the center of symmetry of the function be (a,b).

Then if the point (x,y) is on the image of the function, then the point (2a-x, 2b-y) must also be on the image of the function, so the point (2a-x,2b-y) is substituted into the analytic expression of the function, and reduced to the form y=f(x), and the expression is at this time.

contains a and b, and combines this formula with the original function.

expressions, because these two function expressions represent a function, so there are comparison coefficients, you can get the values of a and b, and naturally find the center of symmetry.

The center of symmetry of a function means that the graph of the function rotates 180° around a certain point, and if it can coincide with another graph, then the two graphs are said to be symmetrical with respect to this point, and this point is called the center of symmetry.

The formula for the center of symmetry of the function is f(x) with respect to (a,b) symmetry, then there is f(x)+f(2a-x)=2b, { or f(a+x)+f(a-x)=2b

Specific methods: 1. Symmetry: a function: f(a+x)=f(b-x) holds, f(x) about the straight line x=(a+b) 2 symmetry.

2. F(a+x)+F(b-x)=c holds, and F(X) is symmetrical with respect to the point ((A+B) 2, c 2).

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

4. Proof: Take a point (m,n) on the function, and prove that the point after the symmetrical transformation is still on the function.

5. As the central symmetry formula proves: take a point (m,n) on the function, and the symmetry point is (a+b-m,c-n).

6. 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 symmetry point is also in the function.

Symmetry point coordinate formula: When the line is perpendicular to the x-axis, byAxisymmetryThe property shows that the midpoint of y=b,aa' is on the line x=k, (a+x) 2=k, x=2k-a, so it is easy to find the coordinates of a' (2k-a,b) and so on.

Let the coordinates of the symmetry point a be (a,b). According to the symmetry points a(a,b) and the known points b(c,d), the coordinates of the midpoint between a and b can be represented as ((a+c) 2,(b+d) 2), and this midpoint is on a known straight line. Substituting this point coordinate into a known linear equation.

A binary linear equation with respect to a and b can be obtained.

Because points a and b are symmetrical with respect to a known line, the line ab is perpendicular to the known line. And because of the slope of the straight line where two perpendicular axes intersect.

The product is -1, i.e., k1*k2=-1.

Xianglu Wuguan information:

When a straight line is a general straight line, its general form can be expressed as y=kx+b.

Because the two points A and B are symmetrical with respect to the known straight line, the straight line ab and the known straight line are suspicious of each other. And because the product of the slopes of two perpendicular intersecting lines is -1, that is, k1*k2=-1.

Set the coordinates of the point (a, b), according to the set point (a, b) and the known point (c, d), you can represent the symmetry point of the banquet line mark (a+c 2, b+d 2), and this symmetry point is in a straight line, so substituting this point into a straight line, you can find a, b, that is, the coordinates of the point sought.

The general formula of the straight line is y=kx+b, where k is the slope, so the slope of the straight line y=-x+1 is -1, and the straight line formed by two points of symmetry about the straight line is perpendicular to each other. Since the product of the slopes of two straight lines perpendicular to each other is -1, the slope of ab is -1 -1=1.

Set the coordinates of the point (a, b), according to the set point (a, b) and the known point (c, d), you can represent the symmetry point of the banquet line mark (a+c 2, b+d 2), and this symmetry point is in a straight line, so substituting this point into a straight line, you can find a, b, that is, the coordinates of the point sought.

The general formula of the straight line is y=kx+b, where k is the slope, so the slope of the straight line y=-x+1 is -1, and the straight line formed by two points of symmetry about the straight line is perpendicular to each other. Since the product of the slopes of two straight lines perpendicular to each other is -1, the slope of ab is -1 -1=1.

The symmetry point and axis of symmetry of the angular function are the test points of the college entrance examination, and many candidates always find it difficult to start with such problems.

The center of symmetry is solved so that the value of the point function is zero. There are many ways to find the axis of symmetry, you can draw a diagram, and you can also find it by the symmetry point.

y=sinx The axis of symmetry is x=k + 2 (k is an integer) and the center of symmetry is (k, 0) (k is an integer).

y=cosxThe axis of symmetry is x=k (k is an integer) and the center of symmetry is (k + 2,0) (k is an integer).

y=tanx The center of symmetry is (k, 0) (k is an integer), and there is no axis of symmetry.

It's something to remember.

For the sinusoidal function y=asin( x+ ) let x+ k + 2 solve x to find the key book and find the axis of symmetry, so that x+ k solves x is the abscissa of the center of symmetry, and the ordinate is 0(If the function is of the form y=asin( x+ )k, then the ordinate here is k).

Cosine, tangent function is similar.

1. Set the coordinates of the point a(a,b) bridge blind, according to the set point a(a,b) and the known point b(c,d), the coordinates of the symmetry point c(a+c 2, b+d 2) can be expressed, and this symmetry point is in a straight line. So substituting this point into a straight line, this is a formula.

Then according to the fact that the straight line composed of the point ab is perpendicular to the known straight line, the product of the slopes of the two straight lines is listed as -1, and the second equation can be obtained.

According to these two equations, we can find a, b, that is, the coordinates of the points to be found.

2. Simultaneous binary linear equations (1) and (2) to obtain a system of binary linear equations, and to solve a and b values, that is, the coordinates (a, b) of the symmetry point a to be found.

For example, if we know that the coordinates of the point b are (-2,1), find the coordinates of the symmetry point of the straight line y=-x+1?

Let the coordinates of the symmetry point a be (a,b), then the coordinates of the midpoint c of a and point b(-2,1) are ((a-2) 2,(b+1) 2), and c is on the line y=-x+1. Substituting the coordinates of point c into the equation of a known straight line, b+1 2=-(a-2 2)+1, obtains: a+b=3 (1).

Because the two points a and b are symmetrical with respect to the known straight line y=-x+1, the straight line ab is perpendicular to the known straight line. And because the slope of the straight line is known to be -1, the slope of the straight line ab is 1

ab slope: b-1 a+2=1 (2).

Simultaneous equations (1) and (2), solving the binary system of linear equations yields: a=0, b=3

So the coordinates of the point are (0,3).

1. When the straight line is perpendicular to the x-axis.

From the properties of axisymmetric symmetry, the midpoint of y=b,aa' is on the line x=k, then,a+x) 2=k,x=2k-a

So it is easy to find the coordinates of a' (2k-a,b).

2. When the straight line is perpendicular to the y-axis.

From the properties of axisymmetry, it can be seen that the midpoint of x=a, bb' is on the line y=k, then, y+b) 2=k, y=2k-b

So it's easy to find the coordinates of b' (a,2k-b).

3. When the straight line is a general straight line, that is, its general form can be expressed as y=kx+b, which is transformed into the form of a straight line ax+by+c=0.

a, b) The coordinates of the symmetry point with respect to the line ax+by+c=0 are .

How do you find the symmetry point of the point about the point? Many people are not very clear about how to find the symmetry point of a point, so I will introduce how to find the symmetry point of a point.

1 on a number line.

, the points a, b, c, b on the number line are the midpoints of a, c, a, c represent the number a, c, find the number represented by b.

Solution 2: Because b is the midpoint of AC.

So ab = bc

i.e. x-a=c-x

x=(a+c)/2

3. The core of this principle is that the distance from two points to the midpoint of the number axis is equal, and by extension, to the coordinate axis.

is that the distance between the two points and the symmetry point is equal, and then it is decomposed into the horizontal and vertical coordinates to the symmetry point and the horizontal and vertical coordinates are equally distanced.

About axis symmetry.

1. Coordinate system in the plane at right angles.

, the two points are symmetrical with respect to the x-axis, and the abscissa is equal. From the above principle, it can be seen that half of the sum of the ordinates of two points is equal to 0, so the ordinates are opposite to each other. That is, the two points are symmetrical with respect to the x-axis, the abscissa is equal, and the ordinate is opposite to each other.

2 In a planar Cartesian coordinate system, the two points are symmetrical with respect to the y-axis and the ordinates are equal. From the above principle, it can be seen that half of the sum of the abscissa of two points is equal to 0, so the abscissa is the opposite of each other. That is, the two points are symmetrical with respect to the y-axis, the abscissa is opposite to each other, and the ordinate is equal.

Point-to-point symmetry.

1 About origin symmetry.

In the planar Cartesian coordinate system, it is known that a,b is symmetrical with respect to the origin, a(x1,y1)b(x2,y2). From the above principle, it can be concluded that x1 = -x2 and y1 = -y2.

2 Regarding the symmetry of any point, in the plane Cartesian coordinate system, it is known that c is the symmetry point of a, b, a(x1, y1)b(x2, y2), find the coordinates of point c. Let the coordinates of point c be (a,b), then a=(x1+x2) 2,b=(y1+y2) 2

Solution: Make the points a, b, c, and d respectively with respect to the origin o into central symmetry.

point A 39; 、b #39;、c #39;、d #39;As follows: point A 39; (-3,0),b #39;(0,2),c #39;(2,-3),d #39;(3,-2).It is found that the coordinate relationship between the two points of symmetry of the original brigade concession point into the Middle Ages is:

Both the abscissa and the ordinate are opposites of each other.