How to judge the axis of symmetry, center of symmetry, period by expression?

1. Axis of symmetry.

Basic expression: f(x)=f(-x) is an even function with symmetry at the origin.

The variations are: 1) f(a+x)=f(a-x).

2）f（x）=f（a-x）

3）f（-x）=f（b+x）

4）f（a+x）=f（b-x）

2. The basic expression of the center of symmetry: f(x)+f(-x)=0 is the symmetry of the center of the origin.

odd functions. 3. Periodic function.

The basic expression: f(x)=f(x+t) variation is: f(x+a)=f(x+b).

The sum in parentheses is symmetrical for the fixed value, and the difference for the fixed value is the period.

Axis of symmetry. Basic expression: f(x)=f(-x) is an even function with symmetry at the origin.

The variation is: f(a+x)=f(a-x).

f（x）=f（a-x）

f（-x）=f（b+x）

f（a+x）=f（b-x）

In this way, there is an axis of symmetry for a land banquet like x and -x.

2.The basic expression of the center of symmetry: f(x)+f(-x)=0 is the symmetry of the center of the origin.

odd functions. The basic variation is similar to the one above. Just pay attention to the equation early cavity silver.

location. 3.Periodic function.

Basic expression: f(x) = f(x+t).

The variation is f(x+a)=f(x+b).

Pay attention to the position of symbols and equations.

4.Otherwise, that's just the basics. There are many more complex variations, but they are generally not taken in the college entrance examination, so they will not be introduced.

The above three main things are to see the structure of the basic formula, and you can roughly distinguish the variations.

For example, f(x+1)+f(x+2)=f(x+3) is a periodic function, and 3 is one of the periods.

1. The basic expression of the axis of symmetry: f(x)=f(-x) is an even function of origin symmetry. The variations are:

1）f（a+x）=f（a-x）

2）f（x）=f（a-x）

3）f（-x）=f（b+x）

4）f（a+x）=f（b-x）

2. The basic expression of the center of symmetry: f(x)+f(-x)=0 is the odd function of the symmetry of the center of the origin.

3. The basic expression of the periodic function: f(x)=f(x+t) The variation is: f(x+a)=f(x+b).

The method of judging the opening direction and magnitude, position and axis of symmetry of the quadratic function axis of symmetry is as follows:

1. The quadratic term coefficient a determines the direction and size of the opening of the parabola. When a>0, the parabola opening is upward; When a<0, the parabola opening is downwarda|The larger it is, the smaller the opening of the parabola;a|The smaller the suspicion, the larger the opening of the parabola.

2. The primary term coefficient b and the quadratic term coefficient a jointly determine the position of the axis of symmetry. When a and b have the same sign (i.e., ab>0), the axis of symmetry is to the left of the y-axis; When A and B are different (i.e., AB<0), the axis of symmetry is to the right of the Y axis. (It can be coincidentally recorded as: left and right).

3. First, determine the general formula of the quadratic function: y=ax 2+bx+c, and then divide and determine the values of a, b, and c by the numbers in the general formula y=ax 2+bx+c of the quadratic function, and after determining the values of a, b, and c, the formula for the axis of symmetry can be obtained as x=-b 2a

4. Determine the vertex formula of the quadratic function, if it is the vertex formula y=a(x-h) 2+k, then the axis of symmetry formula of the vertex formula of the quadratic function is: x=h.

The intersection factor of the symmetry axis of the quadratic function with the x,y axis:

1. The constant term c determines the intersection of the quadratic function image and the y-axis.

The quadratic function image intersects the y-axis at point (0,c).

The vertex coordinates are (h,k), which intersect (0,c) with the y-axis.

2、a<0;k>0 or a>0; At k<0, the quadratic function image has two intersections with the x-axis.

When k=0, the quadratic function image has only one intersection point with the x-axis.

a<0;When k<0 or a>0, k>0, the quadratic function image has no intersection with the x-axis.

3. When a>0, the function obtains the minimum value <> at x=h

k, in the range of xh is the increasing function (i.e., y grows larger with the increase of x), the opening of the quadratic function image is upward, and the range of the function is y>k

When a<0, the function achieves a maximum value of <> at x=h

k, in the range of xh is a subtraction function (i.e., y decreases as x increases), the opening of the quadratic function image is downward, the range of the function is y, when h=0, the parabola's pair of hidden axes is the y-axis, and the function is an even function.

For expressions of the form y=ax 2+bx+c, when a≠0, this is the expression of the quadratic function.

When y=0, ax 2+bx+c=0 If the equation has two roots x1 and x2, it can be known according to Vedder's theorem.

x1+x2=-b/a……（1）

And by making y=ax 2+bx+c into vertices, y=a【x+(b 2a)] 2+(4ac-b 2) 4a can see the axis of symmetry x=-b 2a...... of the function（2）

This is very similar to Eq. (1), but only a relationship of coefficients, 2 (-b 2a) = -b a = x1 + x2 ......（3）

This means that the sum of the two is twice the axis of symmetry.

Generally, it can also be expressed in the following forms:

1. Intersection formula: y=a(x-x1)(x-x2)(a≠0) This means that the abscissa of the intersection of the function and the x-axis is x1, x2

According to Eq. (3), it can be concluded that the axis of symmetry of this function is x=(x1+x2) 2, e.g. y=(x-2)(x-4) axis of symmetry is x=(4+2) 2=3;

2. Vertex formula: y=a(x-h) 2+k(a,h,k is constant, a≠0).

Through the vertex formula, it is very intuitive to see that the axis of symmetry of the function x=h

For example: y=6(x+3) 2+9......（4）

The axis of symmetry must not be understood as x=3, and further deformation of (4) is required

y=6【x-(-3)】 2+9, h=-3, then the axis of symmetry is x=-3

3. General formula: y=ax 2+bx+c (a, b, c are constants, a≠ call mu cover 0).

By equation (2), we can get the axis of symmetry of the function x=-b 2a. For general expressions, be sure to write the functions in a power reduction order of x, and then confirm what numbers a, b, and c refer to respectively (including the symbols before the values, which is especially important).

For example: y=3x-5x 2-9

First according to the power of x, y=-5x 2+3x-9, at this time a=-5, b=3, c=-9

So the axis of symmetry x=-b 2a = -3(-10) = 3 10

These are the common forms of quadratic functions.

In total, each form of the quadratic function can be skillfully used, and the axis of symmetry of the function should not be a big problem.

Summary. Finding the symmetry axis, the center of symmetry, and the periodic problem.

Hello, the answer to the question you asked is as above.

I'm a parent and need to check my child's homework and I want detailed steps.

Hold on. Here's an example

The axis of symmetry of sinx is 2+k (k z).

So the axis of symmetry of 1 2sinx is 4+k 2, so that x 4-3 in the curly braces of the first equation is equal to 4+k 2, and x is the axis of symmetry.

The same goes for everything else.

Center of symmetry of the SINX image and the COSX image.

This is the intersection of the image and the x-axis.

That is, let the first equation be equal to 0

It can be concluded that the center of symmetry of sinx is (k,0).

So let's make the first formula in the title.

x 4 - 3 = k in parentheses

x 4 - 3 = k in parentheses

Solve that x is the symmetry center of the function.

The axisymmetric figure is: after folding along a certain straight line, the parts on both sides of the straight line coincide with each other, and the central symmetrical figure is: the figure rotates around a certain point and the original figure after 180°, which coincides with the original figure, which is both an axisymmetric figure and a central symmetrical figure

Straight lines, line segments, two intersecting lines, rectangles, diamonds, squares, circles, etc

There are only axisymmetric figures: celery rays, angular isosceles triangles, equilateral triangles, isosceles trapezoids, etc

There are only center-symmetrical figures: parallelograms, etc

Figures that are neither axisymmetric nor centrally symmetrical have line leaders: unequal triangles, non-isosceles trapezoids, etc