How to determine the axis of symmetry of a quadratic function

The axes of symmetry are all y-axis, vertex coordinates.

All are (0,0), opening, the first one facing up, the second three facing down, <

Let the quadratic function be.

The analytic formula is y=ax 2+bx+c

Then the symmetry axis of the quadratic function is the straight line x=-b 2a, the vertex abscissa is -b 2a, and the vertex ordinate is (4ac-b 2) 4a<

The image is substituted by the origin (0,0) into the function y=ax 2+2x+a-4a 2

0=a-4a^2

a=1 4 or 0 (round).

y=1/4x^2+2x=1/4(x+4)^2-4

Axis of symmetry: x=-4

The opening < upwards

y=ax2+2ax-3a<

OK. A quadratic function is essentially a parabola.

, we write the quadratic function as vertices.

y=k(x-x0) +h(k≠0), then it is a parabola with a vertex of (x0,h) and a focal length of k2. Parabolas can also have other forms, which will be discussed in analytic geometry later.

The problem you are talking about is actually a problem of coordinate rotation, you assume that the coordinates do not move, and the parabola rotates at a certain angle, which is equivalent to the parabola not moving, and the coordinate axis rotates.

Let the rotation angle be (positive counterclockwise, clockwise.

is negative), the rotation center is the coordinate origin, and the coordinate system after rotation.

x'o'y'The relationship between the coordinates of and the original coordinates xoy is.

x=x'cosθ-y'sinθ①

y=x'sinθ+y'cosθ②

Equivalently, there is.

x'=xcosθ+ysinθ③

y'=-xsinθ+ycosθ④

For example, if y=x 2 is x=0, to make y= 3x, tg = 3, = 3

Substituting the formula yields -( 3 2)x+(1 2)y=[(1 2)x+( 3 2)y] 2

The calculated equation is x 2 + 3y 2 + (2 3) xy + (2 3) x-2y = 0. It's complicated, isn't it?

It's over, it's time to throw bricks and stones! <

b/2a<

b/2a,(4ac-b*b)/4a)<

Recipe pushed out:

y=ax^2+bx+c=a[x^2+bx/a+c/a]=

a(x+b/2a)^2+(4ac-b^2)/4a

The axis of symmetry x=-b 2a<

Let the analytic expression of the quadratic function be y=ax 2+bx+c

Then the symmetry axis of the quadratic function is the straight line x=-b 2a, the vertex abscissa is -b 2a, and the vertex ordinate is (4ac-b 2) 4a<

The vertex formula is (-b 2a, (4ac-b 2) 4a).

Intersection barrier: y=a(x-x) (x-x) is limited to parabolas that have intersections a(x,0) and b(x,0) with the x-axis.

where x1,2 = -b b 2 4ac

Vertex style. y=a(x-h)^2+k

The apex of the parabola p(h,k)].

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

Note: In the three forms of mutual transformation, there are the following relationships:

H=-b 2A= (x +x) 2 k=(4ac-b 2) 4a intersection with x-axis: x, x =(b b 2-4ac) 2a

Determine the location factors of the nucleus.

The primary term coefficient b and the quadratic term coefficient.

a. Jointly determine the axis of symmetry.

location. When a>0 and b have the same sign (i.e., ab>0), the axis of symmetry is left on the y-axis; Since the axis of symmetry is on the left, the axis of symmetry is less than 0, i.e. - b 2a.

When a>0 is different from b (i.e., ab0), the axis of symmetry is to the right of the y-axis. Because the axis of symmetry is on the right, the axis of symmetry should be greater than 0, that is, - b 2a>0, so b 2a should be less than 0, so a and b should have different signs.

It can be simply remembered as left and right, that is, when the axis of symmetry is on the left side of the y-axis, a and b have the same sign (i.e., a>0, b>0 or a).

In fact, b has its own geometric meaning: a quadratic function.

This quadratic function image tangent at the intersection of the image and the y-axis.

(a one-time function.

The value of the slope k. It can be obtained by finding a derivative of the quadratic function.

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 parabola.

The direction and size of the opening. 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 it is, 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 pass the general formula of the quadratic function.

y=ax^2+bx+c

After determining the values of a, b, and c, respectively, and determining the values of a, b, and c, the formula for the axis of symmetry can be obtained.

x=-b/2a

4. Determine the vertex formula of the quadratic function.

If it's vertice.

y=a(x-h)^2+k

Then the axis of symmetry of the vertex formula of the quadratic function is:

x=h。Extended Materials.

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).

Vertex coordinates. is (h,k), and intersects (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, which is an increment function in the range of xh.

i.e., y becomes larger as x gets bigger), the opening of the quadratic function image is upward, and the value range of the function.

It's y>k

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

k, which is a subtraction function in the range of xh.

i.e., y becomes smaller as x increases), the opening of the quadratic function image is downward, and the value range of the function is the y-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）

By converting y=ax 2+bx+c to 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≠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.

As follows:

x=-2 b/a is the vertex coordinate formula in a quadratic function, a, b, c are constants, a≠0, and a determines the opening direction of the function. a>0, the opening direction is upward; a<0, the opening direction is downward. The absolute value of a determines the size of the opening.

The larger the absolute value of a, the smaller the opening, and the smaller the absolute value of a, the larger the opening.

2 b/a is the axis of symmetry of the parabola of the quadratic function. The quadratic term coefficient a determines the direction and magnitude of the opening of the parabola. When a>0, the parabola opening is upward; When a<0, the parabola opening is downward

a|The larger it is, the smaller the opening of the toss line;a|The smaller it is, the larger the opening of the parabola.

Quadratic functions. The quadratic function image is an axisymmetric graph, and the only intersection point between the symmetry axis and the quadratic function image is the vertex of the quadratic function image, b with the same sign, and the symmetry axis is on the left side of the y-axis. a, b different signs, the axis of symmetry is on the right side of the y axis. A quadratic function image has a vertex p with coordinates p(h,k).

The quadratic term coefficient a determines the direction and size of the opening of the quadratic function image.

The basic representation of the quadratic function is y=ax +bx+c(a≠0). The highest order must be quadratic and the image of the quadratic function is a parabola with the axis of symmetry parallel to or coincided with the y-axis.

b 2a is the axis of symmetry of a quadratic function.

ax²+bx+c=y

x²+(b/a)x+c/a=y

x²+2×[b/(2a)]x+c/a=y

x²+2×[b/(2a)]x+[b/(2a)]²b/(2a)]²c/a=y

x+b (2a)] b (2a) +4ac (2a) =y to get the axis of symmetry x=-b 2a.

The only intersection point between the axis of symmetry and the quadratic function image is the vertex p of the quadratic function image.

In particular, when b = 0, the axis of symmetry of the quadratic function image is the y-axis (i.e., the straight line x = 0).

A and B are trapped in high wheels with the same number, and the axis of symmetry is on the left side of the y-axis;

a, b different signs, the axis of symmetry is on the right side of the y axis.

It can be judged by vertex formula.

It is enough to generalize the quadratic function into a vertex type cherry fiber. It's not hard either.

It may be that for junior high school students, there is a need for more spine calculations.

The quadratic function ABC10 formulas are as follows:

a>0, the parabola opening is upward; a<0, the parabolic opening is downward. When the parabolic axis of symmetry is on the left side of the y-axis, a,b have the same sign, and when the parabolic axis of symmetry is on the right side of the y-axis, a,b has different signs. c>0, the intersection of the parabola and the y-axis is above the x-axis; At c<0, the intersection of the parabola and the y-axis is below the x-axis.

When a=0, this image is a one-time function. When b=0, the parabolic vertex is on the y-axis. When c=0, the parabola is on the x-axis.

When the parabolic symmetry is on the left side of the y-axis, a,b have the same sign, and when the parabolic symmetry axis is on the right side of the y-axis, a,b have different signs.

The basic representation of a quadratic function is y=ax +bx+c, a≠0. The quadratic function must be quadratic at its highest order, and the image of the quadratic function is a parabola whose axis of symmetry is parallel to or coincides with the y-axis.

The expression of a quadratic function is y=ax +bx+c and a≠0, and it is defined as a quadratic polynomial. If the value of y is equal to zero, we can get an old equation for the height of the quadratic foci. The solution of this equation is called the root of the equation or the zero point of the function.

Quadratic function properties:

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 smaller it is, the larger the opening of the parabola. a|The larger it is, the smaller 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 have different signs (i.e., AB<0) (which can be coincidentally noted: left is the same as right), the axis of symmetry is on the right side of the Y axis.

3. The constant term c determines the intersection of the parabola and the y-axis. The parabola intersects with the y-axis at (0, c).