What are the basic knowledge of quadratic functions in junior high school, and what are the knowledg

The parabola is axisymmetric and the axis of symmetry is a straight line x = -b 2a The only intersection point between the axis of symmetry and the parabola is the vertex p of the parabola. , when b=0, the axis of symmetry of the parabola is the y-axis;

The parabola has a vertex p with coordinates p ( b 2a , 4ac-b 2 4a ).

When -b 2a=0, p is on the y-axis; When δ = b2; -4ac=0, p is on the x-axis.

The quadratic term coefficient a determines the direction and magnitude of the opening of the parabola.

When a 0, the parabola opens upwards; When a 0, the parabola opens downwards.

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

It's complete, I think it's okay to remember a part, just use the abc value in the question.

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

When a>0, the quadratic function image opens upward;

When a, the parabola opens downwards.

a|The larger it is, the smaller the opening of the quadratic function image.

Determine the location factor.

The primary coefficient b and the quadratic coefficient a together determine the position of the axis of symmetry.

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 the left is the same as the 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 significance: the value of the slope k of the analytical formula (primary function) of the tangent of the quadratic function image at the intersection of the quadratic function image with the y-axis. It can be obtained by finding a derivative of the quadratic function.

Vertex coordinates, opening direction, axis of symmetry, increment and decrease of functions, maximum and minimum values Translation The practice of parabola Properties of quadratic functions.

In quadratic functions, x and y are both variables and constants, the value range of the independent variable x is all real numbers, and b and c can be any real numbers, but not 0 real numbers;

If , then becomes, at that time, is a one-time function; At that time, it was a constant function;

To determine whether a function is quadratic, three conditions must be met:

The functional relation must be an integer;

The maximum number of times the independent variable must be reduced to 2;

The coefficient of the quadratic term must not be 0;

That's all there is to it.

Quadratic Function Definition and Definition Expression In general, there is a relationship between the independent variable x and the dependent variable y: y=ax 2+bx+c

a, b, c are constants, a≠0, and a determines the opening direction of the function, when a>0, the opening stool is up, when a<0, the opening direction is down, iai can also determine the size of the opening, the larger the iai, the smaller the opening, the larger the opening. ) is called a quadratic function of x.

Quadratic Functions The right side of an expression is usually a quadratic trinomial.

Quadratic Functions and Unary Quadratic Equations In particular, the quadratic function (hereinafter referred to as the function) y=ax 2+bx+c, when y=0, the quadratic function is a quadratic equation about x (hereinafter referred to as the equation), that is, ax 2+bx+c=0

In this case, whether the function image intersects with the x-axis or not, that is, whether the equation has a real number root or not. The abscissa of the intersection of the function and the x-axis is the root of the equation.

A method for high-decomposition problems of quadratic functions in junior high schoolComplement and cut the form

The main point of such methods is to make the area of the desired figure appropriately supplemented or cut into a figure that is conducive to representing the area.

Rotate

It mainly refers to the image transformation with the vertex of the quadratic function image imitation wheel as the rotation center and the rotation angle of 180°, this kind of rotation will not change the image shape of the quadratic function, and the opening direction is opposite, so the value of a will be the original opposite number, but the vertex coordinates are unchanged, so it is easy to find its analytical formula.

Axisymmetry

This graphical transformation consists of two ways: x-axis symmetry and symmetry with respect to the y-axis.

Quadratic function image An image that is symmetrical with respect to the x-axis, its shape does not change, but the opening is in opposite directions, so the value of a is the opposite of the original. The analytic formula can be determined by finding the new vertex coordinates according to the coordinate characteristics of the vertex symmetry point with respect to the x-axis.

Quadratic function image with respect to y-axis symmetry whose shape and opening direction are unchanged and therefore the value of a is unchanged. However, the vertex position changes, and the analytic formula can be determined by finding the new vertex coordinates based on the coordinate features of the point with respect to the y-axis.

1.The general form of a quadratic function: y=ax2+bx+c(a0)

2.A few concepts about quadratic functions: the image of quadratic functions is a parabola, so it is also called a parabola y=ax2+bx+c; The parabola is symmetrical with respect to the axis of symmetry and bounded by the axis of symmetry, with half of the image uphill and the other half downhill; where c is called the intercept of the quadratic function on the y-axis, that is, the image of the quadratic function must pass through the point (0,c).

3.y=ax20): when y=ax2+bx+c (a0). b=0 and c=0 when the quadratic function is y=ax20);

This quadratic function is a special quadratic function with the following properties:

1) The image is symmetrical with respect to the y-axis; (2) vertice(0,0);

Hall Clever Argument 4Finding the analytic formula of the quadratic function: knowing that the three points on the image of the quadratic function are missing, you can set the analytic formula y=ax2+bx+c, and substitute the coordinates of these three points to solve the ternary linear equation system about a, b, and c, and find the values of a, b, and c, so as to find the analytic formula --- the undetermined coefficient method.

5.The vertex formula of the quadratic function: y=a(x-h)2+k(a The vertex coordinates of the quadratic function (h, k) can be directly obtained from the vertex formula, the axis of symmetry equation x=h and the maximum value of the function ymax = k

6.Finding the analytic formula of the quadratic function: If you know the coordinates of the vertices of the quadratic function (h, k) and the coordinates of another point on the image, you can set the analytic formula as y=a(x -h)2+ k, and then substitute the coordinates of another point to find a, so as to find the analytic formula.

7.Parallel movement of quadratic function images: Quadratic functions should generally be converted into vertices first, and then the parallel movement of images can be judged; When the image of y=a(x-h)2+k moves in parallel, the value of h, k changes, and the value of a does not change, the specific law is as follows:

K value increases = image pans upward;

The k value decreases and the image pans downward;

x-h) value increases = the image is shifted to the left;

x-h) value decreases and the image is panned to the right.

8.The image of the quadratic function y=ax2+bx+c (a0) and the formula for several important points: broad talk.

9.In the quadratic function y=ax2+bx+c(a0), the relationship between the signs and images of a, b, and c:

1) a=parabolic opening upward; 0 parabolic openings downward;

2) c = parabola passing above the origin; c=0 The parabola passes through the origin;

c = parabola passing below the origin;

3) a, b heterogeneous sign = axis of symmetry on the right side of the y axis; a, b with the same sign = axis of symmetry to the left of the y-axis;

b = 0 axis of symmetry is the y axis;

4) B2-4AC=The parabola has two intersections with the x-axis;

b2-4ac =0=The parabola has an intersection with the x-axis (i.e., tangent);

b2-4ac=no intersection between the parabola and the x-axis.

10.Symmetry of the quadratic function image: Knowing the points and symmetry axes on the quadratic function image, the symmetry of the image can be used to find the symmetry of the known point, and this symmetry point must also be on the image.

1. Definition and knowledge points of quadratic functions: functions of the form y=ax 2+bx+c(a≠0, where a, b, c are constants) are quadratic functions.

1), a determines the direction and shape of the opening of the parabola, when a 0, the opening is upward, when a 0 the opening is downward; The higher the value of a, the smaller the opening; When b=0, the axisymmetric parabola is y-axis; When c=0, the parabola passes through the origin; When b and c are 0 at the same time, their vertices are the origins.

2) The coordinates of the intersection of the parabola y=ax2+bx+c(a≠0) and the y-axis are (0,c); The way to find the coordinates of the two intersections of the x-axis is to make y=0, and then solve the equation for ax2+bx+c=0, and the solution of x is the abscissa of the intersection with the x-axis.

2. The analytical formula of the quadratic function y=ax 2+bx+c(a≠0) is found with respect to the x-axis, y-axis, or vertex symmetry.

1) The new analytic expression for x-axis symmetry with the quadratic function y=ax 2+bx+c(a≠0) is y=-ax 2-bx-c, i.e., a, c, and b all become opposites.

2) The new analytic formula for y-axis symmetry is y=ax 2-bx+c, i.e. a and c do not change, and b becomes inverse. That is, a and c do not change, and b becomes the opposite.

Quadratic functions throw the Sun Fog line, and image symmetry is the key;

openings, vertices, and intersections, which determine the pictorial limits;

The opening and size are broken by A, C and Y axis meet, B has a special symbol, and the symbol is associated with A;

The vertex position is found first, the y-axis is used as a reference line, and the left and right are 0, keep in mind that there is no confusion in mind;

The vertex coordinates are the most important, and the general formula is present, the horizontal mark is the axis of symmetry, and the vertical standard function is the most valuable.

If the position of the axis of symmetry is found, the symbols are inverted, general, vertice, and intersection types, and different expressions can be interchanged.

1) General formula: y ax2+bx+c (a, b, c are constants, a≠0).

2) Vertex formula: y a(x-h)2+k (a, h, k are constants, a≠0)

3) Two roots: y a(x-x1)(x-x2), where x1, x2 are the abscissa of the intersection of the parabola and the x-axis, that is, the two roots of the unary quadratic equation ax2+bx+c 0, a≠0

Quadratic function definition.

Definition: In general, there is the following relationship between the independent variable x and the dependent variable y: y = ax 2 + bx + c (a, b, c are constants, a≠0,), and y is called a quadratic function of x.

Three expressions for quadratic functions.

General formula: y=ax 2+bx+c(a,b,c is constant, a≠0);

Vertex formula: y=a(x-h) 2+k(vertex p(h,k));

Images and properties of quadratic functions.

The image of 1 quadratic function is a parabola.

2 The parabola is axisymmetric graphic. The axis of symmetry is the straight line x=-b 2a.

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

3. The quadratic term coefficient a determines the opening direction of the parabola.

When a 0, the parabola opens upwards;

When a 0, the parabola opens downwards.

4. 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.

5. The number of points where the parabola intersects with the x-axis.

b 2-4ac>0, the parabola has 2 intersections with the x-axis;

b 2-4ac=0, the parabola has 1 intersection point with the x-axis;

b 2-4ac<0, the parabola has no intersection with the x-axis.

The property of a quadratic function parabola.

1.A parabola is an axisymmetric figure. The axis of symmetry is the straight line x=-b 2a.

The only intersection point between the axis of symmetry and the parabola is the vertex p of the parabola. In particular, when b = 0, the axis of symmetry of the parabola is the y axis (i.e., the straight line x = 0).

2.The parabola has a vertex p with the following coordinates: p ( b 2a ,(4ac-b 2) 4a ) p is on the y axis when -b 2a = 0; When δ = b 2-4ac = 0, p is on the x-axis.

3.The quadratic term coefficient a determines the direction and magnitude of the opening of the parabola.

When a 0, the parabola opens upwards; When a 0, the parabola opens downwards. a|The larger it is, the smaller the opening of the parabola;

4.The first or row state sub-coefficient b and quadratic coefficient a together 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 band trace y axis.

5.The constant term c determines the intersection of the parabola and the y-axis.

The parabola intersects with the y-axis at (0,c).

The above is the common knowledge points of quadratic functions in junior high school mathematics that I have compiled for you.