Quadratic function is an important content in junior high school mathematics knowledge system, but it can only be regarded as important basic knowledge in senior high school mathematics, thus playing the role of "connecting the preceding with the following", so it is very important to learn the relevant knowledge of quadratic function well.

Our common quadratic resolution functions are mainly divided into: general type; Vertex type; Intersection point (double root type); For some special cases, the other three forms of quadratic function can be used: symmetry method; Undetermined coefficient method; Translation method; Determine the analytical formula of quadratic function more quickly; Therefore, the first three common quadratic resolution functions need to be kept in mind, while the last three focus on methods and need to be used flexibly.

General formula: y=ax+bx+c(a≠0)

If the coordinates of three points on the quadratic function image (or three pairs of corresponding values of the function) (x, y), (x, y) and (x, y) are known, then we can directly determine the values of a, b and c with the help of equations, thus obtaining the resolution function y=ax+bxc(a≠0).

Summary:

Any quadratic function can be arranged in the form of the general formula y=ax+bx+c(a≠0). Given the coordinates of any three points, the quadratic resolution function can be solved by the simultaneous equation of the general formula.

note:

The analytic formula of any quadratic function can be transformed into a general formula or a vertex, but not all quadratic functions can be written as intersections. Only when the parabola intersects the axis, that is, b-4ac≥0, can the analytical expression of the parabola be expressed as the intersection point. At the same time, it should be noted that the three basic forms of any secondary resolution function are interchangeable.

Determination of analytical formula by translation method

Translation after conversion to vertex:

Firstly, the quadratic function is transformed into the form of y = a (x-h)+k (a ≠ 0) by collocation method, and then the new vertex coordinates are determined by the translation of vertices, so that a new resolution function can be written. Finally, on the basis of the original function, follow the law of "adding left and subtracting right, adding up and subtracting down" and translate the solution.

General formula for direct translation:

For the translation of any quadratic function y=ax+bx+c(a≠0), you can also directly use "left plus right minus, up plus down minus" to translate.