Four analytical formulas of quadratic function: 1 general formula, 2 vertex type, 3 intersection type (two formulas) and 4 symmetric point type.

General formula: y=ax? +bx+c(a, B, C are constants, and A is not equal to 0). It is known that the coordinates of any three points on a parabola can be used to find the resolution function.

Top point: y=a(x-h)? +k(a≠0,a, h and k are constants). Vertex coordinates are (h,k); The symmetry axis is a straight line x=h; The position characteristics of vertices and the opening direction of images are related to the function y=ax? The images are the same. When x=h, the maximum value of y = k. Sometimes the topic will point out that you can use collocation to turn the general formula into a vertex.

Example: Given the vertex (1,2) and another arbitrary point (3, 10) of the quadratic function Y, find the analytical formula of Y.

Solution: Let y=a(x- 1)? +2, substitute (3, 10) into the above formula and get y=2(x- 1)? +2。

Intersection point (two types): [only parabola with intersection point with X axis, that is, y=0, that is, B? -4ac≥0]。

It is known that parabola and X axis, that is, y=0, have intersections A (x10) and B(x2, 0). We can set y=a(x-x 1)(x-x2), and then substitute the third point into X and Y to find A..

Symmetrical point formula: If two symmetrical points (x 1, m)(x2, m) on the quadratic function image are known, it is set as: y=a(x-x 1)(x-x2)+m (a≠0), and then the other coordinate is substituted into the formula. ?