The undetermined coefficient method of quadratic function is used to find the analytic expression of quadratic function 1. (1,2), (3,0), (- 2,20) three-point coordinates, find the analytic expression of quadratic function

The undetermined coefficient method of quadratic function is used to find the analytic expression of quadratic function 1. (1,2), (3,0), (- 2,20) three-point coordinates, find the analytic expression of quadratic function

Let y = AX2 + BX + C
Bring in three-point coordinates
2=a+b+c
0=9a+3b+c
20=4a-2b+c
A = 1.5, B = - 7, C = 7.5
So: y = 1.5x-7x + 7.5

Analytic formula of quadratic function by undetermined coefficient method
Coordinates (- 1, - 1) (0, - 2) (1,1)

Let y = ax ^ 2 + BX + C (a is not equal to 0),
Take (- 1, - 1) (0, - 2) (1,1) into the
a*（-1）^2+b*(-1)+c=-1…… ①
a*0^2+b*0+c=-2……………… ②
a*1^2+b*1+c=1…………… … ③
By solving the equations composed of (1), (2) and (3)
a=2,
b=1,
c=-2
Then, the analytic expression of quadratic function is as follows:
y=2x^2+x-2
Comments: 1. It is a common method to solve the quadratic function analytic formula by setting the quadratic function analytic formula as "general form", but it should be noted that (a is not equal to 0), and then solve the equations to get a, B, C
2. According to different types of questions, you can also find the analytic form according to the "vertex form". Please find some questions in this field

Using undetermined coefficient method to find the analytic expression of quadratic function

The steps of solving quadratic function by undetermined coefficient method are as follows
(1) Let's suppose the analytic expression of quadratic function; (2) according to the known conditions, we get the equations of undetermined coefficients; (3) solve the equations, find out the value of undetermined coefficients, and then write out the analytic expression of function
2、 The common forms of quadratic function analytic expression are as follows:
1. General formula: given the values of three points or three pairs of, on a parabola, we usually choose the general formula
2. Vertex form: when the vertex or axis of symmetry of a parabola is known, the vertex form is usually chosen
3. Intersection type: given the abscissa of the intersection of parabola and axis, the intersection type is usually selected

Quadratic function problem, solved by undetermined coefficient method
(1) Given that the image of a quadratic function passes through a point (0,1), its vertex coordinates are (4,9), find the analytic expression of this function
(2) Given that the image of a quadratic function passes through points (- 1,0), (2,0) and (0,2), the analytic expression of the function is obtained

(1) Given that the image of a quadratic function passes through a point (0,1), its vertex coordinates are (4,9), find the analytic expression of this function
Let the analytic formula be y = a (x-4) ² + 9
Substituting (0,1) into
16a+9=1
a=-1/2
So: the analytic formula is: y = (- 1 / 2) (x-4) &# 178; + 9
y=-x²/2+4x+1
(2) Given that the image of a quadratic function passes through points (- 1,0), (2,0) and (0,2), the analytic expression of the function is obtained
Two points (- 1,0), (2,0) passing through X-axis
Let the analytic formula be y = a (x + 1) (X-2)
Substituting (0,2) into
-2a=2
a=-1
The analytical formula is: y = - (x + 1) (X-2)
y=-x²+x+2