Analytic expression of quadratic function in Mathematics 1. On the image of quadratic function y = ax ^ 2 + BX + C, there are two points whose coordinates are (3,5) and (2) (- 2,15), find the analytic expression of this function 2. On the image of quadratic function y = ax ^ 2 + BX + C, there are two points whose coordinates are (m, n) and (m, n) (P, q)

Analytic expression of quadratic function in Mathematics 1. On the image of quadratic function y = ax ^ 2 + BX + C, there are two points whose coordinates are (3,5) and (2) (- 2,15), find the analytic expression of this function 2. On the image of quadratic function y = ax ^ 2 + BX + C, there are two points whose coordinates are (m, n) and (m, n) (P, q)

1. Two coordinates are substituted into the equation to solve the equation and solve the equation: B = - 2-A; C = 11-6a;
So the analytic formula: y = ax ^ 2 - (2 + a) x + 11-6a; considering quadratic function, a is not equal to 0
2. The same method is used to solve the equation
We can get: B = [n-q-a (m ^ 2-P ^ 2)] / (M-P) m should not be equal to P, otherwise n = q is the same point
c=amp+(mq-np)/(m-p)
In fact, if you use the solution of 2, M = 3, n = 5, P = - 2, q = 15 is the solution of 1

Mathematical quadratic function, master solution. Thank you
We know the quadratic functions Y1 and Y2 of X, where the image opening of Y1 is downward, intersects with X axis at a (- 2,0) and B (4,0), the symmetry axis is parallel to y axis, the distance between its vertex and B point is 5, and y2 = - 4 / 9x ^ - 16 / 9x + 2 / 9
(1) Find the function formula of quadratic function Y1;
(2) Let y2 = a (X-H) ^ + K form;
(3) How to translate the image of Y1 to get the image of Y2

From the coordinates of point AB, the axis of symmetry is x = 1
The distance from point B is 5
Let the vertex ordinate be y
There are
（4-1）²+y²=5²
y=4
The vertex coordinates are (1,4)
Let the linear equation be y = ax & sup2; + BX + C
Then the equations can be obtained
4a-2b+c=0
16a+4b+c=0
a+b+c=4
Solution
a=-4/9
b=8/9
c=32/9
y= -4/9x²+8/9x+32/9
Second question:
y2=-4/9x²-16/9x+2/9
=-4/9（x²+4x+4）+16/9+2/9
=-4/9（x+2）²+2
The third question
The formula of Y1 was as follows:
y1= -4/9（x-1）²+4
∴x-1 +3=x+2
Left plus right minus
Move 3 units to the left
4-2=2
Top plus bottom minus
So move 2 units down
Synthesis: move the image of Y1 3 units to the left and 2 units to the down

The application of mathematical quadratic function analytic formula
How to use y = ax & # 178; + BX + C, y = (x + b) &# 178; + K, and what kind of question types are generally used

Y = AX2 + BX + C is generally used to give three known points in the problem, and then solve a, B and C with a system of linear equations of three variables (it is generally difficult to solve three variables)
Y = a (x + H) ² + K, which is generally used to give you a vertex and any point of a quadratic function, can be used
However, for any point with two intersections with the x-axis, I recommend the intersection formula
y=a（x-x1）（x-x2）
The abscissa of the two intersections of X1 and x2
Example: given the quadratic function through the image a (2,0), B (4,0) and C (3,5), find the analytic expression of the quadratic function
Let the analytic expression of the quadratic function be y = a (x-x1) (x-x2)
∵ the quadratic function image passes through a (2,0), B (4,0)
∴y=a（x-2）（x-4）
And ∵ the quadratic function image passes through C (3,5)
∴5=a（3-2）（3-4）
a=-5
The analytic expression of the quadratic function is y = - 5 (X-2) (x-4), which is full score. If you think it is unreliable, simplify it
y=-5x²+30x-40
I hope you can adopt it