Given that the parabola passes through points a (1,0), B (0, - 3), and the axis of symmetry is x = 2, find the analytic expression of the function

Given that the parabola passes through points a (1,0), B (0, - 3), and the axis of symmetry is x = 2, find the analytic expression of the function

y=-x²+4x-3

The length of the line segment cut by the parabola on the x-axis is 2 units, and the maximum value is 3 when x = - 1. The analytical formula of the parabola is obtained

This problem uses the properties of parabola, the parabola in the problem is y = ax ^ 2 + BX + C function
When x = - 1, we get the maximum value, so x = - 1 is the symmetry axis of the parabola
The intersection of parabola and coordinate axis is symmetric about x = - 1, and the length between two points is 2,
That is, (x1 + x2) / 2 = - 1 | x1-x2 | = 2
So one of the solutions is - 2 and the other is 0
Let y = a (x - (- 2)) * (x-0) = ax (x + 2)
Take (- 1,3) into this point and get a = - 3
So the analytic formula of parabola is y = - 3x ^ 2-6x
Hope to help you

It is known that the line y = - x + 3 intersects the X axis at point B, and intersects the Y axis at point C. the parabola y = ax & # 178; + BX + 3 passes through points a, B and C, and the coordinates of point a are (- 1,0)
(1) Finding the function expression of parabola
(2) As shown in Figure 1, the vertex D of the square defg with side length 2 coincides with point B, G is on the x-axis (and on the right side of point d), e and F are in the first quadrant, and the square defg moves to the left along the x-axis at a speed of one unit per second. In the process of motion, let the area product of the overlapping part of the square defg and △ OBC be s, and the motion time be T seconds (0 < T ≤ 3), and find the functional relationship between S and t time
(3) As shown in Figure 2, point P (1, K) is on the straight line BC, point m is on the x-axis, and point n is on the parabola. Is there a parallelogram with AMNP as the vertex? If so, please write the coordinates of point m directly. If not, explain the reason

【1】 By introducing y = 0 into y = - x + 3, we get
x＝3
So B [0,3]
Taking a 〔 - 1,0 〕 and B 〔 3,0 〕 into y = ax square + BX + 3
a＝－1,b＝2
So the expression of parabola is y = - x square + 2x + 3
【2】 S = 1 / 2T square [0 < T ≤ 3]
【3】 [radical 2,0] or [negative radical 2,0]