As shown in the figure, it is known that the image of quadratic function y = AX2 + BX + C (a, B, C are real numbers, a ≠ 0) passes through point C (T, 2) and intersects with X axis at two points a and B. if AC ⊥ BC, then the value of a is______ ．

As shown in the figure, it is known that the image of quadratic function y = AX2 + BX + C (a, B, C are real numbers, a ≠ 0) passes through point C (T, 2) and intersects with X axis at two points a and B. if AC ⊥ BC, then the value of a is______ ．

Let the image of a (x1, 0), B (X2, 0) ∵ quadratic function y = AX2 + BX + C pass through the point C (T, 2), ∵ at2 + Bt + C = 2 ∵ AC ⊥ BC, ∵ Ca · CB = (x1-t, - 2) · (x2-t, - 2) = 0 ∵ x1x2 − t (x1 + x2) + T2 + 4 = 0 ∵ Ca + BTA + T2 + 4 = 0, that is, at2 + Bt + C + 4A = 0 ∵ 4A + 2 = 0 ∵ a = − 12

It is known that the parabola y = a (x + m) ^ 2 + K and the parabola y = (x + 1) ^ 2 + 3 have the same vertex
It is known that parabola y = a (x + m) ^ 2 + K and parabola y = (x + 1) ^ 2 + 3 have the same vertex and pass through point a (0,1)
(1) The analytic expression of the quadratic function and the point P coordinate are obtained
(2) Find the coordinates of point a (0,1) about the axis of symmetry and the area of triangle APB

1. Y = a (x + m) ^ 2 + K has the same vertex as the parabola y = (x + 1) ^ 2 + 3
So, k = 3, M = 1
Y = a (x + m) ^ 2 + K passing through point a (0,1)
So a = - 2
What is point P? Vertex? Focus?

The parabola y = 3ax & sup2; + 2bx = C is known
Given the parabola y = 3ax & sup2; + 2bx = C, if a = b = 1, and when - 1 < x < 1, the parabola and X-axis have and only have one common point, respectively, find the value of Y (which can be expressed by C) when x = - 1 and X = 1, find the symmetry axis of the parabola, draw a sketch (which can not be drawn), and write the value of C (mainly this question)

1. When a = b = 1, the parabola is y = 3x & sup2; + 2x + C
When x = 1, y = 5 + C
When x = - 1, y = 1 + C
2.y=3(x+1/3)²+c-1/3
The axis of symmetry of the parabola is x = - 1 / 3
3.△=2²-4×3×c=4-12c>0 ∴c

Mathematics problem: the vertex of parabola y = ax ^ 2 + BX + C satisfies the following three conditions: 1. In the third quadrant. 2. On the straight line y = x
The vertex of the parabola y = ax ^ 2 + BX + C satisfies the following three conditions: 1. In the third quadrant. 2. On the straight line y = x, 3. The distance to the origin is 2 root sign 2, and the length of the line segment cut by the X axis of the parabola is 8

1. In the third quadrant: explain the coordinates (x, y), X of the vertex