已知抛物线y=ax²+bx+c的顶点为D(1,-4),与Y轴的交点纵坐标为-3,交X轴于A、B两点,点B在点A

(1)
顶点(1,-4):y = a(x - 1)² - 4
x = 0,y = a - 4 = -3,a = 1
y = (x - 1)² - 4 = x² - 2x - 3 = (x + 1)(x - 3)
(2)
y = x² - 2x - 3 = (x + 1)(x - 3)
A(-1,0),B(3,0)
BC的斜率为1
(i)BC与CP垂直
CP的斜率为-1,方程为y = -x - 3
与抛物线联立,得P(1,-4)
(ii) BC与BP垂直
BP的斜率为-1,方程为y = -x + 3
与抛物线联立,得P(-2,5)
(3)
取E(1,e),过E作x轴的平行线,与抛物线交于P,当EP=AB时,ABPE为平行四边形
AB = 4
(i) P在E右侧
P(5,e),e = 5² - 2*5 - 3 = 12
P(5,12)
(ii) P在E左侧
根据对称性,P(-3,12)