如图,抛物线经过A,C,D三点,且三点坐标为A（-1,0）,C（0,5）,D（2,5）,抛物线与x轴的另一个交点为B点

(1)
C,D的纵坐标相同,对称轴为两点横坐标的平均值:x = (0 + 2)/2 = 1
A,B关于对称轴对称,B(3,0)
y = a(x + 1)(x - 3)
x = 0,y = -3a = 5,a = -5/3
y = (-5/3)(x + 1)(x - 3) = -5x²/3 + 10x/3 + 5
(2)
F(0,f),0 < f < 5
DF,BF的斜率分别为(5 - f)/2,-f/3
四边形DFBG为矩形,[(5 - f)/2](-f/3) = -1
f² - 5f + 6 = (f - 2)(f - 3) = 0
f = 2,F(0,2)
或f = 3,F(0,3)
(3)
BD的中点为M(5/2,5/2)
M也是FG的中点
F(0,f),G(u,v)
(0 + u)/2 = 5/2,u = 5
(f + v)/2 = 5/2,v = 5 - f
FG² = (5 - 0)² + (5 - f - f)² = (2f - 5)² + 25
f = 5/2时,FG最小
(4)
E(1,0)
以E为圆心做半径为2的圆,圆与抛物线的交点为P(p,(-5/3)(p + 1)(p - 3)),-1 < p < 3
显然PA⊥PB
PA,PB的斜率分别为u = (-5/3)(p + 1)(p - 3)/(p + 1) = (-5/3)(p - 3)
和v = (-5/3)(p + 1)(p - 3)/(p - 3) = (-5/3)(p +1)
uv = -1
(25/9)(p - 3)(p + 1) = -1
25p² - 50p - 64 = 0
p = 1 ± (√91)/5
满足到E点的距离小于2的所有点的横坐标x的范围:
-1 < x < 1 - (√91)/5 或1 + (√91)/5 < x < 3