1.令y=ax²-2ax-3a=0,(抛物线开口向上,a>0),得,x²-2x-3=0,(x-3)(x+1)=0,x=-1或x=3,
即抛物线y=ax²-2ax-3a与x轴负半轴交点A坐标为(-1,0),
在y=ax²-2ax-3a中令x=0,得y=-3a,即抛物线y=ax²-2ax-3a与y轴负方向交点C坐标为(0,-3a),AO=1,CO=3a,tan∠ACO=AO/CO=1/3a=1/3,a=1,抛物线表达式为y=x²-2x-3.
2.AO=1,CO=3,
AC=√(AO²+CO²)=√10,
C(0,-3),C点关于抛物线对称轴x=-2/-2=1的对称点为C'(2,-3),A(-1,0),A点关于直线y=1的对称点为A'(-1,2),
连接A'C',交直线y=1和抛物线对称轴分别于点E',F',则A'E'=AE',C'F'=CF',A'E=AE,C'F=CF,
AE+EF+CF=A'E+EF+C'F>=A'C',(两点之间线段最短),
点E与E',F与F'分别重合时,四边形ACFE的周长最短,此时四边形的周长=AC+A'C',
分别延长A'A,C'C,交于点D,则D点坐标为(-1,-3),A'D⊥C'D,A'D=2+3=5,C'D=2+1=3,
A'C'=√(A'D²+C'D²)=√34,
四边形ACFE的最短周长=AC+A'C'=√10+√34