y=ax^2-2ax+3
=a(x-1)^2-a+3,
a≤x≤2a+3
a-1≤x-1≤2a+2
1.
2a+2＜0时,即a＜-1
a-1≤x-1≤2a+2
4(a+1)^2≤(x-1)^2≤(a-1)^2
a(a-1)^2≤a(x-1)^2≤4a(a+1)^2
a(a-1)^2-a+3≤a(x-1)^2-a+3≤4a(a+1)^2-a+3
a(a-1)^2-a+3≤y≤4a(a+1)^2-a+3;
2.
a-1＞0时,即a＞1
a-1≤x-1≤2a+2
(a-1)^2≤(x-1)^2≤4(a+1)^2
4a(a+1)^2≤a(x-1)^2≤a(a-1)^2
4a(a+1)^2-a+3≤a(x-1)^2-a+3≤a(a-1)^2-a+3
4a(a+1)^2-a+3≤y≤a(a-1)^2-a+3;
3.
a-1＜0,2a+2＞0时,即-1＜a＜0或0＜a＜1
(1)-1＜a＜0且2a+2＞-(a-1),即-1/3＜a＜0
a-1≤x-1≤2a+2
0≤(x-1)^2≤4(a+1)^2
4a(a+1)^2≤a(x-1)^2≤0
4a(a+1)^2-a+3≤a(x-1)^2-a+3≤-a+3
4a(a+1)^2-a+3≤y≤-a+3;
(2)-1＜a＜0且2a+2＜-(a-1),即-1＜a＜-1/3
a-1≤x-1≤2a+2
0≤(x-1)^2≤(a-1)^2
a(a-1)^2≤a(x-1)^2≤0
a(a-1)^2-a+3≤a(x-1)^2-a+3≤-a+3
a(a-1)^2-a+3≤y≤-a+3;
(3)0＜a＜1且2a+2＞-(a-1),即0＜a＜1
a-1≤x-1≤2a+2
0≤(x-1)^2≤(a-1)^2
0≤a(x-1)^2≤a(a-1)^2
-a+3≤a(x-1)^2-a+3≤a(a-1)^2-a+3
-a+3≤y≤a(a-1)^2-a+3;
(4)0＜a＜1且2a+2＜-(a-1),无解.
综上所述
a＜-1时,a(a-1)^2-a+3≤y≤4a(a+1)^2-a+3;
-1＜a＜-1/3时,a(a-1)^2-a+3≤y≤-a+3;
-1/3＜a＜0时,4a(a+1)^2-a+3≤y≤-a+3;
0＜a＜1时,-a+3≤y≤a(a-1)^2-a+3;
a＞1时,4a(a+1)^2-a+3≤y≤a(a-1)^2-a+3.