解：?$(1)$?∵拋物線?$y=ax2-2ax-3+2a2$?
?$=a(x-1)2+ 2a2-a-3$?
∴拋物線的對(duì)稱軸為直線?$x=1$?
?$(2)$?∵拋物線的頂點(diǎn)在?$x$?軸上
∴?$2a2-a-3=0$?
解得?$a=\frac {3}{2}$?或?$a=-1$?
∴拋物線的解析式為?$y=\frac {3}{2}x2-3x+\frac {3}{2}$?
或?$y=-x2+2x-1$?
?$(3)$?拋物線的對(duì)稱軸為?$x=1$?，?$Q(3$?，?$y_{2})$?關(guān)于
直線?$x=1$?的對(duì)稱點(diǎn)的坐標(biāo)為?$(-1$?，?$y_{2})$?
分情況討論：?$①$?當(dāng)?$a>0$?時(shí)，要使?$y_{1}＜y_{2}$?
則?$-1＜m＜3$?
?$②$?當(dāng)?$a<0$?時(shí)，要使?$y_{1}＜y_{2}$?
則?$m＜-1$?或?$m＞3$?
