【答案】
分析:(1)由于O是AB的中點(diǎn),則OA=OB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/0.png)
;可設(shè)出點(diǎn)B的橫坐標(biāo),結(jié)合B點(diǎn)的縱坐標(biāo)和勾股定理即可求出B點(diǎn)的橫坐標(biāo);
(2)①已知了拋物線(xiàn)的解析式,即可得到拋物線(xiàn)的對(duì)稱(chēng)軸方程,也就得到了C點(diǎn)的橫坐標(biāo);此時(shí)發(fā)現(xiàn)C點(diǎn)橫坐標(biāo)為正數(shù),所以分兩種情況討論:
一、點(diǎn)C在第一象限;在Rt△OBC中,根據(jù)OB的長(zhǎng)及∠B的度數(shù),可求出OC的長(zhǎng),參照(1)的方法即可求出C點(diǎn)的坐標(biāo);若分別過(guò)A、C作x軸的垂線(xiàn),通過(guò)構(gòu)建的相似三角形即可求出A點(diǎn)的坐標(biāo),A、B關(guān)于原點(diǎn)對(duì)稱(chēng),即可得到B點(diǎn)的坐標(biāo);將A、B的坐標(biāo)代入拋物線(xiàn)的解析式中進(jìn)行驗(yàn)證即可;
二、點(diǎn)D在第四象限;方法同一;
②若b=-2am,則函數(shù)的解析式為:y=ax
2-2amx+c=a(x-m)
2-am
2+c;由此可得C點(diǎn)的橫坐標(biāo)為m;在△ABC旋轉(zhuǎn)的過(guò)程中,C點(diǎn)橫坐標(biāo)的取值范圍在區(qū)間[-1,1]之間,由于當(dāng)m=-1或1時(shí),C點(diǎn)在x軸上,A、B同時(shí)處在y軸,所以此時(shí)拋物線(xiàn)不可能同時(shí)經(jīng)過(guò)A、B兩點(diǎn).
解答:解:(1)∵點(diǎn)O是AB的中點(diǎn),
∴OB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/1.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/2.png)
;(1分)
設(shè)點(diǎn)B的橫坐標(biāo)是x(x>0),
則x
2+(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/3.png)
)
2=(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/4.png)
)
2,(1分)
解得x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/5.png)
,x
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/6.png)
(舍去);
∵點(diǎn)B在第一象限,
∴點(diǎn)B的橫坐標(biāo)是
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/7.png)
;(2分)
(2)①當(dāng)a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/8.png)
,b=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/9.png)
,c=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/10.png)
時(shí),得y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/11.png)
(*)
y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/12.png)
;(1分)
以下分兩種情況討論;
情況1:設(shè)點(diǎn)C在第一象限(如圖),
則點(diǎn)C的橫坐標(biāo)為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/13.png)
,OC=OB×tan30°=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/14.png)
=1;(1分)
由此,可求得點(diǎn)C的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/15.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/16.png)
),
根據(jù)∠A=30°,OC⊥AB,
過(guò)C作X軸的垂線(xiàn)交X軸于N,過(guò)點(diǎn)A作垂線(xiàn)交X軸于點(diǎn)M,
則△AOM∽△CON
∴OA:OC=OM:CN=AM:ON=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/17.png)
:1
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/images18.png)
,
∵NO=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/18.png)
,
∴AM=NO×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/19.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/20.png)
,
∴MO=CN×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/21.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/22.png)
,
∴點(diǎn)A(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/23.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/24.png)
),
∵A,B兩點(diǎn)關(guān)于原點(diǎn)對(duì)稱(chēng),
∴點(diǎn)B的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/25.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/26.png)
),
將點(diǎn)A的橫坐標(biāo)代入解析式的右邊,計(jì)算得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/27.png)
,
即等于點(diǎn)A的縱坐標(biāo);
將點(diǎn)B的橫坐標(biāo)代入解析式的右邊,計(jì)算得-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/28.png)
,即等于點(diǎn)B的縱坐標(biāo);
∴在這種情況下,A,B兩點(diǎn)都在拋物線(xiàn)上;
情況2:設(shè)點(diǎn)C在第四象限(如圖),則點(diǎn)C的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/29.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/images31.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/30.png)
),
點(diǎn)A的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/31.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/32.png)
),點(diǎn)B的坐標(biāo)為(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/33.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274022_DA/34.png)
);
經(jīng)計(jì)算,A,B兩點(diǎn)都不在這條拋物線(xiàn)上;
②存在,m的值是1或-1.
y=a(x-m)
2-am
2+c,
因?yàn)檫@條拋物線(xiàn)的對(duì)稱(chēng)軸經(jīng)過(guò)點(diǎn)C,
所以-1≤m≤1;
當(dāng)m=±1時(shí),點(diǎn)C在x軸上,此時(shí)A,B兩點(diǎn)都在y軸上.
因此當(dāng)m=±1時(shí),A,B兩點(diǎn)不可能同時(shí)在這條拋物線(xiàn)上.
點(diǎn)評(píng):此題是二次函數(shù)的綜合題型,主要考查了等腰三角形的性質(zhì)、解直角三角形、勾股定理、圖形的旋轉(zhuǎn)變換等知識(shí).