【答案】
分析:(1)根據(jù)點(diǎn)A在直線y=kx上,即可得出h,m的關(guān)系式.
(2)當(dāng)EF∥x軸時(shí),根據(jù)拋物線的對(duì)稱性可知:FC=CE即C是EF的中點(diǎn),那么AC就是三角形OEF的中位線,因此AC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/0.png)
OF.
(也可通過聯(lián)立直線OA的解析式和拋物線的解析式得出E點(diǎn)的坐標(biāo),當(dāng)EF∥x軸時(shí),E、F縱坐標(biāo)相同,以此來求出h,k的關(guān)系,進(jìn)而表示出A、C、E、F四點(diǎn)坐標(biāo)以此來求出AC與OF的比例關(guān)系).
(3)先求出F到最低位置時(shí),函數(shù)的解析式(F位置最低時(shí),縱坐標(biāo)值最�。�(lián)立兩函數(shù)的解析式求出A、E的坐標(biāo),然后根據(jù)相似三角形OEF和AEC求出OF,AC的比例關(guān)系.
解答:解:(1)∵拋物線頂點(diǎn)(h,m)在直線y=kx上,
∴m=kh;
(2)方法一:解方程組
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/1.png)
,
將(2)代入(1)得到:(x-h)
2+kh=kx,
整理得:(x-h)[(x-h)-k]=0,
解得:x
1=h,x
2=k+h,
代入到方程(2)y
1=hy
2=k
2+hk,
所以點(diǎn)E坐標(biāo)是(k+h,k
2+hk),
當(dāng)x=0時(shí),y=(x-h)
2+m=h
2+kh,
∴點(diǎn)F坐標(biāo)是(0,h
2+kh),
當(dāng)EF和x軸平行時(shí),點(diǎn)E,F(xiàn)的縱坐標(biāo)相等,
即k
2+kh=h
2+kh,
解得:h=k(h=-k舍去,否則E,F(xiàn),O重合),
此時(shí)點(diǎn)E(2k,2k
2),F(xiàn)(0,2k
2),C(k,2k
2),A(k,k
2),
∴AC:OF=k
2:2k
2=1:2.(3分)
方法二:當(dāng)x=0時(shí),y=(x-h)
2+m=h
2+kh,即F(0,h
2+kh),
當(dāng)EF和x軸平行時(shí),點(diǎn)E,F(xiàn)的縱坐標(biāo)相等,
即點(diǎn)E的縱坐標(biāo)為h
2+kh,
當(dāng)y=h
2+kh時(shí),代入y=(x-h)
2+kh,
解得x=2h(0舍去,否則E,F(xiàn),O重合),
即點(diǎn)E坐標(biāo)為(2h,h
2+kh),(1分)
將此點(diǎn)橫縱坐標(biāo)代入y=kx得到h=k(h=0舍去,否則點(diǎn)E,F(xiàn),O重合),
此時(shí)點(diǎn)E(2k,2k
2),F(xiàn)(0,2k
2),C(k,2k
2),A(k,k
2),
∴AC:OF=k
2:2k
2=1:2.
方法三:∵EF與x軸平行,
根據(jù)拋物線對(duì)稱性得到FC=EC,
∵AC∥FO,
∴∠ECA=∠EFO,∠FOE=∠CAE,
∴△OFE∽△ACE,
∴AC:OF=EC:EF=1:2.
(3)當(dāng)點(diǎn)F的位置處于最低時(shí),其縱坐標(biāo)h
2+kh最小,
∵h(yuǎn)
2+kh=[h
2+kh+(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/2.png)
)
2]-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/3.png)
,
當(dāng)h=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/4.png)
,點(diǎn)F的位置最低,此時(shí)F(0,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/5.png)
),
解方程組
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/6.png)
得E(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/7.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/8.png)
),A(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/9.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/10.png)
).
方法一:設(shè)直線EF的解析式為y=px+q,
將點(diǎn)E(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/11.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/12.png)
),F(xiàn)(0,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/13.png)
)的橫縱坐標(biāo)分別代入得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/14.png)
,
解得:p=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/15.png)
,q=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/16.png)
,
∴直線EF的解析式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/17.png)
x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/18.png)
,
當(dāng)x=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/19.png)
時(shí),y=-k
2,即點(diǎn)C的坐標(biāo)為(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/20.png)
,-k
2),
∵點(diǎn)A(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/21.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/22.png)
),
∴AC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/23.png)
,而OF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/24.png)
,
∴AC=2OF,即AC:OF=2.
方法二:∵E(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/25.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/26.png)
),A(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/27.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103418894840856/SYS201311031034188948408009_DA/28.png)
),
∴點(diǎn)A,E關(guān)于點(diǎn)O對(duì)稱,
∴AO=OE,
∵AC∥FO,
∴∠ECA=∠EFO,∠FOE=∠CAE,
∴△OFE∽△ACE,
∴AC:OF=AE:OE=2:1.
點(diǎn)評(píng):本題主要考查了函數(shù)圖象交點(diǎn)、相似三角形的性質(zhì)等知識(shí)點(diǎn).