【答案】
分析:(1)根據(jù)已知條件得出圖象經(jīng)過(guò)(0,0),(4,0)再結(jié)合圖象的對(duì)稱(chēng)性得出頂點(diǎn)坐標(biāo)為(2,4),代入解析式即可;
(2)利用(1)中解析式用x表示出矩形的周長(zhǎng),再結(jié)合矩形的最值問(wèn)題求出最值;
(3)利用平行線的性質(zhì)與軸對(duì)稱(chēng)性質(zhì)得出△DME的周長(zhǎng)最小值,即OD+DM,利用勾股定理求出即可.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/images0.png)
解:(1)由已知可得:頂點(diǎn)P到x軸的距離是4,
即P點(diǎn)的縱坐標(biāo)為4,
∵拋物線與x軸相交于O,M兩點(diǎn),OM=4,
∴M點(diǎn)的坐標(biāo)為(4,0),P點(diǎn)的坐標(biāo)為(2,4),
解析式為:y=a(x-2)
2+4,
將M點(diǎn)的坐標(biāo)為(4,0)代入求出a=-1,
∴解析式為:y=-1(x-2)
2+4=-x
2+4x;
(2)∵點(diǎn)A,D在拋物線上,假設(shè)A點(diǎn)的橫坐標(biāo)為x,縱坐標(biāo)就是-x
2+4x
∴即(x,-x
2+4x),AB=-x
2+4x,
∵OB=x,CM=x
∴BC=4-2x,
∴矩形ABCD的周長(zhǎng)為L(zhǎng)=2(-x
2+4x+4-2x)=-2x
2+4x+8
當(dāng)x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/0.png)
=1時(shí),L有最大值
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/1.png)
=10
(3)存在.
因?yàn)槭钱?dāng)矩形ABCD的周長(zhǎng)最大時(shí),在拋物線的對(duì)稱(chēng)軸上是否存在點(diǎn)E,
使得△DME的周長(zhǎng)最小,即當(dāng)x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/2.png)
=1時(shí),L有最大值
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/3.png)
=10,
∴A(x,-x
2+4x),A的坐標(biāo)為(1,3),連接OD與拋物線的對(duì)稱(chēng)軸直線x=2交于一點(diǎn),
即AB=3,OB=1,CD=3,CM=1,
∴OF=2,OC=3,CD=3,PF∥CD,
∴EF=2,
∵A,D關(guān)于x=2對(duì)稱(chēng),O,M也是關(guān)于x=2對(duì)稱(chēng),則AC一定經(jīng)過(guò)點(diǎn)E.
∴△DME的周長(zhǎng)=EM+DM+EC=DM+EC+AE=AM+DM.
∴點(diǎn)E為滿(mǎn)足條件的點(diǎn),E點(diǎn)坐標(biāo)為(2,2).
∴DM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/5.png)
,
根據(jù)二次函數(shù)對(duì)稱(chēng)性O(shè)D=DE+EM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/7.png)
=3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/8.png)
,
此時(shí)△DME的周長(zhǎng)最小值為:OD+DM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101185356464770015/SYS201311011853564647700022_DA/9.png)
.
點(diǎn)評(píng):此題主要考查了用頂點(diǎn)式求二次函數(shù)解析式,以及二次函數(shù)最值問(wèn)題,以及利用對(duì)稱(chēng)性求線段最短問(wèn)題,本類(lèi)型是中考中重點(diǎn)題型,在多次中考題中出現(xiàn)過(guò).