【答案】
分析:(1)構(gòu)造全等三角形,由全等三角形對應(yīng)線段之間的相等關(guān)系,求出點D、點E的坐標(biāo);
(2)利用待定系數(shù)法求出拋物線的解析式;
(3)本問非常復(fù)雜,須小心思考與計算:
①為求s的表達(dá)式,需要識別正方形(與拋物線)的運動過程.正方形的平移,從開始到結(jié)束,總共歷時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/0.png)
秒,期間可以劃分成三個階段:當(dāng)0<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/1.png)
時,對應(yīng)圖(3)a;當(dāng)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/2.png)
<t≤1時,對應(yīng)圖(3)b;當(dāng)1<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/3.png)
時,對應(yīng)圖(3)c.每個階段的表達(dá)式不同,請對照圖形認(rèn)真思考;
②當(dāng)運動停止時,點E到達(dá)y軸,點E(-3,2)運動到點E′(0,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/4.png)
),可知整條拋物線向右平移了3個單位,向上平移了
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/5.png)
個單位.由此得到平移之后的拋物線解析式,進(jìn)而求出其頂點坐標(biāo).
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/images6.png)
解:(1)由題意可知:OB=2,OC=1.
如圖(1)所示,過D點作DH⊥y軸于H,過E點作EG⊥x軸于G.
易證△CDH≌△BCO,∴DH=OC=1,CH=OB=2,∴D(-1,3);
同理△EBG≌△BCO,∴BG=OC=1,EG=OB=2,∴E(-3,2).
∴D(-1,3)、E(-3,2).
(2)拋物線經(jīng)過(0,2)、(-1,3)、(-3,2),
則
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/6.png)
?
解得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/7.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/8.png)
.
(3)①當(dāng)點D運動到y(tǒng)軸上時,t=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/9.png)
.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/images11.png)
當(dāng)0<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/10.png)
時,如圖(3)a所示.
設(shè)D′C′交y軸于點F
∵tan∠BCO=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/11.png)
=2,又∵∠BCO=∠FCC′
∴tan∠FCC′=2,即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/12.png)
=2
∵CC′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/13.png)
t,∴FC′=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/14.png)
t.?
∴S
△CC′F?=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/15.png)
CC′•FC′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/16.png)
t×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/17.png)
t=5t
2當(dāng)點B運動到點C時,t=1.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/images20.png)
當(dāng)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/18.png)
<t≤1時,如圖(3)b所示.
設(shè)D′E′交y軸于點G,過G作GH⊥B′C′于H.
在Rt△BOC中,BC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/19.png)
∴GH=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/20.png)
,∴CH=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/21.png)
GH=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/22.png)
∵CC′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/23.png)
t,∴HC′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/24.png)
t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/25.png)
,∴GD′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/26.png)
t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/27.png)
∴S
梯形CC′D′G?=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/28.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/29.png)
t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/30.png)
+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/31.png)
t)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/32.png)
=5t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/33.png)
當(dāng)點E運動到y(tǒng)軸上時,t=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/34.png)
.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/images38.png)
當(dāng)1<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/35.png)
時,如圖(3)c所示
設(shè)D′E′、E′B′分別交y軸于點M、N
∵CC′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/36.png)
t,B′C′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/37.png)
,
∴CB′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/38.png)
t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/39.png)
,?∴B′N=2CB′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/40.png)
t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/41.png)
∵B′E′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/42.png)
,∴E′N=B′E′-B′N=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/43.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/44.png)
t
∴E′M=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/45.png)
E′N=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/46.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/47.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/48.png)
t)
∴S
△MNE′?=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/49.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/50.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/51.png)
t)•
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/52.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/53.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/54.png)
t)=5t
2-15t+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/55.png)
∴S
五邊形B′C′D′MN?=S
正方形B′C′D′E′?-S
△MNE′?=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/56.png)
(5t
2-15t+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/57.png)
)=-5t
2+15t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/58.png)
綜上所述,S與x的函數(shù)關(guān)系式為:
當(dāng)0<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/59.png)
時,S=5t
2當(dāng)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/60.png)
<t≤1時,S=5t
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/61.png)
當(dāng)1<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/62.png)
時,S=-5t
2+15t
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/63.png)
②當(dāng)點E運動到點E′時,運動停止.如圖(3)d所示
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/images68.png)
∵∠CB′E′=∠BOC=90°,∠BCO=∠B′CE′
∴△BOC∽△E′B′C
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/64.png)
∵OB=2,B′E′=BC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/65.png)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/66.png)
∴CE′=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/67.png)
∴OE′=OC+CE′=1+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/68.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/69.png)
∴E′(0,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/70.png)
)
由點E(-3,2)運動到點E′(0,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/71.png)
),可知整條拋物線向右平移了3個單位,向上平移了
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/72.png)
個單位.
∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/73.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/74.png)
?
∴原拋物線頂點坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/75.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/76.png)
)
∴運動停止時,拋物線的頂點坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/77.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192650678094575/SYS201311011926506780945022_DA/78.png)
).
點評:本題是非常典型的動線型綜合題,全面考查了初中數(shù)學(xué)代數(shù)幾何的多個重要知識點,包括:二次函數(shù)的圖象與性質(zhì)、待定系數(shù)法求解析式、拋物線與幾何變換(平移)、相似三角形的判定與性質(zhì)、全等三角形的判定與性質(zhì)、正方形的性質(zhì)等.難點在于第(3)問,識別正方形和拋物線平移過程的不同階段是關(guān)鍵所在.作為中考壓軸題,本題涉及考點眾多,計算復(fù)雜,因而難度很大,對考生綜合能力要求很高,具有很好的區(qū)分度.