【答案】
分析:(1)把y=x
2-5x+4化成頂點式,求出頂點C的坐標,y=x
2-5x+4化成(x-1)(x-4),求出A、B的坐標,設AC直線為y=kx+b,把A、C的坐標代入就能求出直線AC的解析式;
(2)設直線BC的解析式是y=ax+c,把B、C的坐標代入就能求出直線BC,點E坐標為(4-t,0),點F坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/0.png)
),求出EF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/1.png)
,F(xiàn)G=2t-3,根據(jù)EF=FG,即可求出t的值;
(3)可分以下幾種情況:①點F在BC上時,如圖1重疊部分是△BEF
2,此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/2.png)
時,點F坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/3.png)
),根據(jù)三角形的面積公式即可求出;②I如圖2,EB≤EH時重疊部分是直角梯形EFKB,此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/4.png)
<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/5.png)
,根據(jù)三角形的面積公式即可求出;II如圖3,EB>EH,點G在BC下方時,重疊部分是五邊形EFKMH,此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/6.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/7.png)
,因為S=S
正方形EFGH-S
△KMG,根據(jù)三角形的面積公式即可求出;Ⅲ.如圖4,點G在BC上或BC上方時,重疊部分是正方形EFGH,此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/8.png)
≤t<3,
根據(jù)正方形的面積公式求出即可.
解答:(1)解:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/images9.png)
∵y=x
2-5x+4=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/9.png)
,
頂點C的坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/10.png)
),
∵y=x
2-5x+4=(x-1)(x-4),
∴點A(1,0),B(4,0),
設AC直線為y=kx+b,得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/11.png)
,
解得:k=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/12.png)
,b=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/13.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/14.png)
,
答:頂點C的坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/15.png)
),直線AC的解析式是
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/16.png)
.
(2)解:設直線BC的解析式是y=ax+c,
把B(4,0),C(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/17.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/18.png)
)代入得:0=4a+c且-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/19.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/20.png)
a+c,
解得:a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/21.png)
,c=-6,
直線BC的解析式為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/22.png)
,
當F在AC邊上,G在BC邊上時,
點E坐標為(4-t,0),點F坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/23.png)
),
得EF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/24.png)
,
而EF=FG,
∵拋物線的對稱軸和等腰△ABC的對稱軸重合,
∴FG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/25.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/26.png)
=2t-3,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/27.png)
=2t-3,
解得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/28.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/images30.png)
答:當點F在AC邊上,G在BC邊上時t的值是
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/29.png)
.
(3)解:點E坐標為(4-t,0)隨著正方形的移動,重疊部分的形狀不同,可分以下幾種情況:
①點F在BC上時,如圖1重疊部分是△BEF,
此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/30.png)
時,點F坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/31.png)
),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/32.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/33.png)
,
②點F在AC上時,點F坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/34.png)
)又可分三種情況:
Ⅰ.如圖2,EB≤EH時重疊部分是直角梯形EFKB(設FG與直線BC交于點K),
此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/35.png)
<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/36.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/37.png)
,
Ⅱ.如圖3,EB>EH,點G在BC下方時,重疊部分是五邊形EFKMH(設FG與直線BC交于點K,GH與直線BC交于點M),
此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/38.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/39.png)
,
點H坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/40.png)
),點M坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/41.png)
),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/42.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/43.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/44.png)
,
∴S=S
EFGH-S
△KMG=(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/45.png)
)
2![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/46.png)
,
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/47.png)
,
Ⅲ.如圖4,點G在BC上或BC上方時,重疊部分是正方形EFGH,此時
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/48.png)
≤t<3,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/49.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/50.png)
t
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/51.png)
t+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/52.png)
,
答:動點E從點B向點A運動過程中,S關于t的函數(shù)關系S=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/53.png)
t
2(0<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/54.png)
)或S=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/55.png)
t
2+9t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/56.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/57.png)
<t≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/58.png)
)或S=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/59.png)
t
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/60.png)
t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/61.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/62.png)
<t<
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/63.png)
)或S=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/64.png)
t
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/65.png)
t+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/66.png)
(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164913705272733/SYS201310221649137052727023_DA/67.png)
≤t<3).
點評:本題主要考查對二次函數(shù)與X軸的交點,用待定系數(shù)法求一次函數(shù)的解析式,解二元一次方程組,三角形的面積,用十字相乘法分解因式,二次函數(shù)圖象上點的坐標特征等知識點的理解和掌握,此題是一個拔高的題目,有一定的難度,用的數(shù)學思想是分類討論思想.