【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(shè)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(shè)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(

),求出EF=

,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

時,點F坐標為(

),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

<t≤

,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

,

,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

≤t<3,
根據(jù)正方形的面積公式求出即可.
解答:(1)解:
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∵y=x
2-5x+4=

,
頂點C的坐標為(

),
∵y=x
2-5x+4=(x-1)(x-4),
∴點A(1,0),B(4,0),
設(shè)AC直線為y=kx+b,得

,
解得:k=-

,b=

,
∴

,
答:頂點C的坐標為(

),直線AC的解析式是

.
(2)解:設(shè)直線BC的解析式是y=ax+c,
把B(4,0),C(

,-

)代入得:0=4a+c且-

=

a+c,
解得:a=

,c=-6,
直線BC的解析式為

,
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(

),
得EF=

,
而EF=FG,
∵拋物線的對稱軸和等腰△ABC的對稱軸重合,
∴FG=

,

=2t-3,
∴
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=2t-3,
解得

,

答:當點F在AC邊上,G在BC邊上時t的值是

.
(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

時,點F坐標為(

),

=

,
②點F在AC上時,點F坐標為(
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)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(shè)FG與直線BC交于點K),
此時

<t≤

,
∴

,
Ⅱ.如圖3,EB>EH,點G在BC下方時,重疊部分是五邊形EFKMH(設(shè)FG與直線BC交于點K,GH與直線BC交于點M),
此時

,

,
點H坐標為(

),點M坐標為(

),
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,

,

,
∴S=S
EFGH-S
△KMG=(
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)
2
,
=

,
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

≤t<3,
∴

=

t
2-

t+

,
答:動點E從點B向點A運動過程中,S關(guān)于t的函數(shù)關(guān)系S=

t
2(0<t≤

)或S=-

t
2+9t-

(

<t≤

)或S=-

t
2+

t-

(

<t<

)或S=

t
2-

t+

(

≤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(xué)思想是分類討論思想.