题目内容
(2009•台州)如图,已知直线y=-
(1)请直接写出点C,D的坐标;
(2)求抛物线的解析式;
(3)若正方形以每秒

(4)在(3)的条件下,抛物线与正方形一起平移,同时D停止,求抛物线上C,E两点间的抛物线弧所扫过的面积.

【答案】分析:(1)可先根据AB所在直线的解析式求出A,B两点的坐标,即可得出OA、OB的长.过D作DM⊥y轴于M,则△ADM≌△BAO,由此可得出MD、MA的长,也就能求出D的坐标,同理可求出C的坐标;
(2)可根据A、C、D三点的坐标,用待定系数法求出抛物线的解析式;
(3)要分三种情况进行讨论:
①当F点在A′B′之间时,即当0<t≤1时,此时S为三角形FBG的面积,可用正方形的速度求出AB′的长,即可求出B′F的长,然后根据∠GFB′的正切值求出B′G的长,即可得出关于S、t的函数关系式.
②当A′在x轴下方,但C′在x轴上方或x轴上时,即当1<t≤2时,S为梯形A′GB′H的面积,可参照①的方法求出A′G和B′H的长,那么梯形的上下底就可求出,梯形的高为A′B′即正方形的边长,可根据梯形的面积计算公式得出关于S、t的函数关系式.
③当D′逐渐移动到x轴的过程中,即当2<t≤3时,此时S为五边形A′B′C′HG的面积,S=正方形A′B′C′D′的面积-三角形GHD′的面积.可据此来列关于S,t的函数关系式;
(4)CE扫过的图形是个平行四边形,经过关系不难发现这个平行四边形的面积实际上就是矩形BCD′A′的面积.可通过求矩形的面积来求出CE扫过的面积.
解答:
解:(1)C(3,2)D(1,3);
(2)设抛物线为y=ax2+bx+c,抛物线过(0,1)(3,2)(1,3),

解得
,
∴y=-
x2+
x+1;
(3)①当点A运动到x轴上时,t=1,
当0<t≤1时,如图1,
∵∠OFA=∠GFB′,
tan∠OFA=
,
∴tan∠GFB′=
,
∴GB′=
t
∴S△FB′G=
FB′×GB′
=
×
t×
=
t2;
②当点C运动到x轴上时,t=2,
当1<t≤2时,如图2,
A′B′=AB=
,
∴A′F=
t-
,
∴A′G=
,
∵B′H=
,
∴S梯形A′B′HG=
(A′G+B′H)×A′B′
=
=
t-
;
③当点D运动到x轴上时,t=3,
当2<t≤3时,如图3,
∵A′G=
,
∴GD′=
,
∵S△AOF=
×1×2=1,OA=1,△AOF∽△GD′H
∴
,
∴
,
∴S五边形GA′B′C′H=(
)2-(
=-
t2+
t-
;

(4)∵t=3,BB′=AA′=3
,
∴S阴影=S矩形BB′C′C=S矩形AA′D′D
=AD×AA′=
×3
=15.
点评:本题着重考查了待定系数法求二次函数解析式、图形平移变换、三角形相似等重要知识点,(3)小题中要根据正方形的不同位置分类进行讨论,不要漏解.
(2)可根据A、C、D三点的坐标,用待定系数法求出抛物线的解析式;
(3)要分三种情况进行讨论:
①当F点在A′B′之间时,即当0<t≤1时,此时S为三角形FBG的面积,可用正方形的速度求出AB′的长,即可求出B′F的长,然后根据∠GFB′的正切值求出B′G的长,即可得出关于S、t的函数关系式.
②当A′在x轴下方,但C′在x轴上方或x轴上时,即当1<t≤2时,S为梯形A′GB′H的面积,可参照①的方法求出A′G和B′H的长,那么梯形的上下底就可求出,梯形的高为A′B′即正方形的边长,可根据梯形的面积计算公式得出关于S、t的函数关系式.
③当D′逐渐移动到x轴的过程中,即当2<t≤3时,此时S为五边形A′B′C′HG的面积,S=正方形A′B′C′D′的面积-三角形GHD′的面积.可据此来列关于S,t的函数关系式;
(4)CE扫过的图形是个平行四边形,经过关系不难发现这个平行四边形的面积实际上就是矩形BCD′A′的面积.可通过求矩形的面积来求出CE扫过的面积.
解答:

(2)设抛物线为y=ax2+bx+c,抛物线过(0,1)(3,2)(1,3),

解得

∴y=-


(3)①当点A运动到x轴上时,t=1,
当0<t≤1时,如图1,
∵∠OFA=∠GFB′,

tan∠OFA=

∴tan∠GFB′=

∴GB′=

∴S△FB′G=

=




②当点C运动到x轴上时,t=2,
当1<t≤2时,如图2,
A′B′=AB=

∴A′F=



∴A′G=

∵B′H=

∴S梯形A′B′HG=

=



③当点D运动到x轴上时,t=3,
当2<t≤3时,如图3,
∵A′G=

∴GD′=


∵S△AOF=

∴

∴

∴S五边形GA′B′C′H=(


=-




(4)∵t=3,BB′=AA′=3

∴S阴影=S矩形BB′C′C=S矩形AA′D′D
=AD×AA′=


点评:本题着重考查了待定系数法求二次函数解析式、图形平移变换、三角形相似等重要知识点,(3)小题中要根据正方形的不同位置分类进行讨论,不要漏解.

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