【答案】
分析:(1)先求出直線y=kx-4與y軸的交點(diǎn)C的坐標(biāo),再設(shè)經(jīng)過(guò)點(diǎn)A(-1,0)和點(diǎn)B(3,0)的二次函數(shù)的解析式為y=a(x+1)(x-3),然后將C點(diǎn)坐標(biāo)代入,運(yùn)用待定系數(shù)法即可求出這個(gè)二次函數(shù)的解析式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/0.png)
x
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/1.png)
x-4;
(2)先利用配方法求出二次函數(shù)y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/2.png)
x
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/3.png)
x-4的頂點(diǎn)D的坐標(biāo),再將D點(diǎn)坐標(biāo)代入y=kx-4,求出k的值,得到直線CD的解析式,再求出CD與x軸交點(diǎn)E的坐標(biāo),根據(jù)三角形面積公式可得△AEC的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/4.png)
AE•OC=4;設(shè)直線BC與拋物線的對(duì)稱軸交于點(diǎn)F,運(yùn)用待定系數(shù)法求出直線BC的解析式,令x=1,求出y的值,得到F點(diǎn)坐標(biāo)及DF的長(zhǎng)度,根據(jù)三角形面積公式可得△BCD的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/5.png)
DF•OB=4,從而得出△AEC的面積與△BCD的面積相等;
(3)過(guò)點(diǎn)A作AG⊥BC于G,易得AB=4,OC=4,運(yùn)用勾股定理求出BC=5,AC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/6.png)
,根據(jù)三角形面積公式得出△ABC的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/7.png)
AB•OC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/8.png)
BC•AG,則AG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/9.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/10.png)
,在Rt△ACG中根據(jù)三角函數(shù)的定義即可求出sin∠ACB的值.
解答:解:(1)∵y=kx-4,
∴當(dāng)x=0時(shí),y=-4,即C點(diǎn)坐標(biāo)為(0,-4).
設(shè)經(jīng)過(guò)點(diǎn)A(-1,0)和點(diǎn)B(3,0)的二次函數(shù)的解析式為y=a(x+1)(x-3),
將C(0,-4)代入,得-4=-3a,
解得a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/11.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/images12.png)
∴這個(gè)二次函數(shù)的解析式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/12.png)
(x+1)(x-3),即y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/13.png)
x
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/14.png)
x-4;
(2)△AEC的面積與△BCD的面積相等,理由如下:
∵y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/15.png)
x
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/16.png)
x-4=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/17.png)
(x-1)
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/18.png)
,
∴對(duì)稱軸為直線x=1,頂點(diǎn)D的坐標(biāo)為(1,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/19.png)
).
將D(1,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/20.png)
)代入y=kx-4,
得-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/21.png)
=k-4,解得k=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/22.png)
,
∴y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/23.png)
x-4,
當(dāng)y=0時(shí),-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/24.png)
x-4=0,解得x=-3,
∴E點(diǎn)坐標(biāo)為(-3,0),AE=2,
∴△AEC的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/25.png)
AE•OC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/26.png)
×2×4=4.
設(shè)直線BC與拋物線的對(duì)稱軸交于點(diǎn)F,如圖,
易求直線BC的解析式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/27.png)
x-4,
當(dāng)x=1時(shí),y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/28.png)
×1-4=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/29.png)
,
∴F點(diǎn)坐標(biāo)為(1,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/30.png)
),DF=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/31.png)
-(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/32.png)
)=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/33.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/images35.png)
∴△BCD的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/34.png)
DF•OB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/35.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/36.png)
×3=4,
∴△AEC的面積與△BCD的面積相等;
(3)如圖,過(guò)點(diǎn)A作AG⊥BC于G.
∵A(-1,0),B(3,0),C(0,-4),
∴AB=4,OC=4,BC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/37.png)
=5,AC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/38.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/39.png)
.
∵△ABC的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/40.png)
AB•OC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/41.png)
BC•AG,
∴AG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/42.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/43.png)
,
∴sin∠ACB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/44.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/45.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101193831033862526/SYS201311011938310338625027_DA/46.png)
.
點(diǎn)評(píng):本題是二次函數(shù)的綜合題型,其中涉及到的知識(shí)點(diǎn)有運(yùn)用待定系數(shù)法求一次函數(shù)、二次函數(shù)的解析式,拋物線的頂點(diǎn)坐標(biāo),三角形的面積,勾股定理,銳角三角函數(shù)的定義,綜合性較強(qiáng),難度適中.