在平面直角坐標系中,已知二次函數y=a(x-1)2+k(a>0)的圖象與x軸相交于點A,B(點A在點B的左邊),頂點為C,點D在這個二次函數圖象的對稱軸上,若四邊形ABCD是一條邊長為4且有一個內角為120°的菱形,求此二次函數的關系式?
【答案】
分析:此題分為當∠ACB=120°時與當∠DAC=120°時去分析,由四邊形ACBD是菱形,可得AB⊥CD,又由AC=CB=4,即可求得點C與B的坐標,繼而求得此二次函數的關系式.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/images0.png)
解:當∠ACB=120°時,
∵四邊形ACBD是菱形,
∴AB⊥CD,
∵AC=CB=4,
得C(1,-2),B(1+2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/0.png)
,0),
代入y=a(x-1)
2+k中,
∴a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/1.png)
,k=-2,
∴y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/2.png)
(x-1)
2-2.
當∠DAC=120°時,由四邊形ACBD是菱形得,
得C(1,-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/3.png)
),B(3,0),
代入y=a(x-1)
2+k中,
∴a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/4.png)
,k=-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/5.png)
.
∴y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/6.png)
(x-1)
2-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/7.png)
.
由圖形的對稱性可知:y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/8.png)
(x-1)
2-2或y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/9.png)
(x-1)
2-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/10.png)
也符合題意,
∵a>0,
∴二次函數的關系式為:y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/11.png)
(x-1)
2-2或y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/12.png)
(x-1)
2-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161153498954205/SYS201310221611534989542026_DA/13.png)
.
點評:此題考查了菱形的性質,以及待定系數法求二次函數的解析式的知識.此題難度適中,解題的關鍵是注意方程思想與分類討論思想的應用.