已知拋物線y=3x2+2x+n,
(1)若n=-1,求該拋物線與x軸的交點坐標;
(2)當-1<x<1時,拋物線與x軸有且只有一個公共點,求n的取值范圍.
【答案】
分析:(1)把n=-1,y=0代入拋物線解析式,通過解一元二次方程可求得交點坐標.
(2)分3種情況.第1種:△=0,n=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/0.png)
;
第2種:把x=-1代入函數(shù)使y大于0,且把x=1代入函數(shù),使y小于0,解這個不等式,可得n的取值范圍;
第3種:把x=-1代入函數(shù)使y小于0,且把x=1代入函數(shù),使y大于0,解這個不等式組,可得n的取值范圍.
綜合這三個結果即可得n的范圍.在2,3種情況下必須保證△大于0.
解答:解:(1)當n=-1時,拋物線為y=3x
2+2x-1,
方程3x
2+2x-1=0的兩個根為:x=-1或x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/1.png)
.
∴該拋物線與x軸交點的坐標是(-1,0)和(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/2.png)
);(2分)
(2)∵拋物線與x軸有公共點,
∴對于方程3x
2+2x+n=0,判別式△=4-12n≥0,
∴n≤
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/3.png)
.(3分)
①當n=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/4.png)
時,由方程3x
2+2x+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/5.png)
=0,解得x
1=x
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/6.png)
.此時拋物線為y=3x
2+2x+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/7.png)
與x軸只有一個公共點(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/8.png)
);(4分)
②當n<
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/9.png)
時,
x
1=-1時,y
1=3-2+n=1+n;
x
2=1時,y
2=3+2+n=5+n;
由已知-1<x<1時,該拋物線與x軸有且只有一個公共點,考慮其對稱軸為x=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/10.png)
,
應有y
1≤0,且y
2>0即1+n≤0,且5+n>0.(5分)
解得:-5<n≤-1.(6分)
綜合①,②得n的取值范圍是:n=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163639591444313/SYS201310221636395914443022_DA/11.png)
或-5<n≤-1.(7分)
點評:考查二次函數(shù)y=ax
2+bx+c的圖象與x軸交點的個數(shù)的判斷.