函數(shù)y=mx2+(m-1)x+1過的第三象限,則m的取值范圍是 .
【答案】
分析:分別根據(jù)m=0時,函數(shù)式一次函數(shù),以及m>0和m<0時,此函數(shù)是二次函數(shù),再利用△>0,以及根與系數(shù)關(guān)系求出m的取值范圍即可.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/images0.png)
解:當m=0時y=x+1不經(jīng)過第三象限.
當(1)m>0時,令y=0,則0=mx
2+(m-1)x+1,
△=(m-1)
2-4m,
=m
2-6m+1>0,
則m>3+2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/0.png)
或m<3-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/1.png)
,
且滿足x
1+x
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/2.png)
=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/3.png)
<0,且x
1•x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/5.png)
=>0,
解得:m>1,
故m>3+2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/6.png)
或1<m<3-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/7.png)
,
當m<0時,令y=0,則0=mx
2+(m-1)x+1,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/images9.png)
△=(m-1)
2-4m>0
=m
2-2m+1-4m
=m
2-6m+1>0
則m>3+2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/8.png)
(不合題意舍去)或m<3-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/9.png)
,
此時m<0.
總述:m>3+2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/10.png)
或1<m<3-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/11.png)
或m<0.
故答案為:m>3+2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/12.png)
或1<m<3-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191334011334992/SYS201311011913340113349013_DA/13.png)
或m<0.
點評:此題主要考查了二次函數(shù)的性質(zhì)以及一次函數(shù)性質(zhì),利用數(shù)形結(jié)合得出m的取值范圍是解題關(guān)鍵.