已知拋物線y=ax2+bx+c(a>0)的頂點是C(0,1),直線l:y=-ax+3與這條拋物線交于P、Q兩點,與x軸、y軸分別交于點M和N.
(1)設(shè)點P到x軸的距離為2,試求直線l的函數(shù)關(guān)系式;
(2)若線段MP與PN的長度之比為3:1,試求拋物線的函數(shù)關(guān)系式.
【答案】
分析:(1)由于拋物線的頂點為C(0,1),因此拋物線的解析式中b=0,c=1.即拋物線的解析式為y=ax
2+1.已知了P到x軸的距離為2,即P點的縱坐標為2.可根據(jù)直線l的解析式求出P點的坐標,然后將P點坐標代入拋物線的解析式中即可求得a的值,也就能求出直線l的函數(shù)關(guān)系式.
(2)本題要根據(jù)相似三角形來求.已知了線段MP與PN的長度之比為3:1,如果過P作x軸的垂線,根據(jù)平行線分線段成比例定理即可得出P點的縱坐標的值.進而可仿照(1)的方法,先代入直線的解析式,然后再代入拋物線中即可求出a的值,也就求出了拋物線的解析式.
解答:解:(1)∵拋物線的頂點是C(0,1),
∴b=0,c=1,
∴y=ax
2+1.
如圖1,∵a>0,直線l過點N(0,3),
∴M點在x軸正半軸上.
∵點P到x軸的距離為2,
即點P的縱坐標為2.
把y=2代入y=-ax+3
得,x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/0.png)
,
∴P點坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/1.png)
,2).
∵直線與拋物線交于點P,
∴點P在y=ax
2+1上,
∴2=a•(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/2.png)
)
2+1,
∴a=1.
∴直線l的函數(shù)關(guān)系式為y=-x+3.
(2)如圖1,若點P在y軸的右邊,記為P
1過點P
1作P
1A⊥x軸于A,
∵∠P
1MA=∠NMO,
∴Rt△MP
1A∽Rt△MNO,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/images3.png)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/3.png)
.
∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/4.png)
,
∴MP
1=3P
1N,MN=MP
1+P
1N=4P
1N
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/5.png)
,
即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/6.png)
,
∵ON=3,
∴P
1A=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/7.png)
,
即點P
1的縱坐標為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/8.png)
.
把y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/9.png)
代入y=-ax+3,
得x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/10.png)
,
∴點P
1的坐標為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/11.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/12.png)
).
又∵點P
1是直線l與拋物線的交點,
∴點P
1在拋物線y=ax
2+1上,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/13.png)
=a•(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/14.png)
)
2+1,
∴a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/15.png)
.
拋物線的函數(shù)關(guān)系式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/16.png)
x
2+1.
如圖2,若點P在y軸的左邊,記為P
2.作P
2A⊥x軸于A,
∵∠P
2MA=∠NMO,
∴Rt△MP
2A∽Rt△MNO,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/17.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/18.png)
.
∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/19.png)
,
∴MP
2=3P
2N,MN=MP
2-P
2N=2P
2N,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/images21.png)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/20.png)
,即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/21.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/22.png)
,
∵ON=3,
∴P
2A=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/23.png)
,即即點P
2的縱坐標為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/24.png)
.
由P
2在直線l上可求得P
2(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/25.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/26.png)
),
又∵P
2在拋物線上,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/27.png)
=a•(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/28.png)
)
2+1,
∴a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/29.png)
.
∴拋物線的函數(shù)關(guān)系式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101138391465112/SYS201311031011383914651022_DA/30.png)
x
2+1.
點評:本題主要考查了一次函數(shù)與二次函數(shù)解析式的確定以及函數(shù)圖象交點等知識.