【答案】
分析:(1)根據(jù)以點(diǎn)G(4,0)為圓心,2為半徑的圓與x軸交于A、B兩點(diǎn),求得點(diǎn)A的坐標(biāo)為(2,0),點(diǎn)B的坐標(biāo)為(6,0),然后代入拋物線y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/0.png)
x
2+bx+c求得函數(shù)的解析式即可;
(2)首先求得拋物線與y軸的交點(diǎn)點(diǎn)C的坐標(biāo),然后將y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/1.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/2.png)
x-2配方成y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/3.png)
(x-4)
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/4.png)
的形式,從而求得頂點(diǎn)坐標(biāo),即可作出函數(shù)的圖象;
(3)根據(jù)F(8,m)在拋物線y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/5.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/6.png)
x-2上,求得點(diǎn)F的坐標(biāo),連接AF,則與拋物線的對(duì)稱軸的交點(diǎn)為點(diǎn)P,此時(shí)PF+PB的最小,然后利用勾股定理求得AF的長(zhǎng)即為最小值;
(4)連接EG,根據(jù)OE是⊙G的切線,得到∠OEG=90°,然后利用勾股定理求得OE的長(zhǎng)即可,進(jìn)而得出E點(diǎn)坐標(biāo),求出即可.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/images7.png)
解:(1)∵以點(diǎn)G(4,0)為圓心,2為半徑的圓與x軸交于A、B兩點(diǎn),
點(diǎn)A的坐標(biāo)為(2,0),點(diǎn)B的坐標(biāo)為(6,0),
∵拋物線y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/7.png)
x
2+bx+c過(guò)點(diǎn)A和點(diǎn)B,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/8.png)
,
解得:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/9.png)
,
∴此拋物線的函數(shù)關(guān)系式為:y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/10.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/11.png)
x-2;
(2)∵C點(diǎn)為拋物線與y軸的交點(diǎn),
∴當(dāng)x=0時(shí),y=-2,
∴點(diǎn)C的坐標(biāo)為(0,-2);
∵y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/12.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/13.png)
x-2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/14.png)
(x
2-8x)-2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/15.png)
(x-4)
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/16.png)
,
∴此拋物線的頂點(diǎn)坐標(biāo)為(4,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/17.png)
),如圖:
(3)∵點(diǎn)F(8,m)在拋物線y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/18.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/19.png)
x-2上,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/images21.png)
∴點(diǎn)F的坐標(biāo)為(8,-2),
連接AF,則與拋物線的對(duì)稱軸的交點(diǎn)為點(diǎn)P,此時(shí)PF+PB的最小,
∴PA=PB,
∴PF+PB=PA+PF=AF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/20.png)
=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/21.png)
;
∴PF+PB的最小值為2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/22.png)
;
(4)連接EG,作ER⊥OB,ET⊥y軸,
∴EG=2,
∵OE是⊙G的切線,
∴∠OEG=90°,
∴OE=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/23.png)
.
∵EG=2,OG=4,
∴∠EOG=30°,
∴∠EGO=90°-∠EOG=90°-30°=60°,
∴RG=1,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/images26.png)
∴ER=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/24.png)
,OR=3,
∴ET=3,
∴△COE的面積為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/25.png)
×2×3=3,
∴△COQ的面積為3,
當(dāng)Q點(diǎn)橫坐標(biāo)為3時(shí),
y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/26.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/27.png)
x-2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/28.png)
;
∴Q點(diǎn)的坐標(biāo)為:(3,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/29.png)
),
當(dāng)Q點(diǎn)橫坐標(biāo)為-3時(shí),
y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/30.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/31.png)
x-2=0;
y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/32.png)
,
∴Q點(diǎn)的坐標(biāo)為:(-3,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/33.png)
),
∴點(diǎn)Q的坐標(biāo)為:(-3,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/34.png)
),(3,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022162451663441774/SYS201310221624516634417023_DA/35.png)
).
點(diǎn)評(píng):本題考查了二次函數(shù)的綜合知識(shí),特別是題目中與幾何知識(shí)結(jié)合起來(lái),更是中考中的常見(jiàn)考題.