题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061905-14277.png)
(1)求抛物线的解析式.
(2)若点D(2,2)是抛物线上一点,那么在抛物线的对称轴上,是否存在一点P,使得△BDP的周长最小,若存在,请求出点P的坐标,若不存在,请说明理由.
注:二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061905-17328.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-48217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-55480.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-54529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-62337.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-94122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-92138.png)
解析
将C(0,3)代入
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-15280.png)
将A(-2,0)代入
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-71180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-75101.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
∴抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-94122.png)
(2)如图:连接AD,与对称轴相交于P,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-34188.png)
由于点A和点B关于对称轴对称,则即BP+DP=AP+DP,当A、P、D共线时BP+DP=AP+DP最小。
设AD的解析式为y=kx+b,
将A(-2,0),D(2,2)分别代入解析式得,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061908-37623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061908-87689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
∵二次函数的对称轴为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061908-70310.png)
∴当x=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-92138.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061906-85593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-92138.png)
(1)根据OC=3,可知c=3,于是得到抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061907-71180.png)
(2)由于BD为定值,则△BDP的周长最小,即BP+DP最小,由于点A和点B关于对称轴对称,则即BP+DP=AP+DP,当A、P、D共线时BP+DP=AP+DP最小。
核心考点
试题【 如图,抛物线与x轴交于A、B两点,与y轴交于点C,且OA=2,OC=3.(1)求抛物线的解析式.(2)若点D(2,2)是抛物线上一点,那么在抛物线的对称轴上,】;主要考察你对二次函数定义等知识点的理解。[详细]
举一反三
(1)求此抛物线的解析式;
(2)写出顶点坐标及对称轴;
(3)若抛物线上有一点B,且
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061859-50825.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061900-60871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061852-40313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061852-97815.png)
①
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061853-38489.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061853-44645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061853-93142.png)
③
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061853-14850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061853-74533.png)
⑤当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061854-68927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061854-59979.png)
其中,正确的说法有 (请写出所有正确说法的序号).
(1)试确定此二次函数的解析式;
(2)求出抛物线的顶点C的坐标;
(3)判断点P(-2,3)是否在这个二次函数的图象上?如果在,请求出△PAB的面积;如果不在,试说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061843-44274.png)
X | … | -1 | 0 | 1 | 3 | … |
y | … | -3 | 1 | 3 | 1 | … |
则下列判断中正确的是( )
A.抛物线开口向上 B.抛物线与y轴交于负半轴
C.当x=4时,y>0 D.方程
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061844-78092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061830-11373.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061830-13880.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-44299.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-61756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-78396.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-32527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-54561.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-36268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-10229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-94283.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-68072.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-49676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-32527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-95756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-20348.png)
(1)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061831-49676.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-64253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-20348.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-64253.png)
(3)若在直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061830-11373.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-49057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-20400.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-29685.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-64253.png)
(4)在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-64253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-48897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-64253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061832-97220.png)
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