题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061624-15367.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061624-38669.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061624-38669.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061624-38669.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-11799.jpg)
(1)求这条抛物线的解析式;
(2)P为线段OM上一点,过点P作PQ⊥
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061624-38669.png)
(3)随着点P的运动,四边形PQCO的面积S有最大值吗?如果S有最大值,请求出S的最大值并指出点Q的具体位置和四边形PQCO的特殊形状;如果S没有最大值,请简要说明理由;
(4)随着点P的运动,是否存在t的某个值,能满足PO=OC?如果存在,请求出t的值。
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-27539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-64212.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-48798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-21644.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-58094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-10870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-65505.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-49053.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061627-98142.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061627-39695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061627-64864.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061627-36858.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061627-71356.png)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061628-51483.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061628-93005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061628-88572.png)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061628-47963.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061628-93005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061628-77058.png)
∴设抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061629-94961.png)
将点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061629-88188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061629-39913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061629-28658.png)
∴抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061629-16549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-27539.png)
(2)设直线OM的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061629-65482.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061630-27069.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061630-35499.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061630-90428.png)
则点P
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061630-39155.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061630-40867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061631-29097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061631-40955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061631-58406.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061631-83309.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-48798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-64212.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-48798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-21644.png)
(1)随着点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-79754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-70050.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-41404.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-80624.png)
从图像可看出,随着点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-79754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-35556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-80457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-78767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-78767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-41404.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-65252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-80457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-41404.png)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-45322.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-58094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-10870.png)
因而
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-15625.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-45322.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-36608.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-86806.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-98631.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061632-70050.png)
(4)随着点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061633-65252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-65505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-25181.png)
设点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061635-34770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061635-75689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061635-20142.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061635-22958.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-25181.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061636-15603.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061636-38801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061625-21644.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061636-29507.png)
∴当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061626-65505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061634-25181.png)
(1)x=O和x=4时,y的值相等,即可得到函数的对称轴是x=2,把x=2和x=3分别代入直线y=4x-16就可以求出抛物线上的两个点的坐标,并且其中一点是顶点,利用待定系数法,设出函数的顶点式一般形式,就可以求出函数的解析式;
(2)根据待定系数法可以求出直线OM的解析式,设OQ的长为t,即P,Q的横坐标是t,把x=t代入直线OM的解析式,就可以求出P点的纵坐标,得到PQ的长,四边形PQCO的面积S=S△COQ+S△OPQ,很据三角形的面积公式就可以得到函数解析式;
(3)从图象可看出,随着点P由O→M运动,△COQ的面积与△OPQ的面积在不断增大,即S不断变大,显当然点P运动到点M时,S最值;
(4)在直角△OPQ中,根据勾股定理就可以求出点P的坐标.
核心考点
试题【如图,抛物线与轴交于A、B两点(点A在点B左侧),与y轴交于点C,且当=O和=4时,y的值相等。直线y=4x-16与这条抛物线相交于两点,其中一点的横坐标是3,】;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
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061619-95717.png)
A.开口向下,顶点坐标![]() | B.开口向上,顶点坐标![]() |
C.开口向下,顶点坐标![]() | D.开口向上,顶点坐标![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061612-32561.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061612-73176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061609-86650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061609-17476.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061603-70944.png)
(1)求抛物线解析式及顶点坐标;
(2)设点E(x,y)是抛物线上一动点,且位于第四象限,四边形OEAF是以OA为对角线的平行四边形,求四边形OEAF的面积S与x之间的函数关系式,并写出自变量x的取值范围;
(3)①当四边形OEAF的面积为24时,请判断OEAF是否为菱形?
②是否存在点E,使四边形OEAF为正方形?若存在,求出点E的坐标;若不存在,请说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061603-28725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019061600-22844.png)
A.(4,0) | B.(-4,0) | C.(0,-4) | D.(0,4) |
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