题目
题型:四川省中考真题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165822-38125.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165822-54537.gif)
(1)若点C是点B关于x轴的对称点,求经过O、C、A三点的抛物线的函数表达式;
(2)在(1)中,抛物线上是否存在一点P,使以P、O、C、A为顶点的四边形为梯形?若存在,求出点P的坐标;若不存在,请说明理由;
(3)若将点O、点A分别变换为点Q(-2k ,0)、点R(5k,0)(k>1的常数),设过Q、R两点,且以QR的垂直平分线为对称轴的抛物线与y轴的交点为N,其顶点为M,记△QNM的面积为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165822-77091.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165823-35669.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165823-44060.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165823-61010.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165823-87877.gif)
答案
在
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165823-73475.gif)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165824-51456.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165824-39206.gif)
又由勾股定理,得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165824-65626.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165824-67743.gif)
∵点B在第一象限内,
∴点B的坐标为(4,3),
∴点B关于x轴对称的点C的坐标为(4,-3),
设经过
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165825-90523.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165825-56081.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165825-20302.gif)
∴经过
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165825-23262.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165825-22698.gif)
(2)假设在(1)中的抛物线上存在点P,使
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165826-56112.gif)
①∵点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165826-19767.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165826-49651.gif)
∴过点C作直线OA的平行线与抛物线交于点P1
则直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165826-51284.gif)
对于
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165826-34402.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165827-54645.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165827-97282.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165827-26186.gif)
而点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165827-98920.gif)
在四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165828-22256.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165828-16493.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165828-15476.gif)
∴点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165828-26788.gif)
②若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165828-24106.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165829-43601.gif)
将点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165829-71326.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165829-91197.gif)
∴直线CO的函数表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165829-69271.gif)
于是可设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165830-15386.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165830-52208.gif)
将点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165830-57149.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165830-78862.gif)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165830-93664.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165831-86355.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165831-44488.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165831-61408.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165831-76878.gif)
而点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165832-90570.gif)
过点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165832-71711.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165832-90141.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165832-13268.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165833-27823.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165833-99908.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165833-52919.gif)
∴在四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165833-69108.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165834-34311.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165834-97564.gif)
∴点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165834-66391.gif)
③若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165834-50497.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165834-96485.gif)
将点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165835-78648.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165835-56991.gif)
∴直线CA的函数表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165835-11961.gif)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165835-95436.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165836-71097.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165836-45563.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165836-66763.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165836-62295.gif)
而点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165837-14835.gif)
过点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165837-72150.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165837-14504.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165837-61984.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165838-51012.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165838-98169.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165838-48339.gif)
∴在四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165838-85501.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165839-59577.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165839-25218.gif)
∴点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165839-76071.gif)
综上可知,在(1)中的抛物线上存在点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165839-31289.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165839-15359.gif)
(3)由题知,抛物线的开口可能向上,也可能向下,
①当抛物线开口向上时,则此抛物线与y轴的负半轴交于点N,
可设抛物线的函数表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165840-79339.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165840-49112.gif)
如图,过点M作
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165840-15268.gif)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165840-32259.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165841-18220.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165841-39579.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165841-29824.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165841-74612.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165842-80077.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165842-26309.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165842-53260.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165842-37068.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165842-28271.gif)
②当抛物线开口向下时,则此抛物线与y轴的正半轴交于点N,
同理,可得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165843-48697.gif)
综上可知,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165843-98166.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165843-63834.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165843-15460.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165844-60461.gif)
核心考点
试题【如图,在平面直角坐标系xOy中,△OAB的顶点A的坐标为(10,0),顶点B在第一象限内,且,sin∠OAB=。(1)若点C是点B关于x轴的对称点,求经过O、C】;主要考察你对二次函数的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165814-23702.gif)
(2)求此抛物线的表达式;
(3)求△ABC的面积;
(4)若点E是线段AB上的一个动点(与点A、点B不重合),过点E作EF∥AC交BC于点F,连接CE,设AE的长为m,△CEF的面积为S,求S与m之间的函数关系式,并写出自变量m的取值范围;
(5)在(4)的基础上试说明S是否存在最大值,若存在,请求出S的最大值,并求出此时点E的坐标,判断此时△BCE的形状;若不存在,请说明理由。
(1)设x天后每千克该野生菌的市场价格为y元,试写出y与x之间的函数关系式;
(2)若存放x天后,将这批野生菌一次性出售,设这批野生菌的销售总额为P元,试写出P与x之间的函数关系式;
(3)李经理将这批野生茵存放多少天后出售可获得最大利润W元?(利润=销售总额-收购成本-各种费用)
(1)若△AOB的外接圆与y轴交 于点D,求D点坐标;
(2)若点C的坐标为(-1,0),试猜想过D、C的直线与△AOB的外接圆的位置关系,并加以说明;
(3)二次函数的图象经过点O和 A且顶点在圆上,求此函数的解析式。
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165743-29549.gif)
(1)求C1的坐标;
(2)求经过三点O,C1,C的抛物线的解析式;
(3)若⊙P的半径为R,圆心P在(2)的抛物线上运动,⊙P与两坐标轴都相切时,求⊙P半径R的值。
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020165743-80176.gif)
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