题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045442-68549.png)
(1)如图1,求抛物线y=(x﹣2)2+1的伴随直线的表达式.
(2)如图2,若抛物线y=a(x﹣m)2+n(m>0)的伴随直线是y=x﹣3,伴随四边形的面积为12,求此抛物线的表达式.
(3)如图3,若抛物线y=a(x﹣m)2+n的伴随直线是y=﹣2x+b(b>0),且伴随四边形ABCD是矩形.用含b的代数式表示m、n的值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-83262.png)
(2)抛物线的表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-13510.png)
(3)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-34584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-64058.png)
解析
试题分析:(1)由题意可知:A(0,5),B(2,1),
设伴随直线AB的表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-92629.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045444-10721.png)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045444-14174.png)
∴抛物线y=(x﹣2)2+1的伴随直线的表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-83262.png)
(2)令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045444-27606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045444-37051.png)
由题意可知:顶点B(m,n)在伴随直线y=x﹣3上,
∴n=m-3,
∴B(m,m-3),
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045445-39971.png)
∵点A、B关于原点O的对称点分别为C、D,
∴C(0,3) ,D(-m,-m+3),
过点B作
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045445-62769.png)
∵ m>0,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045445-63651.png)
∵伴随四边形ABCD的面积为12,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045445-95368.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045445-47547.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045446-54980.png)
∴B(2,-1),
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045446-45573.png)
把A(0,-3)代入
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045446-45573.png)
得:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045446-69555.png)
∴抛物线的表达式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-13510.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045446-53438.png)
(3)∴伴随直线AB;y=﹣2x+b(b>0)与x轴、y轴分别交于点F (
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045446-11602.png)
∴C(0,-b)
∵伴随四边形ABCD是矩形,
∴顶点B(m,n)在y轴右侧的直线y=﹣2x+b上,
∠ABC=90º,
∴B(m,-2m+b),
过点B作
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045445-62769.png)
∴E(0,-2m+b),
∴tan
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045447-89993.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045447-79254.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045447-40368.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-34584.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045443-64058.png)
点评:本题考查一次函数,二次函数,矩形,解答本题的关键是用待定系数法求一次函数,二次函数的解析式子,熟悉矩形的性质,本题难度较大
核心考点
试题【已知抛物线y=a(x﹣m)2+n与y轴交于点A,它的顶点为点B,点A、B关于原点O的对称点分别为C、D.若A、B、C、D中任何三点都不在一直线上,则称四边形AB】;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
x | 0 | 1 | 2 | 3 | 4 |
y | 3 | 0 | -2 | 0 | 3 |
A.y=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045438-39630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045439-28007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045439-91261.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045434-77541.jpg)
A.1 | B.2 | C.3 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-49039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-55584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-62986.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-49039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-24754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-12618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-24754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-12618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-55875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-30418.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-62166.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-75085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-21114.png)
(1)求出一元二次函数的关系式;
(2)点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-86987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-35470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-86987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045427-49039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-91300.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-25941.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045428-45484.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-32952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-77106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-77106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-71185.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-71185.png)
(3)在(2)的条件下,当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-39343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045429-32952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-24383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-48716.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-26403.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-33727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-95230.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045416-24817.jpg)
(1)当顶点A运动至与原点重合时,顶点C是否在该抛物线上?
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-48716.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-26403.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045416-65240.png)
(3)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045415-48716.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045411-31500.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045411-59380.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045411-60587.png)
A.第一象限 | B.第二象限 | C.第三象限 | D.第四象限 |
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