题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045227-55208.png)
(1)求抛物线解析式及点D坐标;
(2)点E在x轴上,若以A,E,D,P为顶点的四边形是平行四边形,求此时点P的坐标;
(3)过点P作直线CD的垂线,垂足为Q,若将△CPQ沿CP翻折,点Q的对应点为Q′.是否存在点P,使Q′恰好落在x轴上?若存在,求出此时点P的坐标;若不存在,说明理由.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045227-97615.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-52402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-41014.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-61200.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-56914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045229-74431.png)
解析
试题分析:(1)∵抛物线y=ax2+bx+2经过A(﹣1,0),B(4,0)两点,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045229-58173.png)
解得:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045229-89018.png)
∴y=﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045227-97615.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-52402.png)
当y=2时,﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045229-51446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-52402.png)
即:点D坐标为(3,2).
(2)A,E两点都在x轴上,AE有两种可能:
①当AE为一边时,AE∥PD,
∴P1(0,2),
②当AE为对角线时,根据平行四边形对顶点到另一条对角线距离相等,
可知P点、D点到直线AE(即x轴)的距离相等,
∴P点的纵坐标为﹣2,
代入抛物线的解析式:﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
解得:x1=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-25271.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-79166.png)
∴P点的坐标为(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-79166.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-61200.png)
综上所述:p1(0,2);p2(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-41014.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-61200.png)
(3)存在满足条件的点P,显然点P在直线CD下方,设直线PQ交x轴于F,
点P的坐标为(a,﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-35256.png)
①当P点在y轴右侧时(如图1),CQ=a,
PQ=2﹣(﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
又∵∠CQ′O+∠FQ′P=90°,∠COQ′=∠Q′FP=90°,
∴∠FQ′P=∠OCQ′,
∴△COQ′~△Q′FP,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045231-16492.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045231-45233.png)
∴Q′F=a﹣3,
∴OQ′=OF﹣Q′F=a﹣(a﹣3)=3,CQ=CQ′=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045231-19392.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045231-44723.png)
此时a=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045231-15187.png)
②当P点在y轴左侧时(如图2)此时a<0,,﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
PQ=2﹣(﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-84000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045230-69630.png)
又∵∠CQ′O+∠FQ′P=90°,∠CQ′O+∠OCQ′=90°,
∴∠FQ′P=∠OCQ′,∠COQ′=∠Q′FP=90°,
∴△COQ′~△Q′FP,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045232-20101.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045232-69320.png)
∴OQ′=3,
CQ=CQ′=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045232-45857.png)
此时a=﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045232-60193.png)
综上所述,满足条件的点P坐标为(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-56914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045228-30227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045229-74431.png)
点评:本题考查二次函数,相似三角形,本题需要考生掌握待定系数法,会用待定系数法求解析式,掌握相似三角形的判定方法,会证明两个三角形相似
核心考点
试题【如图,抛物线y=ax2+bx+2交x轴于A(﹣1,0),B(4,0)两点,交y轴于点C,与过点C且平行于x轴的直线交于另一点D,点P是抛物线上一动点.(1)求抛】;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/20191019/20191019045213-63336.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045213-91402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-59929.png)
x | …… | 0 | 1 | 2 | 3 | 4 | 5 | …… |
y | …… | 4 | 1 | 0 | 1 | 4 | 9 | …… |
(2)点A(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-38541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-82437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-97069.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-37190.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-91779.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-63016.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-82437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045214-37190.png)
(3)若将此图象沿x轴向右平移3个单位,请写出平移后图象所对应的函数关系式: ;
(4)设点P1(m,y1)、P2(m+1,y2)、P3(m+2,y3)都在二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045213-63336.png)
温度![]() | …… | -4 | -2 | 0 | 2 | 4 | 4.5 | …… |
植物每天高度增长量![]() | …… | 41 | 49 | 49 | 41 | 25 | 19.75 | …… |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045204-95024.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045204-95659.png)
(1)请你选择一种适当的函数,求出它的函数关系式,并简要说明不选择另外两种函数的理由;
(2)温度为多少时,这种植物每天高度的增长量最大?
(3)如果实验室温度保持不变,在10天内要使该植物高度增长量的总和超过250mm,那么实验室的温度
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045204-95659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045152-70161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045152-18711.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045153-48877.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045153-33285.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045153-13876.jpg)
(1)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045153-48877.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045153-44368.png)
(2)①若点P的坐标为(-2,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045154-76641.png)
②试证明:对于直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045152-70161.png)
(3)设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045152-70161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045154-39602.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045147-82304.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045147-82304.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045147-13327.png)
(1)若图①仅看作符合条件的一种情况,求出所有符合条件的点D的坐标;
(2)在图①中,若点P以每秒1个单位长度的速度沿直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019045147-82304.png)
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