题目
题型:福建省期末题难度:来源:
(1)求抛物线的解析式;
(2)设抛物线的顶点为D,点P在抛物线的对称轴上,且
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192316-85951.gif)
(3)点Q在直线BC上方的抛物线上,且点Q到直线BC的距离最远,求点Q坐标.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192316-38723.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192317-17032.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192317-97736.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192317-27937.gif)
∴抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192317-97127.gif)
(2) 由.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192318-53740.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192318-39367.gif)
可得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192318-61567.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192318-84209.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192319-82968.gif)
如图1,设抛物线对称轴与轴交于点F,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192319-95911.gif)
过点A作
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192319-16856.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192319-96423.gif)
可得,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192320-99783.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192320-31620.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192320-64894.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192320-74371.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192321-39742.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192321-91169.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192321-73027.gif)
解得PF=2.
点P在抛物线的对称轴上,点P的坐标为(-2,2)或(-2,-2).
(3)设直线BC的解析式,直线BC经过B(-3,0),C(0,-3),
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192321-83726.gif)
解得:k=-1,b=-3, ∴直线BC的解析式
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192321-86717.gif)
设点Q(m,n),过点Q作QH⊥BC于H,并过点Q作 QS∥y轴交直线BC于点S,
则S点坐标为(m,-m-3) ∴QS=n-(-m-3)=n+m+3
∵点Q(m,n)在抛物线y=-x2-4x-3上,
∴n=-m2-4m-3
∴QS=-m2-4m-3+m+3
=-m2-3m
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192322-72372.gif)
当m=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192322-44795.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192322-88675.gif)
∵BO=OC,∠BOC=90°,
∴∠OCB=45°
∵QS∥y轴, ∴∠QSH=45°
∴△QHS是等腰直角三角形
∴当斜边QS最大时QH最大.
∵当m=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192322-98235.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192323-76145.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192323-45942.gif)
∴Q(
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192323-51880.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192323-53536.gif)
∴Q点的坐标为(
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192324-31893.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192324-80603.gif)
核心考点
试题【在平面直角坐标系中,抛物线与轴交于A(-1,0),B(-3,0)两点,与轴交于点.(1)求抛物线的解析式;(2)设抛物线的顶点为D,点P在抛物线的对称轴上,且,】;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/20191020/20191020192303-37657.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192303-98404.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192303-22459.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192303-29613.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-17646.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-36094.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-17982.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-35023.gif)
(1)试直接写出D点的坐标;
(2)已知点B与点D在经过原点的抛物线上,点P在第一象限内的该抛物线上移动,过P点作
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-67312.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-16073.gif)
①若以O、P、Q为顶点的三角形与
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-33171.gif)
②试问在抛物线的对称轴上是否存在一点T,使得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192304-15588.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192305-22789.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192253-63080.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192253-76991.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192253-97149.gif)
(2)试求水池的总容积V与
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192253-43052.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192254-75351.gif)
(3)如果房屋的墙壁可利用的长度为10.5m,请利用函数图象与性质求V的最大值。
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192240-87184.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192240-28425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192240-57420.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192241-41948.gif)
(2)点P从点C出发沿着CB以每秒1个单位长度的速度向点B匀速运动,过点P作PE∥AC交AB于B,设运动时间为t秒。用含t的代数式表示△PBE的面积S;
(3)在(2)的条件下点P的运动过程中,将△PBE沿着PE折叠(如图所示),点B在平面内的落点为
点D。当△PDE与△ABC重叠部分的面积等于
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192241-96266.gif)
(1)求直线和抛物线的解析式;
(2)设抛物线的顶点为D,连接DA、DB,求S△DAB?
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020192235-54831.gif)
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