2024九年级下·云南·专题练习
1 . 在平面直角坐标系中,
,
.
、
两点,且与
轴交于点
,求此抛物线的顶点坐标;
(2)如图,小敏发现所有过
、
两点的抛物线如果与
轴负半轴交于点
,
为抛物线的顶点,那么
与
的面积比不变,请你求出这个比值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d57173ef4cd72eb270686875dfd623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f36488f7dff759ace2a27e56c3857056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7597c6e4b7c7215651b1ef03bf2c1e11.png)
(2)如图,小敏发现所有过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dddb32a7a5c157fdf8aa049b2d665b4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90b40c2b0ab8e1cfe5112d428b4b829f.png)
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2 . 已知抛物线
与直线
.
(1)求证:直线与抛物线总有公共点;
(2)若直线与抛物线两交点的横坐标为
,
,且
,抛物线
与
轴交于
,
两点(
在
的右侧),点
在抛物线
上,且在直线
下方,连接
交
于点
,连接
,
,记
的面积为
,
的面积为
,
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a04f498af850423af9f8f808e601bafc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dba30672cd6bfd08367fddf4437d3e87.png)
(1)求证:直线与抛物线总有公共点;
(2)若直线与抛物线两交点的横坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f926c6da13074f338d7201efbd3fd750.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c89bdd4cadbcef0e3f96af0d53f8a570.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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3 . 如图,在平面直角坐标系中,二次函数
的图象与轴交于A,B点,与y轴交于点
,点B的坐标为
,点P是抛物线上一个动点.
(2)若P点在第一象限运动,当P运动到什么位置时,
的面积最大?请求出点P的坐标和
面积的最大值;
(3)连接
,并把
沿
翻折,那么是否存在点P,使四边形
为菱形;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58d88bbd34102b55fa928e8ff83f0d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/850e01debd266a398b597a1876dbb3ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb442fb1fe1a5d9768cf11c3e1b7ec5b.png)
(2)若P点在第一象限运动,当P运动到什么位置时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4234cf4d8159f5df0333bcc269ffe099.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4234cf4d8159f5df0333bcc269ffe099.png)
(3)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a952b29b279065f01066a5a6bc92d53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ee780a7d0e8811fb04d955304543fce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdc93e193fad261689949a52819753f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85ef67d769dd41ac295abf1f53f4d4c2.png)
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5卷引用:专题12 二次函数与几何问题(三)(四大热点题型归纳)-2024年中考数学二轮热点题型归纳与变式演练(云南专用)
(已下线)专题12 二次函数与几何问题(三)(四大热点题型归纳)-2024年中考数学二轮热点题型归纳与变式演练(云南专用)2023年山东省青岛市高新区中考数学二模模拟试题2024年广东省深圳市南山外国语学校中考二模数学试题2024年广东省深圳市南山区桃源中学中考二模数学试题数学(深圳卷)-【试题猜想】2024年中考考前最后一卷
4 . 如图,已知二次函数
的图象与直线
相交于点A和点B,点A在x轴上,点B在y轴上,抛物线的顶点为P.
(2)在直线
下方的抛物线上是否存在点Q,使得
,若存在,请求出点Q的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58d88bbd34102b55fa928e8ff83f0d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c608def11fa0e2b34f05592ef1d11fd8.png)
(2)在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22177a05152b82d6e26fa7acd109aa76.png)
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5 . 综合与探究
如图1,二次函数
的图象与
轴交于
,
(点
在点
的左侧)两点,与
轴交于点
.直线
经过
,
两点,连接
.
(2)在抛物线上是否存在除点
外的点
,使得
?若存在,请求出此时点
的坐标;若不存在,请说明理由.
(3)如图2,将
沿
轴正方向平移得到
(点A,O,C的对应点分别为
,
,
),
,
分别交线段
于点E,F,当
与
的面积相等时,请直接写出
与
重叠部分的面积.
如图1,二次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/114da91d9b7124b283b3edcd50daf60d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f590ca2bde213675bffe68ed4017f957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)在抛物线上是否存在除点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38397641e807f6b2286ecf8e5bebc095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(3)如图2,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bbf9680f74a9ac5d934304654ce2771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab206477d4305b2b7f3bd7a7040bd5c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12fe32dfbd66709875c5b9f79c9496da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24cfad1ba71a78d8f415335cde2f8c52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd36eaf776b9c76e6a5b13d7953c6399.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e642f209374b934bd85c61f135b6a4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2c8d46d5444746abfea21ab5cfcb55c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab206477d4305b2b7f3bd7a7040bd5c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7ffcd1925a2b1259221c6a476152f7.png)
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(已下线)专题11 二次函数与几何问题(二)(六大热点题型归纳)-2024年中考数学二轮热点题型归纳与变式演练(云南专用) 2024年山西省朔州市多校中考一模数学试题2024山西省中考模拟卷数学试题3
6 . 如图
,在平面直角坐标系中,直线
交两坐标轴于
、
两点,二次函数
图象经过
,
,
三点且
.
(2)在抛物线的对称轴上是否存在点
?使得
的长度最短
若存在,求出点
的坐标;若不存在,请说明理由.
(3)在直线上方抛物线上是否存在点
?使得
的面积有最大值
若存在,求出点
的坐标及此时
的面积;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/812eed46a589bde8b7c78a81a8cf9b9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90385c676848de67293e3ed6bc000fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/804bf446decc902afd6d8218bc65da5f.png)
(2)在抛物线的对称轴上是否存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a51a8d10345f567e0f2daf18529d42f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e1d0f65817ba32a732040518f41440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)在直线上方抛物线上是否存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe51b50eca45db5d8ca5f4949c56d137.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e1d0f65817ba32a732040518f41440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe51b50eca45db5d8ca5f4949c56d137.png)
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7 . 如图抛物线
经过点
,点
,与y轴交于点C,抛物线的顶点为D.
(2)抛物线上是否存在点p,使
的面积是
面积的3倍,若存在,请直接写出点p的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e59da5115d0dafea24822245f92c48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d57173ef4cd72eb270686875dfd623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/163775ddc1073c5742cdc8660637ac63.png)
(2)抛物线上是否存在点p,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c025ee3317be1099b7bf03a11e37ed4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
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3卷引用:专题10 二次函数与几何问题(一)(六大热点题型归纳)-2024年中考数学二轮热点题型归纳与变式演练(云南专用)
(已下线)专题10 二次函数与几何问题(一)(六大热点题型归纳)-2024年中考数学二轮热点题型归纳与变式演练(云南专用)2023年宁夏回族自治区吴忠市第三中学中考数学模拟试题(一)2023年宁夏吴忠市第三中学高中阶段招生模拟考试数学模拟预测题
8 . 如图,抛物线
过点
和点
,与
轴交于点
,抛物线的对称轴
交
轴于点
,交抛物线于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/15/b7ebbfcb-7180-4b07-b5f4-462422565f74.png?resizew=171)
(1)求抛物线的表达式及
点的坐标;
(2)点
是直线
上的点,若
的面积与
的面积相等,求点
的坐标;
(3)点
在第四象限,且为抛物线上的点,若四边形
是梯形,求点
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e59da5115d0dafea24822245f92c48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/311497849126f1aaf1da0ec75602eabf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76d03fa28c117649b0fdfe17eed7b583.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/15/b7ebbfcb-7180-4b07-b5f4-462422565f74.png?resizew=171)
(1)求抛物线的表达式及
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(3)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f293d54f30e7ffbe3b84b357262e65e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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9 . 如图,在平面直角坐标系中,抛物线
与直线
交于A、C两点,直线l与y轴交于点B,抛物线上的动点D在直线l的上方.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/8/e07ef1dc-6981-4c05-93ed-f84b1cca9ba5.png?resizew=446)
(1)请直接写出点A、点C的坐标;
(2)连接
、
,当三角形
的面积最大时,求点D的坐标?
(3)连接
、
,当三角形
以
边为直角边时,求点D的坐标?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e442a4a5aad13013e653ba19d48d133c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4597737b444f53a1b83c097885c48fc7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/8/e07ef1dc-6981-4c05-93ed-f84b1cca9ba5.png?resizew=446)
(1)请直接写出点A、点C的坐标;
(2)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
(3)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
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10 . 如图,在平面直角坐标系中,抛物线
与x轴交于
,
两点,交y轴于点C.
的解析式;
(2)设抛物线
关于坐标原点对称的抛物线为
,点A,B的对应点分别为
,
,抛物线
的顶点为E.则在x轴下方的抛物线
上是否存在点F,使得
.若存在,求出F点的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/994b512beacb73fb44b3ee3b8606f498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d57173ef4cd72eb270686875dfd623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f36488f7dff759ace2a27e56c3857056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)设抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddaa7f60a49df04948968746e66c4bcd.png)
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