抛物线
与
轴交于
、
两点.与
轴交于点
、点
在抛物线上.
![](https://img.xkw.com/dksih/QBM/2022/11/22/3114868951801856/3117927391846400/STEM/5f41b982a387409ca53bca9bc292ec9b.png?resizew=530)
(1)求抛物线的解析式.
(2)如图1,连接
、
,点
在对称轴左侧的抛物线上,若
,求点
的坐标.
(3)如图2,过点
的直线
,点
是直线
上方抛物线上一动点,过点
作
,垂足为点
,连接
,
,
,
.当四边形
的面积最大时,求点
的坐标及四边形
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96dde9c4aad0536c069127df0d4b12f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/311497849126f1aaf1da0ec75602eabf.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/62855ff21228ecea8e28fd0fbeb814ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00948ef06a3e3d6853e626ae47a25e6.png)
![](https://img.xkw.com/dksih/QBM/2022/11/22/3114868951801856/3117927391846400/STEM/5f41b982a387409ca53bca9bc292ec9b.png?resizew=530)
(1)求抛物线的解析式.
(2)如图1,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a458aa4e438a9b1e6cd54827c30ce76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)如图2,过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3b129de312166ddc54f822b3c88be9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9e312bc0401546bc01231f7720a928a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f53330c107f8245290a5a42c3d356acd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf702adb116c1e46569eb7050d029f49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de6b96f44e4f6bfc49f9abd52cbf4971.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23cb23f9075fe81e63edfa600f0dffd0.png)
更新时间:2022-11-26 13:42:25
|
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【推荐1】如图,已知抛物线
交x轴的正半轴于点A,交y轴于点B,动点
在x轴上,过点C作x轴的垂线交线段
于点D,交该抛物线于点P,连接
交
于点E.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/25/355fbb58-6b77-40bf-84f6-a08ff909cf5d.png?resizew=152)
(1)求点A,B的坐标.
(2)当
时,求线段
的长.
(3)当
是以
为腰的等腰三角形时,求m的值.(直接写出答案即可)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ce7d85ec793a1055adb54ba02164ec4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0335684e3cb39e6920971150b03807e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/25/355fbb58-6b77-40bf-84f6-a08ff909cf5d.png?resizew=152)
(1)求点A,B的坐标.
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94f16d5ed858699bfea5039a7bf8ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022f0dc17db5fcc83b204dc845447ffc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
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【推荐2】一条笔直的公路上依次有A、B、C三地,甲车从A地驶往C地,乙车从A地驶往B地,两车同时出发并以各自的速度匀速行驶,乙车中途因汽车故障停下来修理,修好后立即以原速的两倍继续前进到达B地:如图是甲、乙两车与A地的距离y(千米)与出发时间x(小时)之间的大致图象.
(1)直接写出A、C两地之间的距离______
;B、C两地之间的距离______
;
(2)写出乙车修理好之后与A地的距离y(千米)与出发时间x(小时)之间的关系式______.
(3)乙_____小时追上甲;
(4)当两车相距40千米时,甲车行驶了______小时.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/3/284da33f-f2fa-4292-8b2a-fd203537c1b9.png?resizew=317)
(1)直接写出A、C两地之间的距离______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5fe30c67ac20cd4e8b9cc2d0d420a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5fe30c67ac20cd4e8b9cc2d0d420a7b.png)
(2)写出乙车修理好之后与A地的距离y(千米)与出发时间x(小时)之间的关系式______.
(3)乙_____小时追上甲;
(4)当两车相距40千米时,甲车行驶了______小时.
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【推荐1】如图,抛物线与
轴交于
、
两点,与
轴交于点
,且
,
,
.
(1)求抛物线的解析式;
(2)
为
的中点,动点
从
点出发,先到达
轴上的点
,再走到抛物线对称轴上的点
,最后返回到点
.要使动点
走过的路程最短,请找出点
、
的位置,写出坐标,并求出最短路程.
(3)点Q是抛物线上位于x轴上方的一点,点R在x轴上,是否存在以点Q为直角顶点的等腰直角三角形
?若存在,求出点
的坐标,若不存在,请说明理由.
![](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/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9774f83067ed956a551bc41adcce0469.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9466d03bc916a9169eaf39863d59fceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c26ea9e9e42ced4f9b7540b368fbd171.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/27/99ad01fa-496d-4310-b488-90c73671d5be.png?resizew=465)
(1)求抛物线的解析式;
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5a122e25cf4eb9f03ffe5ec823bfc31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(3)点Q是抛物线上位于x轴上方的一点,点R在x轴上,是否存在以点Q为直角顶点的等腰直角三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866d879b5b894173c3029e9192d6bc29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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【推荐2】在平面直角坐标系xOy中,已知抛物线
与x轴交于A(1,0),B(3,0),与y轴交于点C(0,-3).
![](https://img.xkw.com/dksih/QBM/2022/4/28/2974195992109056/2981388120842240/STEM/d88c79b2-67b6-499a-9c31-d88a607b6b62.png?resizew=353)
(1)求抛物线的函数表达式;
(2)点D为抛物线上一点,过点D作x轴的垂线,垂足为E,若
,求点D的坐标;
(3)点P为抛物线上一点,若
,求点P的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90385c676848de67293e3ed6bc000fe.png)
![](https://img.xkw.com/dksih/QBM/2022/4/28/2974195992109056/2981388120842240/STEM/d88c79b2-67b6-499a-9c31-d88a607b6b62.png?resizew=353)
(1)求抛物线的函数表达式;
(2)点D为抛物线上一点,过点D作x轴的垂线,垂足为E,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/859d442b82760f9fe529f7098def3c21.png)
(3)点P为抛物线上一点,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b44ac8c782bf34396ff8c8d2fe6022b6.png)
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【推荐1】如图,已知抛物线
与x轴交于点A(1,0)和B,与y轴交于点C,对称轴为
.
![](https://img.xkw.com/dksih/QBM/2021/6/17/2745182325661696/2745543158800384/STEM/3bd35957-0192-4f33-a118-77fdedc3beca.png?resizew=463)
(1)求抛物线的解析式;
(2)如图1,若点P是线段BC上的一个动点(不与点B,C重合),过点P作y轴的平行线交抛物线于点Q,连接OQ.当线段PQ长度最大时,判断四边形OCPQ的形状并说明理由.
(3)如图2,在(2)的条件下,D是OC的中点,过点Q的直线与抛物线交于点E,且
.在y轴上是否存在点F,使得
为等腰三角形?若存在,求点F的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86f94568a5d4e4865c5e9de9995c3a69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a61d77911527508524874b212a0937d.png)
![](https://img.xkw.com/dksih/QBM/2021/6/17/2745182325661696/2745543158800384/STEM/3bd35957-0192-4f33-a118-77fdedc3beca.png?resizew=463)
(1)求抛物线的解析式;
(2)如图1,若点P是线段BC上的一个动点(不与点B,C重合),过点P作y轴的平行线交抛物线于点Q,连接OQ.当线段PQ长度最大时,判断四边形OCPQ的形状并说明理由.
(3)如图2,在(2)的条件下,D是OC的中点,过点Q的直线与抛物线交于点E,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/163a88553a193e3e31f42a42def91c65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2ea13010e2399194be2a681310543e.png)
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【推荐2】阅读材料:
已知两数的和为4,求这两个数的积的最大值.
(1)解:设其中一个数为x,则另一个数为(4﹣x),令它们的积为y,则:
y=x(4﹣x)
=﹣x2+4x
=﹣(x﹣2)2+4.
∵﹣1<0,
∴y最大值=4.
问题解决:
(1)若一个矩形的周长为20cm,则它面积的最大值为 cm2.
(2)观察下列两个数的积,猜想哪两个数积最大,并用二次函数的知识说明理由:
99×1.98×2.97×3.96×4,…,50×50.
拓展应用:
(3)若m、n为任意实数,则代数式(m﹣2n)(8﹣m+2n)的最大值是 ,此时,m和n之间的关系式是 .
已知两数的和为4,求这两个数的积的最大值.
(1)解:设其中一个数为x,则另一个数为(4﹣x),令它们的积为y,则:
y=x(4﹣x)
=﹣x2+4x
=﹣(x﹣2)2+4.
∵﹣1<0,
∴y最大值=4.
问题解决:
(1)若一个矩形的周长为20cm,则它面积的最大值为 cm2.
(2)观察下列两个数的积,猜想哪两个数积最大,并用二次函数的知识说明理由:
99×1.98×2.97×3.96×4,…,50×50.
拓展应用:
(3)若m、n为任意实数,则代数式(m﹣2n)(8﹣m+2n)的最大值是 ,此时,m和n之间的关系式是 .
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【推荐1】如图,抛物线
交x轴于点A(﹣3,0)和点B,交y轴于点C(0,3).
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/19/6392232d-8284-4f30-9254-060148ef93aa.png?resizew=413)
(1)求抛物线的函数表达式;
(2)若点P在抛物线上,且
,求点P的坐标;
(3)如图b,设点Q是线段AC上的一动点,作DQ⊥x轴,交抛物线于点D,求线段DQ长度的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58d88bbd34102b55fa928e8ff83f0d52.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/19/6392232d-8284-4f30-9254-060148ef93aa.png?resizew=413)
(1)求抛物线的函数表达式;
(2)若点P在抛物线上,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08c3a53bcac6a30db6284862d39584ba.png)
(3)如图b,设点Q是线段AC上的一动点,作DQ⊥x轴,交抛物线于点D,求线段DQ长度的最大值.
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【推荐2】直线
与x轴、y轴分别相交于A、B两点,与抛物线
交于点B,如图所示.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/25/ec7281da-9107-4047-8591-fa4608299534.png?resizew=299)
(1)求该抛物线的解析式;
(2)已知点M是抛物线上的一个动点,并且点M在第一象限内,连接
,设点M的横坐标为m,四边形
的面积为S,求S与m的函数表达式,并求出S的最大值;
(3)若点D在平面内,点C在直线
上,平面内是否存在点D使得以O,B,C,D为顶点的四边形是菱形.若存在,请直接写出点D的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f79ed52c39de817f13b4d0c1a4d8bfcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edd914af3ed6407458c356f2c4c93b2c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/25/ec7281da-9107-4047-8591-fa4608299534.png?resizew=299)
(1)求该抛物线的解析式;
(2)已知点M是抛物线上的一个动点,并且点M在第一象限内,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3a1b29fe1f6f93f6ba0605fcd2c27f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e089d6bfbe51464227f195f6115d8272.png)
(3)若点D在平面内,点C在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
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