1 . 【问题提出】
在数学兴趣小组的研讨中,小明提出自己遇到的问题:解不等式
.
【问题探究】
数学老师启发小明尝试从“函数图象”的角度解决这个问题:
如图1,在平面直角坐标系中,分别画出函数
和函数
的图象,从函数角x度看,解不等式
相当于求双曲线
在抛物线
上方的点的横坐标的取值范围.
的解集为______.
【类比探究】
(2)受此启发,小明尝试解不等式
.经过分析,小明发现需要借助函数
和函数______的图象来求解.请在图2中画出相应的函数图象,并得出不等式
的解集为______.
【拓展应用】
(3)小明想借助函数图象进一步研究不等式,于是尝试解不等式组
,并进行了一些准备,如图3所示.请根据小明的思路分析,直接写出该不等式组的解集______.
在数学兴趣小组的研讨中,小明提出自己遇到的问题:解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a22111a0d5ae443e8698cfb9d6a31d95.png)
【问题探究】
数学老师启发小明尝试从“函数图象”的角度解决这个问题:
如图1,在平面直角坐标系中,分别画出函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a22111a0d5ae443e8698cfb9d6a31d95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a22111a0d5ae443e8698cfb9d6a31d95.png)
【类比探究】
(2)受此启发,小明尝试解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f33e93287dfb8996247c0aef587ad13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/646e11d5bff57e56ce82c2339f2d71ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f33e93287dfb8996247c0aef587ad13.png)
【拓展应用】
(3)小明想借助函数图象进一步研究不等式,于是尝试解不等式组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0801989dcd3d03ce110ba1df0e2fdf.png)
您最近一年使用:0次
2 . 【定义】在平面直角坐标系中,有一条直线
,对于任意一个函数图像,把该图像在直线
上的点以及直线
右边的部分向上平移
个单位长度(
),再把直线
左边的部分向下平移
个单位长度,得到一个新的函数图像,则这个新函数叫做原函数关于直线
的“
分移函数”.例如:函数
关于直线
的“
分移函数”为
.
【概念理解】
(1)① 已知点
、
、
,其中在函数
关于直线
的“
分移函数”图像上的点有_________ ;
② 已知点
在函数
关于直线
的“
分移函数”图像上,求
的值.
【拓展探究】
(2)若二次函数
关于直线
的“
分移函数”与
轴有三个公共点,是否存在
,使得这三个公共点的横坐标之和为
,若存在请求出
的值,若不存在,请说明理由.
【深度思考】
(3)已知
,
,
,
,若函数
关于直线
的“
分移函数”图像与四边形
的边恰好有
个公共点,请直接写出
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de8610232c77741a37463feba1a66c94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3e913f3a5279472f33c246635fab43b.png)
【概念理解】
(1)① 已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf74dfac8cd1661fc3ad7aba83769b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a850efc958e96bf650d2bcb599ce4ad9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae83f3afd31f5de73cc6e2b256207f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/026808536f6b6d265c778e23836fbf13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
② 已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cf29b4f94402634b15bddc40918db2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/548deab72d7f7d1318558eb1f0f2c88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
【拓展探究】
(2)若二次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682c90df1732eb1757b96cddb450f9d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55aa0a20848c37c1892c567b2315e04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c9d84ca9a49ef591a72b85d4c502baf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
【深度思考】
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15e193e00a5d2b4654735c1f850d9dc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b83a87f55e2ee372668e82c4a0d4697a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/965b4d8367fcc1707a614c90bf172dc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e06b3f3fef180034147a0a0e675c1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ce1066b6ec99c38bf6fdaae5dcb7d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
您最近一年使用:0次
2023-09-26更新
|
388次组卷
|
2卷引用:2023年江苏省盐城市鹿鸣路初级中学中考一模数学试题
3 . 定义:当
(
,
为常数,
)时,函数
最大值与最小值之差恰好为
,我们称函数
是在
上的“雅正函数”,“
”的值叫做该“雅正函数”的“雅正值”.
【初步理解】
(1)试判断下列函数是在
上的“雅正函数”为______.(填序号)
①
;②
;③
.
【尝试应用】
(2)若一次函数
(
,
为常数,
)和反比例函数
(
为常数,
)都是在
上的“雅正函数”,求
的值.
【拓展延伸】
(3)若二次函数
是在
(
,
为常数,
)上的“雅正函数”,雅正值是3.
①求
、
的值;
②若该二次函数图象与
轴交于点
,
(点
在点
的左侧),与
轴交于点
.点
为二次函数
图象上一点,且点
的横坐标为
,点
、点
是线段
上的两个动点(点
在点
的左侧),分别过点
、点
作
轴的平行线交抛物线于点
、点
,如果
,其中
为常数.试探究:是否存在常数
,使得
为定值.如果存在,请求出
的值;如果不存在,请说明理由.参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53241838f18d2c148e4927726f151c77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb7961cbe98aac6a5fdee94582c341b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/023782cf6bc34246c16caf4cc6209868.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39c883d61ef6f0c7642f1fd883ae4674.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/023782cf6bc34246c16caf4cc6209868.png)
【初步理解】
(1)试判断下列函数是在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95842ad442c7f6d5ec4b32939b929e63.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cab0906bff33675959408a1d1b6fa823.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498bcfd1aac03057dcd0adef2dd113a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6ef9db1f5ca96d4618d9bf385f71a2d.png)
【尝试应用】
(2)若一次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0c143e0781facdd4e4b6db9854e0ad2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e06744e0406608a46242aa6c2b194870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb79a1b0c6dc78e6de17de6bed477fb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce662cd099fa16b8b4f900cd1f90177e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa0c0d5ff048d05f8e478977c582222d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b881044b5c73db6fcce110525741b02.png)
【拓展延伸】
(3)若二次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a74eedbb26267f0066d7bf26559f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39c883d61ef6f0c7642f1fd883ae4674.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b140e221ddf537b8964fff8557cca0.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
②若该二次函数图象与
![](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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a74eedbb26267f0066d7bf26559f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6513f1fcfb91b3460a08daa41de18f23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c00f5c9394ad8784b26ce80062409460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c322f108e17d6805f3daf4312820135.png)
您最近一年使用:0次
4 . 【问题提出】在数学兴趣小组的研讨中,小蒙提出了自己遇到的问题:解不等式 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f41789f4b8b6666cb435a92cabf7f56.png)
如图1,在平面直角坐标系中,分别画出函数.
和函数
的图象,从函数角度看,解不等式
相当于求抛物线.
在双曲线
下方的点的横坐标的取值范围.
(1)观察图1,可知两个图象的交点坐标为______ ,所以
的解为______.
【类比探究】受此启发,小蒙尝试解不等式
经过分析,小蒙发现需要借助函数
和函数 的图象来求解.
(2)请先完成上面的填空,再在图2中画出相应的函数图象,写出不等式
的解集并说明理由.
【拓展应用】小蒙想借助函数图象进一步研究不等式,于是尝试解不等式组
并进行了一些准备,如图3所示.
(3)请根据小蒙的思路分析,直接写出该不等式组的解集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f41789f4b8b6666cb435a92cabf7f56.png)
如图1,在平面直角坐标系中,分别画出函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd14cc1a3f03fa2c1df2a8f12cfb298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbb5c7e6458a6730c7cb4099f1c50a33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd14cc1a3f03fa2c1df2a8f12cfb298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
(1)观察图1,可知两个图象的交点坐标为______ ,所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbb5c7e6458a6730c7cb4099f1c50a33.png)
【类比探究】受此启发,小蒙尝试解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/982000bf40412aa81303f64b23a06148.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/646e11d5bff57e56ce82c2339f2d71ce.png)
(2)请先完成上面的填空,再在图2中画出相应的函数图象,写出不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb87313c31e83d86824ac25c1438ded0.png)
【拓展应用】小蒙想借助函数图象进一步研究不等式,于是尝试解不等式组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c63ca258e722a9c4c04cb5ec547e194.png)
(3)请根据小蒙的思路分析,直接写出该不等式组的解集.
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5 . 请仔细阅读下面的材料,并完成相应的任务.
任务:
(1)利用图象法解上述材料中的方程,下列叙述错误的是( )
A.利用图象法解方程体现了数形结合思想
B.画出抛物线
和直线
,观察图象交点的横坐标,也可得出该方程的根
C.画出抛物线
和直线
,观察图象交点的横坐标,也可得出该方程的根
D.画出抛物线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070d1ea22a92808dad7489438c239629.png)
和直线
,观察图象交点的横坐标,也可得出该方程的根
(2)请你利用图象法解方程
,把函数图象画在图3的平面直角坐标系中,并写出解方程的分析过程.
(3)若方程
无实数根,从图形的角度看就是抛物线
与直线 无交点,此时a的取值范围是 ;
(4)拓展迁移:方程
的根的情况是 .
利用图象法解一元二次方程 数学活动课上,王老师提出这样一个问题:我们曾经利用一次函数的图象解一元一 次方程,类比前面的学习经验,我们能否利用二次函数的图象解一元二次方程呢? 例如,解方程: ![]() 王老师倡导同学们以小组为单位进行合作探究,同学们经过几分钟热烈的讨论交流, 智慧小组率先展示了他们的方法:将方程进一步变形为 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 善思小组受智慧小组的启发,展示了他们的方法:画出二次函数 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
(1)利用图象法解上述材料中的方程,下列叙述错误的是( )
A.利用图象法解方程体现了数形结合思想
B.画出抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b9f0b9e53a83e68f5fec944f343119.png)
C.画出抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81e6893063df9d4188f3b7002664097d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9ed5bd57fd5573d09d3aea1d3a4e4db.png)
D.画出抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070d1ea22a92808dad7489438c239629.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bed04a438788f040f5f45d70ff4d7e2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f58bc0c5316829c338bf1908aefbf4b3.png)
(2)请你利用图象法解方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b28cc80cd1cc1138665b44d724d8f53.png)
(3)若方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/431233c969a20525bce278d5922e4860.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa990b6bed988fb1fe1487aa2a7d85f4.png)
(4)拓展迁移:方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7606d8bb2cee6ca8d6f706e0cf9cd625.png)
您最近一年使用:0次
名校
6 . 问题呈现:探究二次函数
(其中
,m为常数)的图像与一次函数
的图像公共点.
(1)问题可转化为:二次函数
的图像与一次函数
______的图像的公共点.
(2)问题解决:在如图平面直角坐标系中画出
的图像.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/19/2e21b71a-012e-4eef-afbb-8b3956e38aed.png?resizew=184)
(3)请结合(2)中图像,就m的取值范围讨论两个图像公共点的个数.
(4)问题拓展:若二次函数
(其中
,m为常数)的图像与一次函数
的图像有两个公共点,则m的取值范围为______.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ad8b6089ecdf8ccde0e18e02c5c647.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/836d745b71ec18b1135e8bbf6990bffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c31c4f39399ec245a67db2933ed639f2.png)
(1)问题可转化为:二次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf14aab90c72f1a57ec55a581c28806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070d1ea22a92808dad7489438c239629.png)
(2)问题解决:在如图平面直角坐标系中画出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf14aab90c72f1a57ec55a581c28806.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/19/2e21b71a-012e-4eef-afbb-8b3956e38aed.png?resizew=184)
(3)请结合(2)中图像,就m的取值范围讨论两个图像公共点的个数.
(4)问题拓展:若二次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3ecc6a3e907c42c236de968817f3867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48698531fa3e1aa7736aed548367bfb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9427de5d22d37859a110dbb234313c1.png)
您最近一年使用:0次
2022-01-29更新
|
820次组卷
|
6卷引用:江苏省南京市鼓楼区金陵汇文学校2021-2022学年九年级上学期期末数学试题
江苏省南京市鼓楼区金陵汇文学校2021-2022学年九年级上学期期末数学试题江苏省南京市鼓楼区2021-2022学年九年级上学期期末数学试题(已下线)专题22.57 《二次函数》挑战综合(压轴)题分类专题(二)(专项练习)-2022-2023学年九年级数学上册基础知识专项讲练(人教版)(已下线)专题5.51 《二次函数》挑战综合(压轴)题分类专题(二)(专项练习)-2022-2023学年九年级数学下册基础知识专项讲练(苏科版)(已下线)专题2.57 二次函数(挑战综合(压轴)题分类专题)(二)(专项练习)-2022-2023学年九年级数学下册基础知识专项讲练(北师大版)14-二次函数图像与性质及与a、b、c的关系
真题
7 . 某数学兴趣小组运用《几何画板》软件探究y=ax2(a>0)型抛物线图象.发现:如图1所示,该类型图象上任意一点M到定点 F(0,
)的距离MF,始终等于它到定直线l:y=﹣
上的距离MN(该结论不需要证明),他们称:定点F为图象的焦点,定直线l为图象的准线,y=﹣
叫做抛物线的准线方程.其中原点O为FH的中点,FH=2OF=
,例如,抛物线y=
x2,其焦点坐标为F(0,
),准线方程为l:y=﹣
.其中MF=MN,FH=2OH=1.
请分别直接写出抛物线y=2x2的焦点坐标和准线l的方程: , .
(2)【技能训练】
如图2所示,已知抛物线y=
x2上一点P到准线l的距离为6,求点P的坐标;
(3)【能力提升】
如图3所示,已知过抛物线y=ax2(a>0)的焦点F的直线依次交抛物线及准线l于点A、B、C.若BC=2BF,AF=4,求a的值;
(4)【拓展升华】
古希腊数学家欧多克索斯在深入研究比例理论时,提出了分线段的“中末比”问题:点C将一条线段AB分为两段AC和CB,使得其中较长一段AC是全线段AB与另一段CB的比例中项,即满足:
=
=
.后人把
这个数称为“黄金分割”把点C称为线段AB的黄金分割点.
如图4所示,抛物线y=
x2的焦点F(0,1),准线l与y轴交于点H(0,﹣1),E为线段HF的黄金分割点,点M为y轴左侧的抛物线上一点.当
=
时,请直接写出△HME的面积值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afd5daa9a0b9a14a4a44b02bae65b586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afd5daa9a0b9a14a4a44b02bae65b586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afd5daa9a0b9a14a4a44b02bae65b586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e28f005da664fc2f7ceacf1e32261e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
请分别直接写出抛物线y=2x2的焦点坐标和准线l的方程: , .
(2)【技能训练】
如图2所示,已知抛物线y=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ca8b26c3ad6d892590290a2304126bd.png)
(3)【能力提升】
如图3所示,已知过抛物线y=ax2(a>0)的焦点F的直线依次交抛物线及准线l于点A、B、C.若BC=2BF,AF=4,求a的值;
(4)【拓展升华】
古希腊数学家欧多克索斯在深入研究比例理论时,提出了分线段的“中末比”问题:点C将一条线段AB分为两段AC和CB,使得其中较长一段AC是全线段AB与另一段CB的比例中项,即满足:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/746853ea6d76bd7cccc6bdd6c739aed7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9e7e01495050cca64f635c2c2658157.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/029d393bb07b7140905b85f550519de4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/029d393bb07b7140905b85f550519de4.png)
如图4所示,抛物线y=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b69d5b90c263274ccbdf1a0386e15110.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
您最近一年使用:0次
2022-06-28更新
|
1169次组卷
|
9卷引用:2022年湖北省鄂州市中考数学真题
2022年湖北省鄂州市中考数学真题(已下线)专题22 与二次函数相关的压轴题-2022年中考数学真题分项汇编(全国通用)(第2期)(已下线)2022年湖北省随州市中考数学真题变式题21-24题(已下线)2022年湖北省鄂州市中考数学真题变式题21-24题(已下线)2022年湖北省武汉市中考数学真题变式题21-24题(已下线)二次函数的综合题03综合测(已下线)2023年新疆维吾尔族自治区中考数学真题变式题20-23题2024年广东省珠海市梅华中学中考一模数学试题2024年广东省惠州市惠阳区中考二模数学试题
真题
8 . 【问题情境 建构函数】
(1)如图1,在矩形
中,
是
的中点,
,垂足为
.设
,试用含
的代数式表示
.
(2)在上述表达式中,
与
成函数关系,其图像如图2所示.若
取任意实数,此时的函数图像是否具有对称性?若有,请说明理由,并在图2上补全函数图像.
(3)在“
取任意实数”的条件下,对上述函数继续探究,得出以下结论:①函数值
随
的增大而增大;②函数值
的取值范围是
;③存在一条直线与该函数图像有四个交点;④在图像上存在四点
,使得四边形
是平行四边形.其中正确的是__________.(写出所有正确结论的序号)
【抽象回归 拓展总结】
(4)若将(1)中的“
”改成“
”,此时
关于
的函数表达式是__________;一般地,当
取任意实数时,类比一次函数、反比例函数、二次函数的研究过程,探究此类函数的相关性质(直接写出3条即可).
(1)如图1,在矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f74c3b0f7dead2845b967c419404d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e97bd4e9a6cfde753bfbd6e36136c57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbe6fea5bc81509d46cf9e3c4d49f588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(2)在上述表达式中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)在“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4d4f4de104ad982e8dbcd395a19891.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4496fe22b40bc63581998e6b7ef6783.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
【抽象回归 拓展总结】
(4)若将(1)中的“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b246acb593f661622d106a5b4deb84d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a7f846ce54225b1590103a39669cf69.png)
您最近一年使用:0次
2023-06-17更新
|
2411次组卷
|
9卷引用:2023年江苏省连云港市中考数学真题
2023年江苏省连云港市中考数学真题(已下线)专题14 几何综合题(37题)-学易金卷:2023年中考数学真题分项汇编(全国通用)(已下线)专题14反比例函数与几何压轴问题(优选真题60道)-学易金卷:三年(2021-2023)中考数学真题分项汇编【全国通用】(已下线)专题32 函数与几何综合问题(共25题)-学易金卷:2023年中考数学真题分项汇编(全国通用)(已下线)2023年江苏省连云港市中考数学真题变式题22-27题(已下线)专题6 类比思想(已下线)第5讲 探究题江苏省南京市2023-2024学年九年级下学期期中数学试题2024年江苏省南京市联合体中考数学一模试题
9 . 阅读材料:小明同学在平面直角坐标系中研究中点时,发现了一个有趣的结论:若
,
是平面直角坐标系内两点,
是
的中点,则有结论
,
.这其实就是中点坐标公式,有了这个公式可以解决很多坐标系中求中点坐标的问题.
已知:二次函数
的函数图像上分别有
,
两点,其中
,
,
分别在对称轴的异侧,
是
中点,
是
中点.利用阅读材料解决如下问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/25/baf53fe8-54c4-48f8-a269-c80b8ed1f409.png?resizew=477)
概念理解:
(1)如图1,若
,求出
,
的坐标.
解决问题:
(2)如图2,点
是
关于
轴的对称点,作
轴交抛物线于点
.延长
至
,使得
.试判断
是否在
轴上,并说明理由.
拓展探究:
(3)如图3,
是一个动点,作
轴交抛物线于点
.延长
至
,使得
.
①令
,试探究
值是否为定值,若是,求出这个定值;若不是,请说明理由.
②在①条件下,
轴上一点
,抛物线上任意一点
,连接
,
,直接写出
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6f5adf13b4214666292dd64b947741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af405a054bfe7fb7ce40e48d816467e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3feccf154671abf1114e77c8cb03c83c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c45b5bbd5fb7706c6f7c24df34fc145.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19600a8ab63d28c2f4b2f7ea345fa9ab.png)
已知:二次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.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/5590337b3868db8523eeb7f448efcf05.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/25/baf53fe8-54c4-48f8-a269-c80b8ed1f409.png?resizew=477)
概念理解:
(1)如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a16a46f03535c5af872fa8fa9f21a431.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
解决问题:
(2)如图2,点
![](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/fdd69e51abb06a0e6e43a878c7120bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c5ae265994fd876bbacbf6f5cfbfa71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
拓展探究:
(3)如图3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a66c5f00b5b38a2d052354b5611970e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdd69e51abb06a0e6e43a878c7120bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c5ae265994fd876bbacbf6f5cfbfa71.png)
①令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9cb2babf873aa67865ea6670032eb4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f15802a8683020919d225220fcc8681.png)
②在①条件下,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4effc718edc9ada7580cb3a4019fc28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53b95463a97c60db3250cb641bf6523d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e35f43501c77172f66c49ca109396574.png)
您最近一年使用:0次
解题方法
10 . 综合与实践
操作探究
(1)如图1,将矩形
折叠,使点
与点
重合,折痕为
,
与
交于点
.请回答下列问题:
①与
全等的三角形为______,与
相似的三角形为______.并证明你的结论:(相似比不为1,只填一个即可):
②若连接
、
,请判断四边形
的形状:______.并证明你的结论;
拓展延伸
(2)如图2,矩形
中,
,
,点
、
分别在
、
边上,且
,将矩形折叠,使点
与点
重合,折痕为
,
与
交于点
,连接
.
①设
,
,则
与
的数量关系为______;
②设
,
,请用含
的式子表示
:______;
③
的最小值为______.
操作探究
(1)如图1,将矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00870385ca7f3214e2971779eb4c7904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00870385ca7f3214e2971779eb4c7904.png)
②若连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9766e5eb6796dafc5ffe212afdfc43c0.png)
拓展延伸
(2)如图2,矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4817c12128c0d81f54e9529cb0a3588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0215e13a9fb5574d5194aeb9507a98aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce6c0e9de83f2e64ae33609fc08459d.png)
①设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/417af7fd8533934a0d5b280408e15d4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87aac645b73f767aa10cdd3359842798.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce629f210d757b89a726d2ae851f9ed6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da49a7a7feead3a9584e503cfde55a81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce6c0e9de83f2e64ae33609fc08459d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/23/ddea6cbc-e094-40e0-ad34-b89a895a05ce.png?resizew=409)
您最近一年使用:0次