反比例函数经典例题.pdf
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1、反比例函数一、基础知识1.定义:一般地,形如xky(k 为常数,ok)的函数称为反比例函数。xky还可以写成kxy12.反比例函数解析式的特征:等号左边是函数 y,等号右边是一个分式。分子是不为零的常数k(也叫做比例系数 k),分母中含有自变量x,且指数为 1.比例系数0k自变量x的取值为一切非零实数。函数 y 的取值是一切非零实数。3.反比例函数的图像图像的画法:描点法 列表(应以 O为中心,沿 O的两边分别取三对或以上互为相反的数)描点(有小到大的顺序)连线(从左到右光滑的曲线)反比例函数的图像是双曲线,xky(k 为常数,0k)中自变量0 x,函数值0y,所以双曲线是不经过原点,断开的两
2、个分支,延伸部分逐渐靠近坐标轴,但是永远不与坐标轴相交。反比例函数的图像是是轴对称图形(对称轴是xy或xy)。反比例函数xky(0k)中比例系数k 的几何意义是:过双曲线xky(0k)上任意引x轴 y 轴的垂线,所得矩形面积为k。4反比例函数性质如下表:k 的取值图像所在象限函数的增减性ok一、三象限在每个象限内,y 值随x的增大而减小ok二、四象限在每个象限内,y 值随x的增大而增大5.反比例函数解析式的确定:利用待定系数法(只需一对对应值或图像上一个点的坐标即可求出 k)6“反比例关系”与“反比例函数”:成反比例的关系式不一定是反比例函数,但是反比例函数xky中的两个变量必成反比例关系。7
3、.反比例函数的应用二、例题【例 1】如果函数222kkkxy的图像是双曲线,且在第二,四象限内,那么的值是多少?【解析】有函数图像为双曲线则此函数为反比例函数xky,(0k)即kxy1(0k)又在第二,四象限内,则0k可以求出的值【答案】由反比例函数的定义,得:01222kkk解得0211kkk或1k1k时函数222kkkxy为xy1【例 2】在反比例函数xy1的图像上有三点1x,1y,2x,2y,3x,3y。若3210 xxx则下列各式正确的是()A213yyy B 123yyy C 321yyy D 231yyy【解析】可直接以数的角度比较大小,也可用图像法,还可取特殊值法。解法一:由题意
4、得111xy,221xy,331xy3210 xxx,213yyy所以选 A 解法二:用图像法,在直角坐标系中作出xy1的图像描出三个点,满足3210 xxx观察图像直接得到213yyy选 A 解法三:用特殊值法213321321321,1,1,211,1,2,0yyyyyyxxxxxx令【例 3】如果一次函数的图像与反比例函数xmnymnmxy30相交于点(221,),那么该直线与双曲线的另一个交点为()【解析】12132212213nmmnnmxxmnynmxy解得,相交于与双曲线直线文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1
5、E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H
6、10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D
7、1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7
8、T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q
9、8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E
10、6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2
11、Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7221111121,122211yxyxxyxyxyxy得解方程组双曲线为直线为11,另一个点为【例 4】如图,在AOBRt中,点 A是直线mxy与双曲线xmy在第一象限的交点,且2AOBS,则m的值是 _.图解:因为直线mxy与双曲线xmy过点 A,设 A点的坐标为AAyx,.则有AAAAxmymxy,.所以AAyxm.又点A在第一象限,所以AAAAyyABxxOB
12、,.所以myxABOBSAAAOB212121.而已知2AOBS.所以4m.三、练习题1.反比例函数xy2的图像位于()A第一、二象限 B 第一、三象限 C 第二、三象限 D 第二、四象限2.若 y 与x成反比例,x与z成正比例,则 y 是z的()A、正比例函数 B、反比例函数 C、一次函数D、不能确定3.如果矩形的面积为6cm2,那么它的长ycm 与宽xcm 之间的函数图象大致为()文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8
13、H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6
14、D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z
15、7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3
16、Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3
17、E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR
18、2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4
19、G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7o y x y x o y x o y x o A B C D4.某气球内充满了一定质量的气体,当温度不变时,气球内气体的气压P(kPa)是气体体积 V(m3)的反比例函数,其图象如图所示当气球内气压大于 120 kPa时,气球将爆炸为了安全起见,气球的体积应()A、不小于54m3 B、小于54m3 C、不小于45m3 D、小于45m3 5如图,A、C是函数xy1的图象上的任意两点,过A作x轴的垂线,垂足为B,过 C作 y 轴的垂线,垂足为D,记 RtAOB 的面积为 S
20、1,RtCOD 的面积为 S2则()A S1S2 B S1 S2C S1=S2 D S1与 S2的大小关系不能确定6 关于 x 的一次函数 y=-2x+m和反比例函数 y=1nx的图象都经过点 A(-2,1).求:(1)一次函数和反比例函数的解析式;(2)两函数图象的另一个交点B的坐标;(3)AOB的面积7.如图所示,一次函数yaxb 的图象与反比例函数ykx的图象交于 A、B两点,与 x 轴交于点 C已知点 A的坐标为(2,1),点 B的坐标为(12,m)(1)求反比例函数和一次函数的解析式;(2)根据图象写出使一次函数的值小于反比例函数的值的x 的取值范围OCABOyxABCD文档编码:C
21、R2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT4G3Q8H10X5 ZI4D3E6D1O7文档编码:CR2Z7T1E7O6 HT
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