反比例函数(提高)知识讲解.pdf

精品资料欢迎下载反比例函数（提高）【学习目标】1.理解反比例函数的概念和意义，能根据问题的反比例关系确定函数解析式2.能根据解析式画出反比例函数的图象，初步掌握反比例函数的图象和性质3.会用待定系数法确定反比例函数解析式，进一步理解反比例函数的图象和性质【要点梳理】要点一、反比例函数的定义一般地，形如kyx(k为常数，0k)的函数称为反比例函数，其中x是自变量，y是函数，定义域是不等于零的一切实数.要点诠释：（1）在kyx中，自变量x是分式kx的分母，当0 x时，分式kx无意义，所以自变量x的取值范围是，函数y的取值范围是0y.故函数图象与x轴、y轴无交点；（2）kyx()可以写成()的形式，自变量x的指数是1，在解决有关自变量指数问题时应特别注意系数这一条件.（3）kyx()也可以写成的形式，用它可以迅速地求出反比例函数的比例系数k，从而得到反比例函数的解析式.要点二、确定反比例函数的关系式确定反比例函数关系式的方法仍是待定系数法，由于反比例函数kyx中，只有一个待定系数k，因此只需要知道一对xy、的对应值或图象上的一个点的坐标，即可求出k的值，从而确定其解析式.用待定系数法求反比例函数关系式的一般步骤是：（1）设所求的反比例函数为：kyx(0k)；（2）把已知条件（自变量与函数的对应值）代入关系式，得到关于待定系数的方程；（3）解方程求出待定系数k的值；（4）把求得的k值代回所设的函数关系式kyx中.要点三、反比例函数的图象和性质-第 1 页，共 5 页精品p d f 资料 可编辑资料-精品资料欢迎下载1、反比例函数的图象特征：反比例函数的图象是双曲线，它有两个分支，这两个分支分别位于第一、三象限或第二、四象限；反比例函数的图象关于原点对称，永远不会与x轴、y轴相交，只是无限靠近两坐标轴.要点诠释：（1）若点(ab，)在反比例函数kyx的图象上，则点(ab，)也在此图象上，所以反比例函数的图象关于原点对称；（2）在反比例函数(k为常数，0k)中，由于，所以两个分支都无限接近但永远不能达到x轴和y轴2、反比例函数的性质（1）如图 1，当0k时，双曲线的两个分支分别位于第一、三象限，在每个象限内，y值随x值的增大而减小；（2）如图 2，当0k时，双曲线的两个分支分别位于第二、四象限，在每个象限内，y值随x值的增大而增大；要点诠释：反比例函数的增减性不是连续的，它的增减性都是在各自的象限内的增减情况，反比例函数的增减性都是由反比例系数k的符号决定的；反过来，由双曲线所在的位置和函数的增减性，也可以推断出k的符号.要点四、反比例函数()中的比例系数k的几何意义过双曲线xky(0k)上任意一点作x轴、y轴的垂线，所得矩形的面积为k.过双曲线xky(0k)上任意一点作一坐标轴的垂线，连接该点和原点，所得三角形的面积为2k.-第 2 页，共 5 页精品p d f 资料 可编辑资料-文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1精品资料欢迎下载要点诠释：只要函数式已经确定，不论图象上点的位置如何变化，这一点与两坐标轴的垂线和两坐标轴围成的面积始终是不变的.【典型例题】类型一、反比例函数定义1、k为何值时，2221()kkykk x是反比例函数？【答案与解析】解：由222110kkkk得0201kkkk或且2k【总结升华】根据反比例函数关系式的一般式(0)kykx，也可以写成1(0)ykxk，后 一 种 写 法 中x的 次 数 为 1，可 知 此 函 数 为 反 比 例 函 数，必 须 具 备 两 个 条 件，2211kk且20kk，二者缺一不可类型二、确定反比例函数的解析式2、已知12yyy，1y与x成正比例，2y与x成反比例，且当x1 时，y7；当x2 时，y 8(1)y与x之间的函数关系式；(2)自变量的取值范围；(3)当x4 时，y的值【答案与解析】解：(1)1y与x成正比例，设111(0)yk x k2y与x成反比例，设222(0)kykx2121kyyyk xx把17xy与28xy分别代入上式，得12217,28.2kkkk123,4.kk所以y与x的函数解析式为43yxx-第 3 页，共 5 页精品p d f 资料 可编辑资料-文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1精品资料欢迎下载(2)自变量的取值范围是x0(3)当x4 时，434134y【总结升华】注意，比例系数要分别用1k和2k表示，不能用成同一个比例系数k.举一反三：【变式】已知y与23x成反比例，且41x时，2y，求y与x的函数关系式【答案】解：因为y与23x成反比例，所以23kyx，且21234k，解得5k.所以y与x的函数关系式为523yx.类型三、反比例函数的图象和性质3、若 A(1x，1y)、B(2x，2y)在函数12yx的图象上，当1x、2x 满足 _时，12yy.【答案】120 xx或120 xx或210 xx；【解析】12yx的图象在一、三象限，在每个象限内，随着x的增大，函数值y减小，所以120 xx或120 xx时，12yy.当B 点在三象限，A 点在一象限，即210 xx，也满足12yy.【总结升华】反比例函数的增减性是在每个象限内讨论的，A、B 两点要分成同在一象限、同在三象限和分属一、三象限讨论，这样才能把情况考虑完整.举一反三：【变式】如图所示，正比例函数2ykx与反比例函数1kyx在同一坐标系中的图象不可能是（）【答案】D；提示：对于D项，由正比例函数2ykx的图象经过第二、第四象限，得k0，由-第 4 页，共 5 页精品p d f 资料 可编辑资料-文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1精品资料欢迎下载反比例函1kyx的图象位于第一、第三象限，得k1，k不存在，故 D项错误解决这类图象问题的一般解法是先根据函数表达式的大致图象来确定函数表达式中字母系数的符号或范围，再根据字母系数的符号或范围确定另一个函数图象的大致位置类型四、反比例函数综合4、如图所示，已知双曲线(0)kykx，经过 RtOAB斜边 OB的中点 D，与直角边AB交于点 C，DE OA，3OBCS，求反比例函数的解析式【答案与解析】解：过点D作 DM AB于点 M DMOA，BDM BOA 在 BDM 和 EOD 中90DMBOEDBDMBOAODDBBDM DOE(AAS)，12DMOEOA，12BMDEAB设 D(ab，)，则 B(2ab，2)12ODEAOCSSab，3OBCABDESS梯形即(2)32bba，解得：2ab反比例函数的解析式为2yx【总结升华】本题欲求解析式有两个思路可考虑，一个是求 D点或 C点的坐标，另一个就是求 DOE或 AOC的面积，从条件看，求D点或 C点坐标的可能不大，于是从求DOE或 AOC的面积入手思考，由于D、C、B三点坐标间的特殊关系，设出D点的坐标就可以将B、C 两点的坐标表示出来，然后运用3OBCS求出 D 点两坐标之积，就不难求出解析式-第 5 页，共 5 页精品p d f 资料 可编辑资料-文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 ZR8Y5J6T4Q1文档编码:CM8E8A1Y10H1 HG10I10K3Z6K2 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