高一数学必修一函数的奇偶性.ppt

课前热身 1 1 2.函数f(x)=2x2-mx+3，当x(-,-1时是减 函数，当x(-1,+)时是增函数，则f(2)= _. 19 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 利用函数的单调性求函数的最值 3、求函数f(x)x 的最大值和最小值 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. x y o 1 2 3 4 5 -1 123-1-2-3 观察下图，思考并讨论以下问题： （1）这两个函数图象有什么共同特征吗？ （2）相应的两个函数值对应表是如何体现这些特征的？ 一、引入新课 x y o 1 2 3 -1 12-13 x -3 -2 -1 0123 f(x)=|x| 32 1 0123 x -3 -2 -1 0123 f(x)=x2 94 1 0149 这两个点的坐标 有什么关系?(x,f(x)(-x,f(-x) 函数的图象关于y轴对称 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 当自变量任取两个互为相反数的值时， 对应的函数值相等。 x-3-2 -10123 f(x)=x29410149 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 二、新课讲解 一般地,如果对于函数f(x)的定义域内任意 一个x,都有f(-x)=f(x), 那么函数f(x) 就叫做 偶函数. 思考:定义中“任意一个x,都有f(- x)=f(x)成立”说明了什么？ 说明f(-x)与f(x)都有意义 ， 即-x、x必须同时属于定义域 ， 因此偶函数的定义域关于原点对称。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 7 7 练习1:判断下面两个函数是否是偶函数?并说明理由. (1)f(x)=5x2+3, x-3,2; (2)f(x)= 判断函数是否为偶函数，必须首先 讨论函数的定义域是否关于原点对称 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. x y o 1 2 3 4 5 -1 123-1-2-3 观察下图，思考并讨论以下问题： （1）这两个函数图象有什么共同特征吗？ （2）相应的两个函数值对应表是如何体现这些特征的？ x y o 1 2 3 -1 12-13 x -3 -2 -1 0123 f(x)=x 32 1 0123 x -3 -2 -1 0123 f(x)=1/x / / 这两个点的坐标 有什么关系?(x,f(x) (-x,f(-x) 函数的图象关于原点对称 -3 -2 -1 0123 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 思考:那么关于原点对称的点的坐标之间有什么关系呢？ 当自变量任取两个互为相反数的值时， 对应的函数值互为相反数。 x -3-2 -1 0123 f(x)=x -3-2 -1 0123 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 1010 一般地,如果对于函数f(x)的定义域内任意 一个x,都有f(-x)=-f(x), 那么函数f(x) 就叫做 奇函数. 思考:定义中“任意一个x,都有f(- x)=f(x)成立”说明了什么？ 说明f(-x)与f(x)都有意义 ， 即-x、x必须同时属于定义域 ， 因此奇函数的定义域关于原点对称的。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 1111 由此可见，定义域关于原点对称是 函数具有奇偶性的前提条件。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 1212 2、既不是奇函数也不是偶函数的函数称为非奇非 偶函数。 1、如果函数f(x)是奇函数或偶函数，就说函数f(x) 具有奇偶性.函数的奇偶性是函数的整体性质。 3、定义域关于原点对称是函数具有奇偶性的前提 条件。 4、具有奇偶性的函数的图象的特征： (1).偶函数的图象关于y轴对称 (2).奇函数的图象关于原点对称 函数奇偶性定义中应注意： 奇偶性是对函数的整个定义域而言的. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. A B D E A1 B1 C1 D1 E1 C H Ox y 例 已知函数 y=f(x) 是偶函数，它在y轴右边的图象 如下图所示，画出函数 y=f(x) 在y轴左边的图象。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 已知函数 y=f(x) 是奇函数，它在y轴右边的图象 如下图所示，画出函数 y=f(x) 在y轴左边的图象。 O x y A B C D E A1 B1 C1 D1 E1 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 四、例题讲解 解： (1)对于函数f(x)=x4, 其定义域为(- ,+ ) 对定义域内的每一个x,都有 f(-x)=(-x)4=x4=f(x) 函数f(x)=x4为偶函数. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 判断函数奇偶性的一般步骤： 1、看函数的定义域是否关于原点对称，若不对称， 则得出结论：该函数无奇偶性。若定义域对称，则 2、计算f（-x），若等于f（x），则函数是偶函数；若 等于-f（x），则函数是奇函数。若两者都不满足，则 函数既不是奇函数也不是偶函数。 注意：1、若可以作出函数图象的，直接观察图象是否 关于y轴对称或者关于原点对称。 2、判断函数奇偶性的方法： 定义法 图象法 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. (1) (2)(3 ) (4 ) 偶函数 非奇非偶函数奇函数 非奇非偶函数 判断下列函数的奇偶性 o o o o x x x x y y y y y 0 y x 偶函数 y x 0 y 是奇函数也是偶函数 (5 ) (6 ) 函数按是否有奇偶性可分为四类 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copy