函數(shù)對稱性、周期性和奇偶性規(guī)律總結(jié)

1、1函數(shù)對稱性、周期性和奇偶性函數(shù)對稱性、周期性和奇偶性關(guān)嶺民中數(shù)學(xué)組關(guān)嶺民中數(shù)學(xué)組(一)、同一函數(shù)的函數(shù)的奇偶性與對稱性：（奇偶性是一種特殊的對稱性）1、奇偶性:（1）奇函數(shù)關(guān)于（0，0）對稱，奇函數(shù)有關(guān)系式0)()(???xfxf（2）偶函數(shù)關(guān)于y（即x=0）軸對稱，偶函數(shù)有關(guān)系式)()(xfxf??2、奇偶性的拓展:同一函數(shù)的對稱性（1）函數(shù)的軸對稱：函數(shù)關(guān)于對稱)(xfy?ax??)()(xafxaf???也可以寫成或)()(xa

2、fxaf???)2()(xafxf??)2()(xafxf???若寫成：，則函數(shù)關(guān)于直線)()(xbfxaf???)(xfy?對稱22)()(baxbxax??????證明：設(shè)點(diǎn)在上，通過可知，)(11yx)(xfy?)2()(xafxf??，即點(diǎn)上，而點(diǎn))2()(111xafxfy???)()2(11xfyyxa??也在與點(diǎn)關(guān)于x=a對稱。得證。)(11yx)2(11yxa?說明：關(guān)于對稱要求橫坐標(biāo)之和為，縱坐標(biāo)相等。ax?2a∵關(guān)于

3、對稱，∴函數(shù)關(guān)于對稱1111()()axyaxy??與xa?)(xfy?ax??)()(xafxaf???∵關(guān)于對稱，∴函數(shù)關(guān)于對稱1111()(2)xyaxy?與xa?)(xfy?ax??)2()(xafxf??∵關(guān)于對稱，∴函數(shù)關(guān)于對稱1111()(2)xyaxy??與xa?)(xfy?ax??)2()(xafxf???（2）函數(shù)的點(diǎn)對稱：函數(shù)關(guān)于點(diǎn)對稱)(xfy?)(ba?bxafxaf2)()(????或bxfxaf2)()2(

4、????上述關(guān)系也可以寫成bxfxaf2)()2(???若寫成：，函數(shù)關(guān)于點(diǎn)對稱cxbfxaf????)()()(xfy?)22(cba?3注：換種說法：與若滿足，即它們關(guān)于)(xfy?()()ygxfx???)()(xgxf??對稱。0?y2、與關(guān)于Y軸對稱。()yfx?()yfx??證明：設(shè)上任一點(diǎn)為則，所以經(jīng)過點(diǎn)()yfx?11()xy11()yfx?()yfx??11()xy?∵與關(guān)于Y軸對稱，∴與關(guān)于Y軸對11()xy11()

5、xy?()yfx?()yfx??稱。注：因?yàn)榇氲盟越?jīng)過點(diǎn)11()xy?()yfx??111(())()yfxfx????()yfx??11()xy?換種說法：與若滿足，即它們關(guān)于)(xfy?()()ygxfx???)()(xgxf??對稱。0?x()(())()gxfxfx?????3、與關(guān)于直線對稱。()yfx?(2)yfax??xa?證明：設(shè)上任一點(diǎn)為則，所以經(jīng)過點(diǎn)()yfx?11()xy11()yfx?(2)yfax??11(

6、2)axy?∵與關(guān)于軸對稱，∴與關(guān)11()xy11(2)axy?xa?()yfx?(2)yfax??于直線對稱。xa?注：換種說法：與若滿足，即它們)(xfy?()(2)ygxfax???)2()(xagxf??關(guān)于對稱。ax?4、與關(guān)于直線對稱。)(xfy?)(2xfay??ay?證明：設(shè)上任一點(diǎn)為則，所以經(jīng)過點(diǎn)()yfx?11()xy11()yfx?)(2xfay??11(2)xay?∵與關(guān)于軸對稱，∴與關(guān)于直線11()xy11(2