1、函數(shù)對(duì)稱性、周期性和奇偶性規(guī)律函數(shù)對(duì)稱性、周期性和奇偶性規(guī)律一、同一函數(shù)的周期性、對(duì)稱性問題(即函數(shù)自身)1、周期性：對(duì)于函數(shù)，如果存在一個(gè)不為零的常數(shù)T，使得當(dāng)x取定義域內(nèi)的每一個(gè)值時(shí)，都有)(xfy?都成立，那么就把函數(shù)叫做周期函數(shù)，不為零的常數(shù)T叫做這個(gè)函數(shù)的周)()(xfTxf??)(xfy?期。如果所有的周期中存在著一個(gè)最小的正數(shù)，就把這個(gè)最小的正數(shù)叫做最小正周期最小正周期。2、對(duì)稱性定義（略），請(qǐng)用圖形來理解。3、對(duì)稱性：我

2、們知道：偶函數(shù)關(guān)于y（即x=0）軸對(duì)稱，偶函數(shù)有關(guān)系式)()(xfxf??奇函數(shù)關(guān)于（0，0）對(duì)稱，奇函數(shù)有關(guān)系式0)()(???xfxf上述關(guān)系式是否可以進(jìn)行拓展？答案是肯定的探討：（1）函數(shù)函數(shù)關(guān)于關(guān)于對(duì)稱對(duì)稱)(xfy?ax??)()(xafxaf???也可以寫成或)()(xafxaf???)2()(xafxf??)2()(xafxf???簡(jiǎn)證：簡(jiǎn)證：設(shè)點(diǎn)在上，通過可知，)(11yx)(xfy?)2()(xafxf??，即點(diǎn)上，而

3、點(diǎn)與點(diǎn))2()(111xafxfy???)()2(11xfyyxa??也在)(11yx關(guān)于x=a對(duì)稱。得證。)2(11yxa?若寫成：若寫成：，函數(shù)關(guān)于直線對(duì))()(xbfxaf???)(xfy?22)()(baxbxax??????稱（2）函數(shù)函數(shù)關(guān)于點(diǎn)關(guān)于點(diǎn)對(duì)稱對(duì)稱)(xfy?)(ba?bxafxaf2)()(????或bxfxaf2)()2(????上述關(guān)系也可以寫成bxfxaf2)()2(???簡(jiǎn)證：簡(jiǎn)證：設(shè)點(diǎn)在上，即，通過可知

4、，)(11yx)(xfy?)(11xfy?bxfxaf2)()2(???，所以，所以點(diǎn)bxfxaf2)()2(11???1112)(2)2(ybxfbxaf?????也在上，而點(diǎn)與關(guān)于對(duì)稱。得)22(11ybxa??)(xfy?)22(11ybxa??)(11yx)(ba證。若寫成：若寫成：，函數(shù)關(guān)于點(diǎn)對(duì)稱cxbfxaf????)()()(xfy?)22(cba?（3）函數(shù)函數(shù)關(guān)于點(diǎn)關(guān)于點(diǎn)對(duì)稱對(duì)稱:假設(shè)函數(shù)關(guān)于對(duì)稱，即關(guān)于任一個(gè)值，都有

5、兩個(gè)y)(xfy?by?by?x值與其對(duì)應(yīng)，顯然這不符合函數(shù)的定義，故函數(shù)自身不可能關(guān)于對(duì)稱。但在曲線c(xy)=0，by?則有可能會(huì)出現(xiàn)關(guān)于對(duì)稱，比如圓它會(huì)關(guān)于y=0對(duì)稱。by?04)(22????yxyxc4、周期性：（1）函數(shù)滿足如下關(guān)系系，則)(xfy?Txf2)(的周期為A、B、)()(xfTxf???)(1)()(1)(xfTxfxfTxf?????或換種說法：與若滿足，即它們關(guān)于對(duì)稱。)(xfy?)(xgy?axgxf2)

6、()(??ay?5、關(guān)于點(diǎn)(ab)對(duì)稱。)2(2)(xafbyxfy????與換種說法：與若滿足，即它們關(guān)于點(diǎn)(ab)對(duì)稱。)(xfy?)(xgy?bxagxf2)2()(???6、與關(guān)于直線對(duì)稱。)(xafy??)(bxy??2bax??7、函數(shù)的軸對(duì)稱：定理1：如果函數(shù)滿足，則函數(shù)的圖象關(guān)于直線對(duì)??xfy?????xbfxaf?????xfy?2bax??稱.推論1：如果函數(shù)滿足，則函數(shù)的圖象關(guān)于直線對(duì)稱.??xfy?????xa

7、fxaf?????xfy?ax?推論2：如果函數(shù)滿足，則函數(shù)的圖象關(guān)于直線（y軸）對(duì)稱.特??xfy?????xfxf????xfy?0?x別地，推論2就是偶函數(shù)的定義和性質(zhì).它是上述定理1的簡(jiǎn)化.8、函數(shù)的點(diǎn)對(duì)稱：定理2：如果函數(shù)滿足，則函數(shù)的圖象關(guān)于點(diǎn)對(duì)??xfy?????bxafxaf2??????xfy???ba稱.推論3：如果函數(shù)滿足，則函數(shù)的圖象關(guān)于點(diǎn)對(duì)稱.??xfy?????0????xafxaf??xfy???0a推論4

8、：如果函數(shù)滿足，則函數(shù)的圖象關(guān)于原點(diǎn)對(duì)稱.特別地，??xfy?????0???xfxf??xfy???00推論4就是奇函數(shù)的定義和性質(zhì).它是上述定理2的簡(jiǎn)化.三、總規(guī)律：定義在Ｒ上的函數(shù)，在對(duì)稱性、周期性和奇偶性這三條性質(zhì)中，只要有兩條存在，則第三??xfy?條一定存在。四、試題1已知定義為R的函數(shù)滿足，且函數(shù)在區(qū)間上單調(diào)遞增.如果??xf????4????xfxf??xf????2，且，則的值（A）.212xx??421??xx???