5 3函數(shù)周期性對稱性必修第一冊學(xué)生版

對于函數(shù)????(??)，如果存在一個不為零的常數(shù)??，使得當(dāng)??取定義域內(nèi)的每一個值時，??(??????(??)都成立，那么把函數(shù)??=??(??)叫做周期函數(shù)，常數(shù)??叫做這個函數(shù)的周期.上圖是三角函數(shù)??(??????????????2??，它就是函數(shù)的最小正周期??，即????，都有??(??2??)=??(??+??)=??(????)，則??=??(??)的周期是??=?????(??+??)=???(??)，則??=??(??)的周期是??=2??；(你可證明試試③若??(??+??)=

，則??=??(??)的周期是??=??(????)=??(????則??=??(??)有對稱軸??=2,,??(??)的圖象關(guān)于點(??+????)對稱 若函數(shù)??=??(??)定義域為??，則兩函數(shù)??=??(????)??=??(????)的圖象關(guān)于直線??=?????對稱2特殊地，函數(shù)??=??(????)與函數(shù)??=??(????)的圖象關(guān)于直??=0對稱若函數(shù)??=??(??)定義域為??，則兩函數(shù)??=??(????)與??=????(????)的圖象關(guān)于點(???????)對稱 特殊地，函數(shù)??=??(????)與函數(shù)??=???(????)圖象關(guān)于點(?????0)對稱2??=??(??)同時關(guān)于直線??=????=??對稱，則函數(shù)??=??(??)??=2|????|；特殊地，若偶函數(shù)??=??(??)的圖像關(guān)于直線??=??對稱，則函數(shù)??=??(??)的周期??=2|??|；??=??(??)同時關(guān)于點(??0??0)對稱，則函數(shù)??=??(??)??=2|????=??(??)同時關(guān)于直線??=??對稱，又關(guān)于點(??0),則函數(shù)??=??(??)??=4|???特殊地，若奇函數(shù)????(??)的圖像關(guān)于直線??=??對稱，則函數(shù)??=??(??)??=1設(shè)??(??)是周期為4的奇函數(shù)，當(dāng)0≤??≤1時，??(??)=??(1??)，則??92【典題2】設(shè)偶函數(shù)??(??)對任意??∈??，都有??(??3)=?

，且當(dāng)??∈[?32]時，??(??)=4????(107.5) 1(★★)已知定義在??上的奇函數(shù)??(??)，滿足??(??+4)=???(??)，且在[0,2]上單調(diào)遞減，則 ??．??(8)<??(11)< ??．??(11)<??(8)<??．??(15)<??(11)< ??．??(15)<??(8)<2(★★)已知??(??)是定義在??上周期為2的函數(shù)，當(dāng)??∈[?1,1]時，??(??)=|??|，那么當(dāng)??∈[?7,?5]??(??) 3(★★★)設(shè)函數(shù)??(??)是定義在??上的奇函數(shù)滿足??(??+1)=???(???1)，若??(?1)>1，??(5)=??2?2???4， 1】定義在??上的函數(shù)??(??)的圖象關(guān)于點(3，0)成中心對稱且對任意的實數(shù)??都有??(??)??(?? 且??(?1)=1,??(0)=?2，則??(1)+??(2)+?+??(2014) 【典題2】已知函數(shù)??(??)

，則 ??．函數(shù)??(??)的圖象關(guān)于??=2對 ??．函數(shù)??(??)的圖象關(guān)于??=4對 ??．??=????(1? ??．??=????(2???．??=log0.1(1? ??．??=log0.1(2?【典題2】下列函數(shù)中，其圖象與函數(shù)??=2??的圖象關(guān)于直線??=1對稱的 1(★★)已知函數(shù)??(??)=????+2的對稱中心為(??,1)，則?? ；?? 2(★★)【多選題】函數(shù)??(??)的圖象關(guān)于直線??=1對稱，那么 ??．??(2???)=?? ??．??(1???)=??(1+??．函數(shù)??=??(??+1)是偶函 ??．函數(shù)??=??(???1)是偶函3(★★★)已知函數(shù)??(??)=??????+????(?????)的圖象關(guān)于直線??=1對稱，則函數(shù)??(1)的值為 4(★★★)已知函數(shù)??(??)=??????，則 ??．????(??)的圖象關(guān)于點(20)??．??=??(??)的圖象關(guān)于直線??=2 5(★★)同一平面直角坐標(biāo)系中，函數(shù)??=2??+1與??=21???的圖象 ??．關(guān)于直線??=??對6(★★★)【多選題】已知定義在??上的函數(shù)??=??(??)滿足條件??(??+2)=???(??)，且函數(shù)??=??(???1)為奇 ??．函數(shù)??=??(??)是周期函 ??．函數(shù)??=??(??)的圖象關(guān)于點(?1,0)對??．函數(shù)??=??(??)為??上的偶函