3.5.3 函數(shù)的周期性和對(duì)稱性-(人教A版2019必修第一冊(cè)) (學(xué)生版)

函數(shù)的周期性和對(duì)稱性一函數(shù)的周期性1概念對(duì)于函數(shù)y=f(x)，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)，f(x+T)=f(x)都成立，那么把函數(shù)y=f(x)叫做周期函數(shù)，常數(shù)T叫做這個(gè)函數(shù)的周期.Eg：上圖是三角函數(shù)fx①函數(shù)圖像可看成由紅色那段圖像玩“分身術(shù)”的向兩邊延申；②紅色圖像的水平長(zhǎng)度為AB=2π，它就是函數(shù)的最小正周期T，即T=2π；(思考：4π是周期么)③整個(gè)函數(shù)，對(duì)于任何x，都有f(x+2π)=f(x).(簡(jiǎn)單說(shuō)來(lái)，兩個(gè)自變量相差2π，它們對(duì)應(yīng)的函數(shù)值均相等)下面兩個(gè)圖像也是周期函數(shù)的圖像！他們的周期是什么？最小正周期呢？2常見的結(jié)論①若f(x+a)=f(x+b)，則y=f(x)的周期是T=a-b.②若f(x+a)=-f(x)，則y=f(x)的周期是T=2a；(你可證明試試)③若fx+a=1fx二函數(shù)的對(duì)稱性1函數(shù)圖象自身的對(duì)稱關(guān)系①軸對(duì)稱：若f(x+a)=f(b-x),則y=f(x)有對(duì)稱軸x=a+b②中心對(duì)稱：若函數(shù)y=f(x)定義域?yàn)镽，且滿足條件f(a+x)+f(b-x)=c(a,b,c為常數(shù))，則函數(shù)y=f(x)的圖象關(guān)于點(diǎn)2兩個(gè)函數(shù)圖象之間的對(duì)稱關(guān)系①軸對(duì)稱若函數(shù)y=f(x)定義域?yàn)镽，則兩函數(shù)y=f(x+a)與y=f(b-x)的圖象關(guān)于直線特殊地，函數(shù)y=f(a+x)與函數(shù)y=f(a-x)的圖象關(guān)于直x=0對(duì)稱.②中心對(duì)稱若函數(shù)y=f(x)定義域?yàn)镽，則兩函數(shù)y=f(a+x)與y=c-f(b-x)的圖象關(guān)于點(diǎn)(b-a2特殊地，函數(shù)y=f(x+a)與函數(shù)y=-f(b-x)圖象關(guān)于點(diǎn)(b-a23周期性與對(duì)稱性拓展①若函數(shù)y=f(x)同時(shí)關(guān)于直線x=a,x=b對(duì)稱，則函數(shù)y=f(x)的周期T=2|b-a|；特殊地，若偶函數(shù)y=f(x)的圖像關(guān)于直線x=a對(duì)稱，則函數(shù)y=f(x)的周期②若函數(shù)y=f(x)同時(shí)關(guān)于點(diǎn)a,0,(b,0)對(duì)稱，則函數(shù)y=f(x)的周期T=2|b-a|③若函數(shù)y=fx同時(shí)關(guān)于直線x=a對(duì)稱，又關(guān)于點(diǎn)b,0對(duì)稱,則函數(shù)y=f(x)的周期T=4|b-a|；特殊地，若奇函數(shù)y=f(x)的圖像關(guān)于直線x=a對(duì)稱，則函數(shù)y=f(x)的周期【題型一】函數(shù)的周期性【典題1】設(shè)f(x)是周期為4的奇函數(shù)，當(dāng)0≤x≤1時(shí)，f(x)=x(1+x)，則f(-9【典題2】設(shè)偶函數(shù)f(x)對(duì)任意x∈R，都有f(x+3)=-1f(x)，且當(dāng)x∈[-3,-2]時(shí)，f(x)=4x，則f(107.5)鞏固練習(xí)1(★★)已知定義在R上的奇函數(shù)f(x)，滿足fx+4=-f(x)，且在[0,2]上單調(diào)遞減，則(A．f(8)<f(11)<f(15)C．f(15)<f(11)<f(8) 2(★★)已知f(x)是定義在R上周期為2的函數(shù)，當(dāng)x∈[-1,1]時(shí)，f(x)=|x|，那么當(dāng)x∈[-7,-5]時(shí)，f(x)=.3(★★★)設(shè)函數(shù)f(x)是定義在R上的奇函數(shù)，滿足fx+1=-f(x-1)，若f-1>1，f5=【題型二】函數(shù)圖象自身的對(duì)稱關(guān)系【典題1】定義在R上的函數(shù)f(x)的圖象關(guān)于點(diǎn)(-34，0)成中心對(duì)稱且對(duì)任意的實(shí)數(shù)x都有fx=-f(x+32)且f【典題2】已知函數(shù)f(x)=2x2xA．函數(shù)f(x)的圖象關(guān)于x=2對(duì)稱 B．函數(shù)f(x)的圖象關(guān)于x=4C．函數(shù)f(x)的圖象關(guān)于(2,2)對(duì)稱 D．函數(shù)f(x)的圖象關(guān)于【題型三】?jī)蓚€(gè)函數(shù)圖象之間的對(duì)稱關(guān)系【典題1】下列函數(shù)中，其圖象與函數(shù)y=lgx的圖象關(guān)于點(diǎn)(1,0)對(duì)稱的是()A．y=lg(1-x) C．y=【典題2】下列函數(shù)中，其圖象與函數(shù)y=2x的圖象關(guān)于直線y=1對(duì)稱的是鞏固練習(xí)1(★★)已知函數(shù)f(x)=ax+2x-6的對(duì)稱中心為(b,1)，則a=；b=2(★★)【多選題】函數(shù)f(x)的圖象關(guān)于直線x=1對(duì)稱，那么()A．f(2-x)=f(x)C．函數(shù)y=f(x+1)是偶函數(shù) D．函數(shù)y=f(x-1)3(★★★)已知函數(shù)f(x)=lnx+ln(a-x)的圖象關(guān)于直線x=1對(duì)稱，則函數(shù)f(1)的值為()A．0 B．1 4(★★★)已知函數(shù)f(x)=lnx4-x，則(A．y=f(x)的圖象關(guān)于點(diǎn)(2,0)對(duì)稱B．y=f(x)C．f(x)在(0,4)上單調(diào)遞減 D．f(x)在(0,2)5(★★)同一平面直角坐標(biāo)系中，函數(shù)y=2x+1與y=21-xA．關(guān)于原點(diǎn)對(duì)稱B．關(guān)于x軸對(duì)稱 C．關(guān)于y軸對(duì)稱D．關(guān)于直線y=x對(duì)稱6(★★★