新高考數(shù)學(xué)一輪復(fù)習(xí) 函數(shù)專項(xiàng)重難點(diǎn)突破專題09 函數(shù)的對(duì)稱性(原卷版)

專題09函數(shù)的對(duì)稱性真題再現(xiàn)一、單選題1．（2022·全國(guó)·統(tǒng)考高考真題）已知函數(shù)SKIPIF1<0的定義域均為R，且SKIPIF1<0．若SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題2．（2022·全國(guó)·統(tǒng)考高考真題）已知函數(shù)SKIPIF1<0及其導(dǎo)函數(shù)SKIPIF1<0的定義域均為SKIPIF1<0，記SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0均為偶函數(shù)，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0三、解答題3．（2023·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)是否存在a，b，使得曲線SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱，若存在，求a，b的值，若不存在，說(shuō)明理由.考點(diǎn)一判斷(證明)函數(shù)的對(duì)稱性一、單選題1．下列函數(shù)的圖象中，既是軸對(duì)稱圖形又是中心對(duì)稱的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．已知角SKIPIF1<0的頂點(diǎn)在原點(diǎn)，始邊與SKIPIF1<0軸的非負(fù)半軸重合，終邊過(guò)點(diǎn)SKIPIF1<0，若函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0圖象的對(duì)稱軸為SKIPIF1<0 B．SKIPIF1<0圖象的對(duì)稱軸為SKIPIF1<0C．SKIPIF1<0圖象的對(duì)稱中心為SKIPIF1<0 D．SKIPIF1<0圖象的對(duì)稱中心為SKIPIF1<03．設(shè)函數(shù)SKIPIF1<0的定義域?yàn)镽，且SKIPIF1<0是奇函數(shù)，則SKIPIF1<0圖像（

）A．關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱 B．關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱C．關(guān)于直線SKIPIF1<0對(duì)稱 D．關(guān)于直線SKIPIF1<0對(duì)稱4．已知函數(shù)SKIPIF1<0，則SKIPIF1<0的圖象（

）A．關(guān)于直線SKIPIF1<0對(duì)稱 B．關(guān)于點(diǎn)SKIPIF1<0對(duì)稱 C．關(guān)于直線SKIPIF1<0對(duì)稱 D．關(guān)于原點(diǎn)對(duì)稱5．已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的函數(shù)，那么函數(shù)SKIPIF1<0的圖象與函數(shù)SKIPIF1<0的圖象之間（

）A．關(guān)于點(diǎn)SKIPIF1<0對(duì)稱 B．關(guān)于直線SKIPIF1<0對(duì)稱C．關(guān)于點(diǎn)SKIPIF1<0對(duì)稱 D．關(guān)于直線SKIPIF1<0對(duì)稱二、多選題6．下列函數(shù)中，哪些函數(shù)的圖像關(guān)于SKIPIF1<0軸對(duì)稱（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．已知SKIPIF1<0是定義在R上的函數(shù)，函數(shù)SKIPIF1<0圖像關(guān)于y軸對(duì)稱，函數(shù)SKIPIF1<0的圖像關(guān)于原點(diǎn)對(duì)稱，則下列說(shuō)法正確的是（

）A．SKIPIF1<0 B．對(duì)SKIPIF1<0，SKIPIF1<0恒成立C．函數(shù)SKIPIF1<0關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱 D．SKIPIF1<08．已知函數(shù)SKIPIF1<0定義域?yàn)镾KIPIF1<0，則下列說(shuō)法正確的是（

）A．若SKIPIF1<0，則函數(shù)SKIPIF1<0圖象關(guān)于SKIPIF1<0對(duì)稱B．函數(shù)SKIPIF1<0與函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱C．函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱D．函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱9．已知SKIPIF1<0是定義在R上的函數(shù)，且SKIPIF1<0，則（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于原點(diǎn)對(duì)稱 B．函數(shù)SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱C．函數(shù)SKIPIF1<0的圖象關(guān)于直線x=1對(duì)稱 D．函數(shù)SKIPIF1<0是以2為周期的周期函數(shù)10．已知函數(shù)SKIPIF1<0的圖像關(guān)于點(diǎn)SKIPIF1<0成中心對(duì)稱圖形的充要條件是函數(shù)SKIPIF1<0為奇函數(shù)，函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0成軸對(duì)稱圖形的充要條件是函數(shù)SKIPIF1<0為偶函數(shù)，則（

）A．函數(shù)SKIPIF1<0的對(duì)稱中心是SKIPIF1<0B．函數(shù)SKIPIF1<0的對(duì)稱中心是SKIPIF1<0C．函數(shù)SKIPIF1<0有對(duì)稱軸D．函數(shù)SKIPIF1<0有對(duì)稱軸三、填空題11．已知函數(shù)SKIPIF1<0在SKIPIF1<0上的最大值與最小值分別為SKIPIF1<0和SKIPIF1<0，則函數(shù)SKIPIF1<0的圖象的對(duì)稱中心是______.考點(diǎn)二利用對(duì)稱性求函數(shù)解析式或函數(shù)值一、單選題1．已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0成中心對(duì)稱，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0，其中a為常數(shù)，若存在SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．0 B．1 C．2 D．SKIPIF1<03．函數(shù)SKIPIF1<0的圖像與函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，其中SKIPIF1<0（

）A．3 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．下列函數(shù)中，其圖象與函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．下列函數(shù)與SKIPIF1<0的圖象關(guān)于原點(diǎn)對(duì)稱的函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0二、多選題6．已知函數(shù)SKIPIF1<0定義域?yàn)镾KIPIF1<0，SKIPIF1<0為偶函數(shù)，SKIPIF1<0為奇函數(shù)，則下列一定成立的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知函數(shù)SKIPIF1<0，則（

）A．函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱B．SKIPIF1<0有三個(gè)零點(diǎn)C．點(diǎn)SKIPIF1<0是曲線SKIPIF1<0的對(duì)稱中心D．曲線SKIPIF1<0與SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱三、填空題8．函數(shù)SKIPIF1<0的圖像關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱，則SKIPIF1<0______.9．奇函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，SKIPIF1<0，則SKIPIF1<0_________.10．已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0______.11．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，若SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，則SKIPIF1<0___.12．給出定義：設(shè)SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，若方程SKIPIF1<0有實(shí)數(shù)解SKIPIF1<0，則稱SKIPIF1<0為函數(shù)SKIPIF1<0的“拐點(diǎn)”，經(jīng)研究發(fā)現(xiàn)所有的三次函數(shù)SKIPIF1<0都有“拐點(diǎn)”，且該“拐點(diǎn)”也是函數(shù)SKIPIF1<0的圖像的對(duì)稱中心，若函數(shù)SKIPIF1<0，則SKIPIF1<0______.13．若函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，則SKIPIF1<0___________.14．已知SKIPIF1<0是定義在R上的函數(shù)SKIPIF1<0的對(duì)稱軸，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0的解析式是_______．四、解答題15．函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，且對(duì)任意實(shí)數(shù)SKIPIF1<0，都有SKIPIF1<0成立.已知當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，求函數(shù)SKIPIF1<0的表達(dá)式；(2)若函數(shù)SKIPIF1<0的最大值為1，當(dāng)SKIPIF1<0時(shí)，求不等式SKIPIF1<0的解集.16．設(shè)SKIPIF1<0同時(shí)滿足條件SKIPIF1<0和對(duì)任意SKIPIF1<0，都有SKIPIF1<0成立.（1）求SKIPIF1<0的解析式；（2）設(shè)函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，且在定義域內(nèi)SKIPIF1<0.若函數(shù)SKIPIF1<0的圖象與SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，求SKIPIF1<0.考點(diǎn)三利用對(duì)稱性研究單調(diào)性一、單選題1．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，且SKIPIF1<0為偶函數(shù)，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，若函數(shù)SKIPIF1<0為偶函數(shù)，且SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．已知定義在R上的函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，若函數(shù)SKIPIF1<0為偶函數(shù)，且SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0在SKIPIF1<0單調(diào)遞減，且SKIPIF1<0，則使得不等式SKIPIF1<0成立的實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．已知函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，滿足對(duì)任意SKIPIF1<0，都有SKIPIF1<0，若SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減，則實(shí)數(shù)a的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0,則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0二、多選題7．已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，SKIPIF1<0，且圖像關(guān)于SKIPIF1<0對(duì)稱，則SKIPIF1<0（

）A．SKIPIF1<0 B．周期SKIPIF1<0C．在SKIPIF1<0單調(diào)遞減 D．滿足SKIPIF1<08．對(duì)于定義在SKIPIF1<0上的函數(shù)SKIPIF1<0，若SKIPIF1<0是奇函數(shù)，SKIPIF1<0是偶函數(shù)，且在SKIPIF1<0上單調(diào)遞減，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0在SKIPIF1<0上單調(diào)遞減9．已知函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，且SKIPIF1<0關(guān)于SKIPIF1<0對(duì)稱，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0為偶函數(shù) D．任意SKIPIF1<0且SKIPIF1<0，都有SKIPIF1<0三、填空題10．已知函數(shù)SKIPIF1<0定義域?yàn)閰^(qū)間SKIPIF1<0，且圖像關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱.當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則滿足SKIPIF1<0的SKIPIF1<0的取值范圍是__________.11．已知函數(shù)SKIPIF1<0，對(duì)于SKIPIF1<0，都有SKIPIF1<0成立，且任取SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的取值范圍是___________.12．已知函數(shù)SKIPIF1<0，則使不等式SKIPIF1<0成立的實(shí)數(shù)t的取值范圍是____.考點(diǎn)四對(duì)稱性的應(yīng)用一、單選題1．已知函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，且在(－∞，SKIPIF1<0]上單調(diào)遞增，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則a，b，c的大小關(guān)系為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．定義在SKIPIF1<0上的函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是增函數(shù)，且函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．給出定義：設(shè)SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù).若方程SKIPIF1<0有實(shí)數(shù)解SKIPIF1<0，則稱SKIPIF1<0為函數(shù)SKIPIF1<0的“拐點(diǎn)”.經(jīng)研究發(fā)現(xiàn)所有的三次函數(shù)SKIPIF1<0都有“拐點(diǎn)”，且該“拐點(diǎn)”也是函數(shù)SKIPIF1<0的圖象的對(duì)稱中心.若函數(shù)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，且SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0成中心對(duì)稱.當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0（

）A．1 B．3 C．SKIPIF1<0 D．SKIPIF1<05．已知SKIPIF1<0是定義在R上的奇函數(shù)，且SKIPIF1<0，函數(shù)SKIPIF1<0.若SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱，則SKIPIF1<0（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．定義在SKIPIF1<0上的