新高考數(shù)學二輪復習 題型歸納演練專題2-2 點對稱+軸對稱+周期+單調(diào)性原卷版

專題2-2點對稱+軸對稱+周期+單調(diào)性目錄TOC\o"1-1"\h\u專題2-2點對稱+軸對稱+周期+單調(diào)性 1 1題型一：利用奇偶性+單調(diào)性解不等式 1題型二：構造奇偶函數(shù)求函數(shù)值 3題型三：奇偶性+周期性 4題型四：對稱性+奇偶性 5題型五：對稱性+周期性+奇偶性（知二推三） 7題型六：三角函數(shù)中的對稱性，周期性，奇偶性與單調(diào)性問題 9 12題型一：利用奇偶性+單調(diào)性解不等式【典型例題】例題1．（2022·河南·新密市第二高級中學高一階段練習）定義在實數(shù)SKIPIF1<0上的奇函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減，且SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·廣東·深圳市燕川中學高一期中）偶函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，且對于任意SKIPIF1<0，均有SKIPIF1<0成立，若SKIPIF1<0，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0為偶函數(shù)，且當SKIPIF1<0時，SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【提分秘籍】1、對于任意SKIPIF1<0，均有SKIPIF1<0成立，注意功能用來判斷函數(shù)的單調(diào)性（有具體函數(shù)時，直接求導可求單調(diào)性）；2、解不等式常涉及到奇偶性，注意配圖解不等式3、涉及到偶函數(shù)時：如果口朝上：誰離對稱軸（SKIPIF1<0）遠，誰的函數(shù)值就大；如果口朝下：誰離對稱軸（SKIPIF1<0）遠，誰的函數(shù)值就小。【變式演練】1．（2022·江西江西·高三階段練習（文））設a為實數(shù)，定義在R上的偶函數(shù)SKIPIF1<0滿足：SKIPIF1<0在SKIPIF1<0上為增函數(shù)，則使得SKIPIF1<0成立的a的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（多選）（2022·江蘇·句容碧桂園學校高一期中）已知定義在SKIPIF1<0上函數(shù)SKIPIF1<0的圖象是連續(xù)不斷的，且滿足以下條件：①SKIPIF1<0，SKIPIF1<0；②SKIPIF1<0，當SKIPIF1<0時，都有SKIPIF1<0；③SKIPIF1<0．則下列選項成立的是（

）A．SKIPIF1<0 B．若SKIPIF1<0，則SKIPIF1<0C．若SKIPIF1<0，則SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0，使得SKIPIF1<03．（2022·廣東·廣州市第五中學高一階段練習）已知偶函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，若SKIPIF1<0，則滿足SKIPIF1<0的x的取值范圍是___________.題型二：構造奇偶函數(shù)求函數(shù)值【典型例題】例題1．（2022·陜西·無高一期中）已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值與最小值分別為SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0在SKIPIF1<0，SKIPIF1<0上的最大值和最小值分別為SKIPIF1<0、SKIPIF1<0，則SKIPIF1<0（

）A．8 B．6 C．4 D．2例題3．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．5 D．7【提分秘籍】對于SKIPIF1<0本身不具有奇偶性，通過構造（通常將尾巴常數(shù)變?yōu)?），構造奇函數(shù)，利用奇函數(shù)的對稱性，求函數(shù)值.【變式演練】1．（2022·河南·高三階段練習（理））已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．5 D．72．（2022·河南省淮陽中學高三階段練習（文））已知函數(shù)SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0上的最大值與最小值之和為______．3．（2022·江西·貴溪市實驗中學高三階段練習（文））已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0______．題型三：奇偶性+周期性【典型例題】例題1．（2021·湖北·高一階段練習）已知函數(shù)SKIPIF1<0是定義在R上的奇函數(shù)，且滿足SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0=（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·河南河南·一模（文））函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，且SKIPIF1<0，則SKIPIF1<0（

）A．-1 B．0 C．1 D．2例題3．（2022·全國·高三階段練習）已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，SKIPIF1<0，若對任意SKIPIF1<0，都有SKIPIF1<0，對任意SKIPIF1<0且SKIPIF1<0，都有SKIPIF1<0，則SKIPIF1<0____________．【提分秘籍】函數(shù)周期性的常用結論與技巧設函數(shù)SKIPIF1<0，SKIPIF1<0.①若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；②若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；③若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；④若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；⑤SKIPIF1<0，則函數(shù)的周期SKIPIF1<0【變式演練】1．（2022·湖南省桃源縣第一中學高三階段練習）函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，滿足SKIPIF1<0，當SKIPIF1<0時，有SKIPIF1<0，求SKIPIF1<0的值（

）A．0 B．1 C．SKIPIF1<0 D．SKIPIF1<02．（2022·河南·新密市第一高級中學高一階段練習）已知函數(shù)SKIPIF1<0對于任意SKIPIF1<0都有SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0在區(qū)間SKIPIF1<0上是單調(diào)遞增的，則SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的大小關系是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．（2022·四川·鄰水縣九龍中學高三階段練習（理））已知偶函數(shù)SKIPIF1<0的定義域為R，滿足SKIPIF1<0，且當SKIPIF1<0，則SKIPIF1<0_______________題型四：對稱性+奇偶性【典型例題】例題1．（2022·寧夏·銀川一中高三階段練習（文））已知函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，且SKIPIF1<0為奇函數(shù)，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0的所有根之和等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·陜西·永壽縣中學高一期中）已知函數(shù)SKIPIF1<0是偶函數(shù)，當SKIPIF1<0時，SKIPIF1<0恒成立，設SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的大小關系為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例題3．（2023·全國·高三專題練習）設函數(shù)SKIPIF1<0，則滿足SKIPIF1<0的SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【提分秘籍】函數(shù)對稱性（異號對稱）（1）軸對稱：若函數(shù)SKIPIF1<0關于直線SKIPIF1<0對稱，則①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0（2）點對稱：若函數(shù)SKIPIF1<0關于直線SKIPIF1<0對稱，則①SKIPIF1<0②SKIPIF1<0③SKIPIF1<0（2）點對稱：若函數(shù)SKIPIF1<0關于直線SKIPIF1<0對稱，則①SKIPIF1<0②SKIPIF1<0③SKIPIF1<0【變式演練】1．（2022·江西·臨川一中高三期中（文））已知定義在R上的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，且在區(qū)間SKIPIF1<0上是減函數(shù)，令SKIPIF1<0，則SKIPIF1<0的大小關系為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·福建省廈門第六中學高三階段練習）設函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，SKIPIF1<0為偶函數(shù)，SKIPIF1<0為奇函數(shù)，當SKIPIF1<0時，SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·全國·高一課時練習）已知SKIPIF1<0是R上的奇函數(shù)，且SKIPIF1<0，當SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0時，SKIPIF1<0，則當SKIPIF1<0時，不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型五：對稱性+周期性+奇偶性（知二推三）【典型例題】例題1．（2022·北京市第十七中學高一期中）已知SKIPIF1<0是定義域為SKIPIF1<0的奇函數(shù)，滿足SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0

）A．0 B．1 C．2 D．2021例題2．（2023·全國·高三專題練習）已知定義域為SKIPIF1<0的函數(shù)SKIPIF1<0存在導函數(shù)SKIPIF1<0，且滿足SKIPIF1<0，則曲線SKIPIF1<0在點SKIPIF1<0處的切線方程可能是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（多選）（2022·全國·高一課時練習）已知定義在SKIPIF1<0上的偶函數(shù)滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0是減函數(shù)，則下列四個命題中正確的是（

）A．SKIPIF1<0B．直線SKIPIF1<0為函數(shù)SKIPIF1<0圖象的一條對稱軸C．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上存在2個零點D．若SKIPIF1<0在區(qū)間SKIPIF1<0上的根為SKIPIF1<0，則SKIPIF1<0【提分秘籍】（1）例1中：奇偶性+對稱性SKIPIF1<0周期性已知SKIPIF1<0是奇函數(shù)，則SKIPIF1<0；又SKIPIF1<0，則SKIPIF1<0關于SKIPIF1<0對稱SKIPIF1<0SKIPIF1<0，綜合考慮SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0（2）例2中：奇偶性+周期性SKIPIF1<0對稱性由SKIPIF1<0和SKIPIF1<0可知SKIPIF1<0SKIPIF1<0SKIPIF1<0關于SKIPIF1<0對稱【變式演練】1．（2022·浙江·高一期中）己知SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，且函數(shù)SKIPIF1<0的圖像關于原點對稱，若SKIPIF1<0，則SKIPIF1<0的值為（

）A．0 B．1 C．SKIPIF1<0 D．22．（2022·江西·臨川一中高三階段練習（理））已知SKIPIF1<0是定義域為SKIPIF1<0的奇函數(shù)，滿足SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·貴州·凱里一中高二期中）已知函數(shù)SKIPIF1<0是SKIPIF1<0上的偶函數(shù)，且SKIPIF1<0的圖象關于點SKIPIF1<0對稱，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．24．（多選）（2022·山東·濰坊七中高三階段練習）設函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，且滿足SKIPIF1<0，SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<0是偶函數(shù) B．SKIPIF1<0為奇函數(shù)C．函數(shù)SKIPIF1<0有SKIPIF1<0個不同的零點 D．SKIPIF1<05．（2023·浙江溫州·模擬預測）定義在R上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0__________，SKIPIF1<0__________．題型六：三角函數(shù)中的對稱性，周期性，奇偶性與單調(diào)性問題【典型例題】例題1．（2022·河南·濮陽南樂一高高三階段練習（文））已知點SKIPIF1<0是函數(shù)SKIPIF1<0圖象的一個對稱中心，其中SKIPIF1<0為常數(shù)且SKIPIF1<0，則以下結論正確的是（

）A．函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0B．將函數(shù)SKIPIF1<0的圖象向右平移SKIPIF1<0個單位所得的圖象關于SKIPIF1<0軸對稱C．函數(shù)SKIPIF1<0在SKIPIF1<0上的最小值為SKIPIF1<0D．若SKIPIF1<0，則SKIPIF1<0例題2．（2022·山西·高二學業(yè)考試）將函數(shù)SKIPIF1<0的圖象向左平移SKIPIF1<0個單位，得到函數(shù)SKIPIF1<0的圖象，那么下列說法正確的是（

）A．函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0 B．函數(shù)SKIPIF1<0的圖象關于點SKIPIF1<0對稱C．函數(shù)SKIPIF1<0為奇函數(shù) D．函數(shù)SKIPIF1<0的圖象關于直線SKIPIF1<0對稱例題3．（多選）（2022·全國·高三專題練習）下列關于函數(shù)SKIPIF1<0的說法正確的是（

）A．在區(qū)間SKIPIF1<0上單調(diào)遞增 B．最小正周期是SKIPIF1<0C．圖象關于點SKIPIF1<0成中心對稱 D．圖象關于直線SKIPIF1<0對稱【提分秘籍】1、三角函數(shù)的對稱性，周期性，奇偶性，單調(diào)性，考查時可能單獨考，也可能以多選的形式綜合在一個題目中考查.2、三角函數(shù)的奇偶性(1)函數(shù)SKIPIF1<0是奇函數(shù)?SKIPIF1<0(SKIPIF1<0)，是偶函數(shù)?SKIPIF1<0(SKIPIF1<0)；(2)函數(shù)SKIPIF1<0是奇函數(shù)?SKIPIF1<0(SKIPIF1<0)，是偶函數(shù)?SKIPIF1<0(SKIPIF1<0)；(3)函數(shù)SKIPIF1<0是奇函數(shù)?SKIPIF1<0(SKIPIF1<0)．3、三角函數(shù)的對稱性(1)函數(shù)SKIPIF1<0的圖象的對稱軸由SKIPIF1<0(SKIPIF1<0)解得，對稱中心的橫坐標由SKIPIF1<0(SKIPIF1<0)解得；(2)函數(shù)SKIPIF1<0的圖象的對稱軸由SKIPIF1<0(SKIPIF1<0)解得，對稱中心的橫坐標由SKIPIF1<0(SKIPIF1<0)解得；(3)函數(shù)SKIPIF1<0的圖象的對稱中心由SKIPIF1<0SKIPIF1<0)解得．【變式演練】1．（2022·湖南·長沙一中高三階段練習）已知函數(shù)SKIPIF1<0的圖象的一條對稱軸與其相鄰的一個對稱中心的距離為SKIPIF1<0，將SKIPIF1<0的圖象向右平移SKIPIF1<0個單位長度得到函數(shù)SKIPIF1<0的圖象.若函數(shù)SKIPIF1<0的圖象在區(qū)間SKIPIF1<0上是增函數(shù)，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（多選）（2022·河南·新密市第一高級中學高一階段練習）已知函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，函數(shù)SKIPIF1<0的圖象關于點SKIPIF1<0對稱，函數(shù)SKIPIF1<0的圖象關于直線SKIPIF1<0對稱，下列結論正確的有（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0D．SKIPIF1<03．（多選）（2022·福建·廈門外國語學校高三期中）將函數(shù)SKIPIF1<0圖像上所有的點向右平移SKIPIF1<0個單位長度，得到函數(shù)SKIPIF1<0的圖像，則下列說法正確的是（

）A．SKIPIF1<0的最小正周期為πB．SKIPIF1<0圖像的一個對稱中心為SKIPIF1<0C．SKIPIF1<0的單調(diào)遞增區(qū)間為SKIPIF1<0D．SKIPIF1<0的圖像與函數(shù)SKIPIF1<0的圖像重合4．（多選）（2022·廣東廣雅中學高三階段練習）設SKIPIF1<0，則下列說法正確的有（

）A．SKIPIF1<0的最小正周期為SKIPIF1<0 B．SKIPIF1<0在SKIPIF1<0上單調(diào)遞增C．SKIPIF1<0的圖象關于SKIPIF1<0軸對稱 D．SKIPIF1<0的圖象關于SKIPIF1<0對稱5．（2022·陜西·蒲城縣蒲城中學高三階段練習（文））將函數(shù)SKIPIF1<0(SKIPIF1<0)的圖象向左平移SKIPIF1<0個單位長度,得到曲線SKIPIF1<0.若SKIPIF1<0關于SKIPIF1<0軸對稱,則SKIPIF1<0的最小值是______．6．（2022·北京海淀·高三期中）若函數(shù)SKIPIF1<0和SKIPIF1<0的圖象的對稱中心完全重合，則SKIPIF1<0__________；SKIPIF1<0__________．一、單選題1．（遼寧省遼陽市2022-2023學年高一上學期期中數(shù)學試題）已知定義在R上的奇函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，且SKIPIF1<0，則滿足SKIPIF1<0的x的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·廣東·深圳科學高中高一期中）己知函數(shù)SKIPIF1<0圖象關于點SKIPIF1<0成中心對稱圖形的充要條件是函數(shù)SKIPIF1<0為奇函數(shù)，則函數(shù)SKIPIF1<0圖象的對稱中心是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·福建·廈門外國語學校高一期中）已知函數(shù)SKIPIF1<0為奇函數(shù)，且在區(qū)間SKIPIF1<0上是增函數(shù)，若SKIPIF1<0，則SKIPIF1<0的解集是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．（2022·貴州·高一期中）已知SKIPIF1<0是定義域為SKIPIF1<0的奇函數(shù)，且SKIPIF1<0為偶函數(shù)，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．0 D．35．（2022·福建泉州·高三期中）已知定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022·福建·高三階段練習）已知函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．（2022·江蘇泰州·高三期中）已知函數(shù)SKIPIF1<0（SKIPIF1<0，SKIPIF1<0），直線SKIPIF1<0和點SKIPIF1<0分別是SKIPIF1<0圖象相鄰的對稱軸和對稱中心，則下列說法正確的是（

）A．函數(shù)SKIPIF1<0為奇函數(shù)B．函數(shù)SKIPIF1<0的圖象關于點SKIPIF1<0對稱C．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上為單調(diào)函數(shù)D．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上有12個零點8．（2022·四川·成都七中模擬預測（文））已知定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0，則下列結論正確個數(shù)為（

）①SKIPIF1<0的一個周期為2

②SKIPIF1<0③SKIPIF1<0

④SKIPIF1<0圖象關于直線SKIPIF1<0對稱A．1 B．2 C．3 D．4二、多選題9．（2022·黑龍江·密山市第四中學高三階段練習）已知函數(shù)SKIPIF1<0，下列說法正確的是（

）A．最小正周期為SKIPIF1<0 B．圖象關于點SKIPIF1<0對稱C．圖象關于直線SKIPIF1<0對稱 D．在區(qū)間SKIPIF1<0的值域為SKIPIF1<010．（2022·黑龍江齊齊哈爾·高三期中）已知SKIPIF1<0，則下列說法錯誤的是（

）A．函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0B．函數(shù)SKIPIF1<0為奇函數(shù)C．函數(shù)SKIPIF1<0在SKIPIF1<0上的值域為SKIPIF1<0D．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的零點個數(shù)為811