新高考數(shù)學(xué)二輪復(fù)習(xí)考點(diǎn)講義第十三講三角函數(shù)圖象及性質(zhì)（原卷版）

第十三講：三角函數(shù)圖象及性質(zhì)【考點(diǎn)梳理】1、用五點(diǎn)法作正弦函數(shù)和余弦函數(shù)的簡圖(1)在正弦函數(shù)SKIPIF1<0，SKIPIF1<0的圖象中，五個(gè)關(guān)鍵點(diǎn)是：SKIPIF1<0．(2)在余弦函數(shù)SKIPIF1<0，SKIPIF1<0的圖象中，五個(gè)關(guān)鍵點(diǎn)是：SKIPIF1<0．2、SKIPIF1<0的圖象與性質(zhì)（1）最小正周期：SKIPIF1<0.（2）定義域與值域：SKIPIF1<0的定義域?yàn)镽，值域?yàn)閇-A,A].（3）最值（SKIPIF1<0）.對(duì)于SKIPIF1<0，SKIPIF1<0（4）對(duì)稱軸與對(duì)稱中心.（SKIPIF1<0）對(duì)于SKIPIF1<0，SKIPIF1<0（5）單調(diào)性.（SKIPIF1<0）對(duì)于SKIPIF1<0，SKIPIF1<0（6）平移與伸縮由SKIPIF1<0的圖象變換得到SKIPIF1<0(SKIPIF1<0，SKIPIF1<0)的圖象的兩種方法(1)先平移后伸縮(2)先伸縮后平移【典型題型講解】考點(diǎn)一：三角函數(shù)的性質(zhì)【典例例題】例1．（多選）（2022·廣東汕頭·高三期末）對(duì)于函數(shù)SKIPIF1<0，x∈R，則（

）A．f(x)的最大值為1 B．直線SKIPIF1<0為其對(duì)稱軸C．f(x)在SKIPIF1<0上單調(diào)遞增 D．點(diǎn)SKIPIF1<0為其對(duì)稱中心例2．（2022·廣東珠?！じ呷谀╆P(guān)于函數(shù)SKIPIF1<0，下列說法正確的是（

）A．函數(shù)SKIPIF1<0的圖象可由函數(shù)SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位得到B．SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱C．SKIPIF1<0的表達(dá)式可以改寫為SKIPIF1<0D．若函數(shù)SKIPIF1<0在SKIPIF1<0的值域?yàn)镾KIPIF1<0，則m的取值范圍是SKIPIF1<0【方法技巧與總結(jié)】研究三角函數(shù)的性質(zhì)，關(guān)鍵式將函數(shù)化為SKIPIF1<0與SKIPIF1<0的形式利用正余弦函數(shù)與復(fù)合函數(shù)的性質(zhì)求解.【變式訓(xùn)練】1．（2022·廣東揭陽·高三期末）已知函數(shù)SKIPIF1<0，則該函數(shù)的增區(qū)間為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<02.（2022·廣東茂名·一模）函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值為______3.已知函數(shù)SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<0為奇函數(shù) B．SKIPIF1<0為偶函數(shù)C．SKIPIF1<0為奇函數(shù) D．SKIPIF1<0為偶函數(shù)4.設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0時(shí)，SKIPIF1<0的最小值為SKIPIF1<0，則（

）A．函數(shù)SKIPIF1<0的周期為SKIPIF1<0B．將函數(shù)SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位，得到的函數(shù)為奇函數(shù)C．當(dāng)SKIPIF1<0，SKIPIF1<0的值域?yàn)镾KIPIF1<0D．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的零點(diǎn)個(gè)數(shù)共有6個(gè)5．設(shè)函數(shù)SKIPIF1<0，SKIPIF1<0，其中SKIPIF1<0，SKIPIF1<0.若SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0的最小正周期大于SKIPIF1<0，則（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<06.若函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，又SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0的最小值為SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<07.（2022·廣東湛江·一模）已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0在區(qū)間SKIPIF1<0上有且只有一個(gè)極大值點(diǎn)，則SKIPIF1<0的最大值為___________.8．（2021·廣東佛山·一模）已知函數(shù)SKIPIF1<0.從下面的兩個(gè)條件中任選其中一個(gè)：①SKIPIF1<0；②若SKIPIF1<0，且SKIPIF1<0的最小值為SKIPIF1<0，SKIPIF1<0，求解下列問題：(1)化簡SKIPIF1<0的表達(dá)式并求SKIPIF1<0的單調(diào)遞增區(qū)間；(2)已知SKIPIF1<0，求SKIPIF1<0的值.考點(diǎn)二：三角函數(shù)的圖象【典例例題】例1．（2022·廣東·金山中學(xué)高三期末）為了得到函數(shù)SKIPIF1<0的圖象，可以將函數(shù)SKIPIF1<0的圖象（

）A．向右平移SKIPIF1<0個(gè)單位長度 B．向右平移SKIPIF1<0個(gè)單位長度C．向左平移SKIPIF1<0個(gè)單位長度 D．向左平移SKIPIF1<0個(gè)單位長度例2．（多選）（2022·廣東中山·高三期末）已知函數(shù)SKIPIF1<0的部分圖象如圖所示，下列說法正確的是（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱B．函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱C．函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減D．函數(shù)SKIPIF1<0圖象向右平移SKIPIF1<0個(gè)單位可得函數(shù)SKIPIF1<0的圖象【方法技巧與總結(jié)】1.圖象變換過程中務(wù)必分清式相位變換，還是周期變換.變換只是相對(duì)于其中的自變量SKIPIF1<0而言的，如果SKIPIF1<0的系數(shù)不是1，就要把這個(gè)系數(shù)提取后再確定變換的單位長度和方向.2.已知函數(shù)SKIPIF1<0的圖象求解析式時(shí)，常采用待定系數(shù)法，由圖中的最高點(diǎn)、最低點(diǎn)或特殊點(diǎn)求A；由函數(shù)的周期確定SKIPIF1<0；確定SKIPIF1<0常根據(jù)“五點(diǎn)法”中的五個(gè)點(diǎn)求解，其中一般把第一個(gè)零點(diǎn)作為突破口，可以從圖象的升降找準(zhǔn)第一個(gè)零點(diǎn)的位置.【變式訓(xùn)練】1．（2022·廣東廣東·一模）將正弦函數(shù)圖象上各點(diǎn)的橫坐標(biāo)縮短為原來的SKIPIF1<0，縱坐標(biāo)不變，得到函數(shù)SKIPIF1<0的圖象．則SKIPIF1<0圖象的一個(gè)對(duì)稱中心為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·廣東韶關(guān)·一模）若將函數(shù)SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位長度，則平移后的函數(shù)圖象的一條對(duì)稱軸為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．（2022·廣東廣州·一模）將函數(shù)SKIPIF1<0的圖象向右平移SKIPIF1<0個(gè)單位，得到函數(shù)SKIPIF1<0的圖象，則下列說法正確的是（

）A．若SKIPIF1<0，則SKIPIF1<0是偶函數(shù)B．若SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減C．若SKIPIF1<0，則SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱D．若SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增4．(多選)（2022·廣東·鐵一中學(xué)高三期末）將函數(shù)SKIPIF1<0的圖象向右平移SKIPIF1<0個(gè)單位長度后得到函數(shù)SKIPIF1<0的圖象，且SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<0為奇函數(shù)B．SKIPIF1<0C．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0在SKIPIF1<0上有4個(gè)極值點(diǎn)D．若SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，則SKIPIF1<0的最大值為55．(多選)（2022·廣東東莞·高三期末）已知函數(shù)SKIPIF1<0，若SKIPIF1<0且對(duì)任意SKIPIF1<0都有SKIPIF1<0，則下列結(jié)論正確的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位后，圖象關(guān)于原點(diǎn)對(duì)稱D．SKIPIF1<0的圖象向右平移SKIPIF1<0個(gè)單位后，圖象關(guān)于SKIPIF1<0軸對(duì)稱6．（多選）（2022·廣東清遠(yuǎn)·高三期末）將函數(shù)SKIPIF1<0圖象上所有的點(diǎn)向右平移SKIPIF1<0個(gè)單位長度后，得到函數(shù)SKIPIF1<0的圖象，若函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0的最小值是SKIPIF1<0B．SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱C．SKIPIF1<0的最小正周期是SKIPIF1<0D．SKIPIF1<0的單調(diào)遞增區(qū)間是SKIPIF1<07．（多選）（2022·廣東惠州·一模）已知函數(shù)SKIPIF1<0（其中SKIPIF1<0，SKIPIF1<0，SKIPIF1<0）的部分圖象如圖所示，則下列結(jié)論正確的是（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0直線對(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在區(qū)間SKIPIF1<0上單調(diào)遞增D．SKIPIF1<0與圖象SKIPIF1<0的所有交點(diǎn)的橫坐標(biāo)之和為SKIPIF1<09．（2022·廣東茂名·二模）已知函數(shù)SKIPIF1<0的部分圖象如圖所示．將函數(shù)SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位得到SKIPIF1<0的圖象，則（）A．SKIPIF1<0） B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．（2022·廣東惠州·二模）已知函數(shù)SKIPIF1<0的部分圖象如圖所示，則下列結(jié)論中正確的是（

）A．SKIPIF1<0的最小正周期為SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0在SKIPIF1<0上單調(diào)遞增 D．SKIPIF1<0為奇函數(shù)【鞏固練習(xí)】一、單選題1．已知直線SKIPIF1<0是函數(shù)SKIPIF1<0的圖象的一條對(duì)稱軸，為了得到函數(shù)SKIPIF1<0的圖象，可把函數(shù)SKIPIF1<0的圖象（

）A．向左平移SKIPIF1<0個(gè)單位長度 B．向右平移SKIPIF1<0個(gè)單位長度C．向左平移SKIPIF1<0個(gè)單位長度 D．向右平移SKIPIF1<0個(gè)單位長度2．已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．5 D．73．已知函數(shù)SKIPIF1<0，則下列結(jié)論錯(cuò)誤的是（

）A．函數(shù)SKIPIF1<0的最小正周期是SKIPIF1<0B．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減C．函數(shù)SKIPIF1<0的圖象可由函數(shù)SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位長度，再向下平移1個(gè)單位長度得到D．函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱4．如圖是函數(shù)SKIPIF1<0的圖象的一部分，則要得到該函數(shù)的圖象，只需要將函數(shù)SKIPIF1<0的圖象（

）A．向左平移SKIPIF1<0個(gè)單位長度 B．向右平移SKIPIF1<0個(gè)單位長度C．向左平移SKIPIF1<0個(gè)單位長度 D．向右平移SKIPIF1<0個(gè)單位長度二、多選題5．已知函數(shù)SKIPIF1<0，則下列說法正確的是（

）A．直線SKIPIF1<0為函數(shù)f(x)圖象的一條對(duì)稱軸B．函數(shù)f(x)圖象橫坐標(biāo)縮短為原來的一半，再向左平移SKIPIF1<0后得到SKIPIF1<0C．函數(shù)f(x)在[－SKIPIF1<0，SKIPIF1<0]上單調(diào)遞增D．函數(shù)SKIPIF1<0的值域?yàn)閇－2，SKIPIF1<0]6．設(shè)函數(shù)SKIPIF1<0，則下列結(jié)論中正確的是（

）A．SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱 B．SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱C．SKIPIF1<0在SKIPIF1<0上單調(diào)遞減 D．SKIPIF1<0在SKIPIF1<0上的最小值為07．已知函數(shù)SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<0B．SKIPIF1<0的圖象關(guān)于原點(diǎn)對(duì)稱C．若SKIPIF1<0，則SKIPIF1<0D．對(duì)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，有SKIPIF1<0成立三、填空題8．寫出一個(gè)同時(shí)具有下列性質(zhì)①②③的函數(shù)SKIPIF1<0___________；已知函數(shù)滿足：①SKIPIF1<0；②SKIPIF1<0；③函數(shù)在SKIPIF1<0上單調(diào)遞減；9．已知函數(shù)SKIPIF1<0的部分圖象如圖所示，則SKIPIF1<0________．10．已知函數(shù)SKIPIF1<0，若SKIPIF1<0，且SKIPIF1<0在SKIPIF1<0上有最大值，沒有最小值，則SKIPIF1<0的最大值為______．四、解答題11．已知函數(shù)SKIPIF1<0(1)求SKIPIF1<0的值；(2)求函數(shù)SKIPIF1<0在SKIPIF1<0上的增區(qū)間和值域．12．設(shè)SKIPIF1<0內(nèi)角A，B，C的對(duì)邊分別為a，b，c，函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求SKIPIF1<0