新高考數(shù)學(xué)一輪復(fù)習(xí)講義 第12講 函數(shù)的圖像（原卷版）

第12講函數(shù)的圖像（精講）題型目錄一覽①作函數(shù)的圖像②函數(shù)圖像的辨識③函數(shù)圖像的應(yīng)用一、知識點梳理一、知識點梳理1．利用描點法作函數(shù)的圖象描點法作函數(shù)圖象的基本步驟是列表、描點、連線，具體為：(1)①確定函數(shù)的定義域；②化簡函數(shù)的解析式；③討論函數(shù)的性質(zhì)(奇偶性、單調(diào)性、周期性、最值等)．(2)列表(找特殊點：如零點、最值點、區(qū)間端點以及與坐標軸的交點等)．(3)描點、連線．2．利用圖象變換法作函數(shù)的圖象(1)平移變換提醒：“左加右減”只針對x本身，與x的系數(shù)無關(guān)，“上加下減”指的是在f(x)整體上加減．(2)對稱變換①y＝f(x)的圖象eq\o(→,\s\up7(關(guān)于x軸對稱))y＝－f(x)的圖象；②y＝f(x)的圖象eq\o(→,\s\up7(關(guān)于y軸對稱))y＝f(－x)的圖象；③y＝f(x)的圖象eq\o(→,\s\up7(關(guān)于原點對稱))y＝－f(－x)的圖象；④y＝ax(a＞0且a≠1)的圖象eq\o(→,\s\up7(關(guān)于直線y＝x對稱))y＝logax(a＞0且a≠1)的圖象．(3)伸縮變換①y＝f(x)的圖象eq\o(→,\s\up27(a＞1，橫坐標縮短為原來的\f(1,a)，縱坐標不變,0＜a＜1，橫坐標伸長為原來的\f(1,a)倍，縱坐標不變))y＝f(ax)的圖象；②y＝f(x)的圖象eq\o(→,\s\up10(a＞1，縱坐標伸長為原來的a倍，橫坐標不變),\s\do10(0＜a＜1，縱坐標縮短為原來的a倍，橫坐標不變))y＝af(x)的圖象．(4)翻轉(zhuǎn)變換①y＝f(x)的圖象eq\o(→,\s\up10(x軸下方部分翻折到上方),\s\do10(x軸及上方部分不變))y＝|f(x)|的圖象；②y＝f(x)的圖象eq\o(→,\s\up10(y軸右側(cè)部分翻折到左側(cè)),\s\do10(原y軸左側(cè)部分去掉，右側(cè)不變))y＝f(|x|)的圖象．【常用結(jié)論】1．函數(shù)圖象自身的軸對稱(1)f(－x)＝f(x)?函數(shù)y＝f(x)的圖象關(guān)于y軸對稱；(2)函數(shù)y＝f(x)的圖象關(guān)于x＝a對稱?f(a＋x)＝f(a－x)?f(x)＝f(2a－x)?f(－x)＝f(2a＋x)；(3)若函數(shù)y＝f(x)的定義域為R，且有f(a＋x)＝f(b－x)，則函數(shù)y＝f(x)的圖象關(guān)于直線x＝eq\f(a＋b,2)對稱．2．函數(shù)圖象自身的中心對稱(1)f(－x)＝－f(x)?函數(shù)y＝f(x)的圖象關(guān)于原點對稱；(2)函數(shù)y＝f(x)的圖象關(guān)于(a，0)對稱?f(a＋x)＝－f(a－x)?f(x)＝－f(2a－x)?f(－x)＝－f(2a＋x)；(3)函數(shù)y＝f(x)的圖象關(guān)于點(a，b)成中心對稱?f(a＋x)＝2b－f(a－x)?f(x)＝2b－f(2a－x)．3．兩個函數(shù)圖象之間的對稱關(guān)系(1)函數(shù)y＝f(a＋x)與y＝f(b－x)的圖象關(guān)于直線x＝eq\f(b－a,2)對稱(由a＋x＝b－x得對稱軸方程)；(2)函數(shù)y＝f(x)與y＝f(2a－x)的圖象關(guān)于直線x＝a對稱；(3)函數(shù)y＝f(x)與y＝2b－f(－x)的圖象關(guān)于點(0，b)對稱；(4)函數(shù)y＝f(x)與y＝2b－f(2a－x)的圖象關(guān)于點(a，b)對稱．二、題型分類精講二、題型分類精講刷真題明導(dǎo)向刷真題明導(dǎo)向一、單選題1．如圖是下列四個函數(shù)中的某個函數(shù)在區(qū)間SKIPIF1<0的大致圖像，則該函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0，則圖象為如圖的函數(shù)可能是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．已知函數(shù)SKIPIF1<0若函數(shù)SKIPIF1<0恰有4個零點，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【答案】D題型一作函數(shù)的圖像策略方法作函數(shù)圖象的兩種常用方法【典例1】已知SKIPIF1<0.（1）畫函數(shù)SKIPIF1<0的圖象；（2）若直線SKIPIF1<0與SKIPIF1<0的圖象有4個不同的交點，求實數(shù)SKIPIF1<0的取值范圍以及所有交點橫坐標之和.【題型訓(xùn)練】一、解答題1．（1）畫函數(shù)SKIPIF1<0的圖象，并寫出單調(diào)增區(qū)間；（2）函數(shù)SKIPIF1<0有兩個零點，求a的取值范圍.2．畫函數(shù)圖象：SKIPIF1<0．3．畫函數(shù)圖象SKIPIF1<0題型二函數(shù)圖像的辨識策略方法辨析函數(shù)圖象的入手點(1)從函數(shù)的定義域，判斷圖象的左右位置；從函數(shù)的值域，判斷圖象的上下位置．(2)從函數(shù)的奇偶性，判斷圖象的對稱性．(3)從函數(shù)的特征點，排除不合要求的圖象．(4)從函數(shù)的單調(diào)性，判斷圖象的變化趨勢．(5)從函數(shù)的周期性，判斷圖象的循環(huán)往復(fù)．【典例1】如圖，函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的圖象大致為（

）A． B．C． D．【題型訓(xùn)練】一、單選題1．函數(shù)SKIPIF1<0的部分圖像大致為（

）A． B．C． D．2．函數(shù)SKIPIF1<0的大致圖象是（

）A． B．C． D．3．已知函數(shù)SKIPIF1<0的部分圖象如圖所示，則它的解析式可能是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．函數(shù)SKIPIF1<0的部分圖象大致為（

）A． B．C． D．5．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的大致圖象為（

）A． B．C． D．6．函數(shù)SKIPIF1<0的部分圖象大致為（

）A． B．C． D．7．函數(shù)SKIPIF1<0的大致圖像為（

）A． B．C． D．8．函數(shù)SKIPIF1<0的圖象大致為（

）A． B．C． D．9．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0的圖像大致為（

）A． B．C． D．10．已知函數(shù)SKIPIF1<0的部分圖象如圖所示，則SKIPIF1<0的解析式可能為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<011．函數(shù)SKIPIF1<0的圖象大致為（

）A． B．C． D．題型三函數(shù)圖像的應(yīng)用策略方法1.利用函數(shù)圖象研究不等式當不等式問題不能用代數(shù)法直接求解但其與函數(shù)有關(guān)時，可將不等式問題轉(zhuǎn)化為兩函數(shù)圖象(圖象易得)的上、下關(guān)系問題，利用圖象法求解．若函數(shù)為抽象函數(shù)，可根據(jù)題目畫出大致圖象，再結(jié)合圖象求解．2.利用函數(shù)圖象研究方程根的個數(shù)當方程與基本函數(shù)有關(guān)時，可以通過函數(shù)圖象研究方程的根，方程f(x)＝0的根就是f(x)的圖象與x軸交點的橫坐標，方程f(x)＝g(x)的根是函數(shù)y＝f(x)與函數(shù)y＝g(x)圖象的交點的橫坐標．【典例1】定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0；當SKIPIF1<0時，SKIPIF1<0；當SKIPIF1<0時，SKIPIF1<0.若對SKIPIF1<0，都有SKIPIF1<0，則SKIPIF1<0的取值范圍是__________.【典例2】對任意SKIPIF1<0，恒有SKIPIF1<0，對任意SKIPIF1<0，現(xiàn)已知函數(shù)SKIPIF1<0的圖像與SKIPIF1<0有4個不同的公共點，則正實數(shù)SKIPIF1<0的值為__________.【題型訓(xùn)練】一、單選題1．已知SKIPIF1<0，當SKIPIF1<0時，函數(shù)SKIPIF1<0的圖象恒在SKIPIF1<0軸下方，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．[0，1]3．已知函數(shù)SKIPIF1<0，若關(guān)于SKIPIF1<0的方程SKIPIF1<0恰有5個不同的實根，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．定義在SKIPIF1<0上的偶函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，關(guān)于SKIPIF1<0的不等式SKIPIF1<0的整數(shù)解有且只有SKIPIF1<0個，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知函數(shù)SKIPIF1<0，若不等式SKIPIF1<0有3個整數(shù)解，則實數(shù)a的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．已知函數(shù)SKIPIF1<0的圖象上恰有3對關(guān)于原點成中心對稱的點，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．已知SKIPIF1<0，函數(shù)SKIPIF1<0，若關(guān)于x的方程SKIPIF1<0有6個解，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．函數(shù)SKIPIF1<0，關(guān)于SKIPIF1<0的方程SKIPIF1<0有2個不相等的實數(shù)根，則實數(shù)a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<09．已知函數(shù)SKIPIF1<0，若關(guān)于x的方程SKIPIF1<0有三個互不相等的實根，則實數(shù)k的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0.若對任意SKIPIF1<0，都有SKIPIF1<0，則SKIPIF1<0的最大值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．已知函數(shù)SKIPIF1<0，SKIPIF1<0的定義域為SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，且SKIPIF1<0，則關(guān)于x的方程SKIPIF1<0有兩解時，實數(shù)a的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0是偶函數(shù)，當SKIPIF1<0時，SKIPIF1<0，若關(guān)于SKIPIF1<0的方程SKIPIF1<0有且僅有SKIPIF1<0個不同實數(shù)根，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0二、填空題13．已知四個函數(shù)：（1）SKIPIF1<0，（2）SKIPIF1<0，（3）SKIPIF1<0，（4）SKIPIF1<0，從中任選SKIPIF1<0個，則事件“所選SKIPIF1<0個函數(shù)的圖象有且僅有一個公共點”的概率為___________.14．已知函數(shù)SKIPIF1<0，則不等式SKIPIF1<0的解集是___________．15．定義在R上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0．若對任意SKIPIF1<0，都有SKIPIF1<0，則t的取值范圍是__________．16．已知函數(shù)SKIPIF1<0，若SKIPIF1<0互不相等，且