Question

(1)寫出函數(shù)y=x²-2x的單調(diào)區(qū)間及其圖像的對(duì)稱軸,觀察：在函數(shù)圖像對(duì)稱軸兩側(cè)的單調(diào)性有什么特點(diǎn)?

(2)寫出函數(shù)y=|x|的單調(diào)區(qū)間及其圖像的對(duì)稱軸,觀察：在函數(shù)圖像對(duì)稱軸兩側(cè)的單調(diào)性有什么特點(diǎn)?

(3)定義在［-4,8］上的函數(shù)y=f(x)的圖像關(guān)于直線x=2對(duì)稱,y=f(x)的部分圖像如圖所示,請(qǐng)補(bǔ)全函數(shù)y=f(x)的圖像,并寫出其單調(diào)區(qū)間,觀察：在函數(shù)圖像對(duì)稱軸兩側(cè)的單調(diào)性有什么特點(diǎn)?

(4)由以上你發(fā)現(xiàn)了什么結(jié)論?試加以證明.

Answer 1

思路分析：本題探討函數(shù)的單調(diào)性的性質(zhì).利用歸納、猜想、證明的方法得到結(jié)論，用定義證明結(jié)論.

解：(1)函數(shù)y=x²-2x的單調(diào)遞減區(qū)間是(-∞,1),單調(diào)遞增區(qū)間是(1,+∞);對(duì)稱軸是直線x=1;區(qū)間(-∞,1)和區(qū)間(1,+∞)關(guān)于直線x=1對(duì)稱,單調(diào)性相反.

(2)函數(shù)y=|x|的單調(diào)遞減區(qū)間是(-∞,0),單調(diào)遞增區(qū)間是(0,+∞);對(duì)稱軸是y軸即直線x=0;區(qū)間(-∞,0)和區(qū)間(0,+∞)關(guān)于直線x=0對(duì)稱,單調(diào)性相反.

(3)函數(shù)y=f(x),x∈［-4,8］的圖像如圖所示.

    函數(shù)y=f(x)的單調(diào)遞增區(qū)間是［-4,-1］,［2,5］;單調(diào)遞減區(qū)間是［5,8］,［-1,2］;區(qū)間［-4,-1］和區(qū)間［5,8］關(guān)于直線x=2對(duì)稱,單調(diào)性相反,區(qū)間［-1,2］和區(qū)間［2,5］關(guān)于直線x=2對(duì)稱,單調(diào)性相反.

(4)可以發(fā)現(xiàn)結(jié)論:如果函數(shù)y=f(x)的圖像關(guān)于直線x=m對(duì)稱,那么函數(shù)y=f(x)在直線x=m兩側(cè)對(duì)稱單調(diào)區(qū)間內(nèi)具有相反的單調(diào)性.證明如下:

    不妨設(shè)函數(shù)y=f(x)在對(duì)稱軸直線x=m的右側(cè)一個(gè)區(qū)間［a,b］上是增函數(shù),區(qū)間［a,b］關(guān)于直線x=m的對(duì)稱區(qū)間是［2m-b,2m-a］.

    由于函數(shù)y=f(x)的圖像關(guān)于直線x=m對(duì)稱,則f(x)=f(2m-x).

    設(shè)2m-b≤x₁＜x₂≤2m-a,則b≥2m-x₁＞2m-x₂≥a,

f(x₁)-f(x₂)=f(2m-x₁)-f(2m-x₂).

又∵函數(shù)y=f(x)在［a,b］上是增函數(shù),∴f(2m-x₁)-f(2m-x₂)＞0.

∴f(x₁)-f(x₂)＞0.

∴f(x₁)＞f(x₂).

∴函數(shù)y=f(x)在區(qū)間［2m-b,2m-a］上是減函數(shù).

∴當(dāng)函數(shù)y=f(x)在對(duì)稱軸直線x=m的右側(cè)一個(gè)區(qū)間［a,b］上是增函數(shù)時(shí),在［a,b］關(guān)于直線x=m的對(duì)稱區(qū)間［2m-b,2m-a］上則是減函數(shù),即單調(diào)性相反.

    因此有結(jié)論:如果函數(shù)y=f(x)的圖像關(guān)于直線x=m對(duì)稱,那么函數(shù)y=f(x)在對(duì)稱軸兩側(cè)的對(duì)稱單調(diào)區(qū)間內(nèi)具有相反的單調(diào)性.