函數(shù)的對(duì)稱(chēng)性

1、函數(shù)的對(duì)稱(chēng)性函數(shù)的對(duì)稱(chēng)性有些函數(shù)有些函數(shù) 其圖像有著優(yōu)美的對(duì)稱(chēng)性，其圖像有著優(yōu)美的對(duì)稱(chēng)性，同時(shí)又有著優(yōu)美的對(duì)稱(chēng)關(guān)系式同時(shí)又有著優(yōu)美的對(duì)稱(chēng)關(guān)系式1-3-1-265432-xx( 1)(1)FF( 2)(2)FF()( )FxF x780 x （偶函數(shù)）（偶函數(shù)）Y=F(x)圖像關(guān)于直線圖像關(guān)于直線x=0對(duì)稱(chēng)對(duì)稱(chēng)知識(shí)回顧知識(shí)回顧l從從”形形”的角度看，的角度看，l從從”數(shù)數(shù)”的角度看，的角度看，F(xiàn)(-x)=F(x)XY1-3-1-265432782x ( )f x f(x)= f(4-x) f(1)= f(0)= f(-2)= f(310)= f(6)f(4-310)0 x4-xY=f(x)圖像關(guān)

2、于直線圖像關(guān)于直線x=2對(duì)稱(chēng)對(duì)稱(chēng)f(3)f(4)l從從”形形”的角度看，的角度看，l從從”數(shù)數(shù)”的角度看，的角度看，xy f(1+x)= f(3-x) f(2+x)= f(2-x) f(x)= f(4-x) 對(duì)于任意的對(duì)于任意的x你還能得到怎樣的等式？你還能得到怎樣的等式？l從從”形形”的角度看，的角度看，l從從”數(shù)數(shù)”的角度看，的角度看，Y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=2對(duì)稱(chēng)對(duì)稱(chēng)1-3-1-26543272x ( )f x0 x4-xYx-2-x1-3-1-26543278x=-1 f(x)= f(-2-x)x思考思考?若若y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=-1對(duì)稱(chēng)對(duì)稱(chēng)Yx-1

3、+x-1-x1-3-1-26543278x=-1 f(-1+x)= f(-1-x)思考思考?若若y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=-1對(duì)稱(chēng)對(duì)稱(chēng) f(x)= f(-2-x)Yx猜測(cè)：若猜測(cè)：若y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng) f(x)=f(2a-x)xa f(a-x)=f(a+x)xa在在y=f(x)圖像上任取一點(diǎn)圖像上任取一點(diǎn)P點(diǎn)點(diǎn)P關(guān)于直線關(guān)于直線x=a的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)P則有則有P的坐標(biāo)應(yīng)滿(mǎn)足的坐標(biāo)應(yīng)滿(mǎn)足y=f(x)也在也在f(x)圖像上圖像上P(x0,f(x0)PP(2a-x0,f(x0) f(x0)=f(2a-x0)即：即： f(x)=f(2a-x)x02a-x0

4、 y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng)（代數(shù)證明）（代數(shù)證明） ()求證求證已知已知 y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng) f(x)=f(2a-x)xa在在y=f(x)圖像上任取一點(diǎn)圖像上任取一點(diǎn)P若點(diǎn)若點(diǎn)P關(guān)于直線關(guān)于直線x=a的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)P也在也在f(x)圖像上圖像上P(x0,f(x0)PP(2a-x0,f(x0) f(x0)=f(2a-x0) f(x)=f(2a-x)x02a-x0 y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng)（代數(shù)證明）（代數(shù)證明） ()已知已知求證求證 y=f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng) 則則y=f(x)圖像關(guān)于直

5、線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng)? f(x)=f(2a-x)P在在f(x)的圖像上的圖像上ly=f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng) f(x)=f(2a-x) f(a-x)=f(a+x)ly=f(x)圖像關(guān)于直線圖像關(guān)于直線x=0對(duì)稱(chēng)對(duì)稱(chēng) f(x)=f(-x)特例：特例：a=0軸對(duì)稱(chēng)性軸對(duì)稱(chēng)性思考？思考？ 若若y=f(x)滿(mǎn)足滿(mǎn)足f(a-x)=f(b+x),則函數(shù)圖像關(guān)于則函數(shù)圖像關(guān)于 對(duì)稱(chēng)對(duì)稱(chēng) a+b2x= 直線直線xa-xxxyoF(-x)+F(x)=0 y=F(x)圖像關(guān)于圖像關(guān)于(0,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)中心對(duì)稱(chēng)性中心對(duì)稱(chēng)性類(lèi)比探究類(lèi)比探究 al從從”形形”的角度看，的角度看，l

6、從從”數(shù)數(shù)”的角度看，的角度看，F(xiàn)(x)+F(2a-x)=0 xyo a y=F(x)圖像關(guān)于圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)l從從”形形”的角度看，的角度看，l從從”數(shù)數(shù)”的角度看，的角度看，中心對(duì)稱(chēng)性中心對(duì)稱(chēng)性類(lèi)比探究類(lèi)比探究x2a-xF(x)+F(2a-x)=0F(a-x)+F(a+x)=0 xyo al從從”形形”的角度看，的角度看，l從從”數(shù)數(shù)”的角度看，的角度看，中心對(duì)稱(chēng)性中心對(duì)稱(chēng)性類(lèi)比探究類(lèi)比探究 a+x a-x y=F(x)圖像關(guān)于圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)baF(a+x)+F(a-x)=2bF(x)+F(2a-x)=2bb中心對(duì)稱(chēng)性中心對(duì)稱(chēng)性 y=F(x)圖像關(guān)于圖

7、像關(guān)于(a,b)中心對(duì)稱(chēng)中心對(duì)稱(chēng)類(lèi)比探究類(lèi)比探究xyo思考？思考？(1)若若y=f(x)滿(mǎn)足滿(mǎn)足f(a-x)+f(b+x)=0, (2)若若y=f(x)滿(mǎn)足滿(mǎn)足f(a-x)+f(b+x)=2c,則函數(shù)圖像關(guān)于則函數(shù)圖像關(guān)于 對(duì)稱(chēng)對(duì)稱(chēng) a+b2( ,0 )點(diǎn)點(diǎn)則函數(shù)圖像關(guān)于則函數(shù)圖像關(guān)于 對(duì)稱(chēng)對(duì)稱(chēng) a+b2( ,C )點(diǎn)點(diǎn) 知識(shí)內(nèi)容：知識(shí)內(nèi)容：函數(shù)圖像的對(duì)稱(chēng)性函數(shù)圖像的對(duì)稱(chēng)性對(duì)稱(chēng)關(guān)系式對(duì)稱(chēng)關(guān)系式 y=F(x)圖像關(guān)于圖像關(guān)于x=a軸對(duì)稱(chēng)軸對(duì)稱(chēng)F(x)=F(2a-x)F(a-x)=F(a+x) y=F(x)圖像關(guān)于點(diǎn)圖像關(guān)于點(diǎn)(a,b)中心對(duì)稱(chēng)中心對(duì)稱(chēng)F(x)+F(2a-x)=2bF(a-x)+F

8、(a+x)=2b 數(shù)學(xué)思想方法數(shù)學(xué)思想方法:1.數(shù)形結(jié)合數(shù)形結(jié)合2.由特殊到一般由特殊到一般3.類(lèi)比思想類(lèi)比思想知識(shí)遷移：知識(shí)遷移：已知對(duì)任意已知對(duì)任意x，有，有f(x+2)=f(-x),當(dāng)當(dāng)x 2,3，y=x求當(dāng)求當(dāng)x -1,0時(shí)，時(shí)，f(x)的解析式？的解析式？謝謝謝謝!奇函數(shù)奇函數(shù)F(-x)=-F(x)即：即：F(-x)+F(x)=0函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(0,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)-xxF(x)+F(2a-x)=0F(a-x)+F(a+x)=0函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)-xxa函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)F(a+x)+F(a-x)=0F(x

9、)+F(2a-x)=0函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)-x x 函數(shù)圖像關(guān)于直線函數(shù)圖像關(guān)于直線x=0對(duì)稱(chēng)對(duì)稱(chēng)F(-x)=F(x) 函數(shù)圖像關(guān)于直線函數(shù)圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng)F(a-x)=F(a+x) x=aF(x)=F(2a-x)函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(0,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)F(-x)=-F(x)F(a-x)+F(a+x)=0F(x)+F(2a-x)=0軸對(duì)稱(chēng)軸對(duì)稱(chēng)中心對(duì)稱(chēng)性中心對(duì)稱(chēng)性a( )f x函數(shù)函數(shù) 圖像關(guān)于圖像關(guān)于 軸對(duì)稱(chēng)軸對(duì)稱(chēng)xa()()f axf axxD 證明：證明： （必要性）（必要性）xxxxxx

10、1-3-1-2654321-3-1-22-xx1-3-1-265432-xx( 1)(1)FF( 2)(2)FF()( )FxF x78( )f x6x (5)(7)ff(4)(8)ff(6)(6)fxfx6x ( 6)( 6)fxfx 思考？若函數(shù)思考？若函數(shù) 圖像關(guān)于圖像關(guān)于 軸對(duì)稱(chēng)，軸對(duì)稱(chēng)，( )f xxa( )f x有怎樣的對(duì)稱(chēng)關(guān)系式？有怎樣的對(duì)稱(chēng)關(guān)系式？0 x xa函數(shù)函數(shù)y=f(x)圖像關(guān)于圖像關(guān)于x=a軸對(duì)稱(chēng)軸對(duì)稱(chēng)證明：證明： （必要性）（必要性）任取任取y=f(x)圖像上一點(diǎn)圖像上一點(diǎn)P(x0,y0)若點(diǎn)若點(diǎn)P關(guān)于直線關(guān)于直線x=a的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)P也在也在f(x)圖像上圖像上

11、分析：分析：P(x0,y0)PP(2a-x0,y0)代入代入y=f(x) Y0=f(2a-x0)y=f(x)圖像上每圖像上每一一點(diǎn)及其關(guān)于點(diǎn)及其關(guān)于x=a對(duì)稱(chēng)點(diǎn)對(duì)稱(chēng)點(diǎn)都在都在y=f(x)圖像上圖像上則則y=f(x)圖像上圖象關(guān)于圖像上圖象關(guān)于x=a對(duì)稱(chēng)對(duì)稱(chēng)則由則由P的任意性可知的任意性可知？ f(x)=f(2a-x) 函數(shù)圖像關(guān)于直線函數(shù)圖像關(guān)于直線x=0對(duì)稱(chēng)對(duì)稱(chēng)F(-x)=F(x) 函數(shù)圖像關(guān)于直線函數(shù)圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng)F(a-x)=F(a+x) x=aF(x)=F(2a-x)函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(0,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)函數(shù)圖像關(guān)于函數(shù)圖像關(guān)于(a,0)中心對(duì)稱(chēng)中心對(duì)稱(chēng)? xa

12、任取任取y=f(x)圖像上一點(diǎn)圖像上一點(diǎn)P(x0,y0)設(shè)設(shè)P是是P關(guān)于直線關(guān)于直線x=a的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)由由f(x)圖像關(guān)于圖像關(guān)于x=a對(duì)稱(chēng)對(duì)稱(chēng)P也在也在y=f(x)圖像上圖像上P(x0,y0)Pf(2a-x0)=f(x0)即：即： f(x)=f(2a-x)x02a-x0P(2a-x0,y0)猜測(cè)：若猜測(cè)：若f(x)圖像關(guān)于直線圖像關(guān)于直線x=a對(duì)稱(chēng)對(duì)稱(chēng) f(x)有怎樣的對(duì)稱(chēng)關(guān)系式？有怎樣的對(duì)稱(chēng)關(guān)系式？ f(x)=f(2a-x) f(a-x)=f(a+x)證明證明: () y0=f(x0)P(x0,f(x0)若若P也在也在f(x)圖像上，圖像上，xa(2a-x0, y0)PP(2a-x0, y0)代入代入y=f(x) f(2a-x0) =f(x0) y0？= f(2a-x0)=y0P在在f