1、三次函數(shù)再探討對(duì)稱中心問題武漢市長(zhǎng)虹中學(xué)郭永清三次函數(shù)存在對(duì)稱中心嗎？三次函數(shù)存在對(duì)稱中心嗎？我們先從幾個(gè)特殊的函數(shù)入手，三次函數(shù)（)是奇函數(shù)，其cxaxxf??3)(0?a圖象關(guān)于對(duì)稱，三次函數(shù)()的圖象關(guān)于點(diǎn)對(duì)稱，那)00(dbxaxxf???3)(0?a)0(d么對(duì)于一般的三次函數(shù)有沒有對(duì)稱中心呢？答案是肯定)0()(23?????adcxbxaxxf的，有對(duì)稱中心，其對(duì)稱中心是。))3(3(abfab??在證明之前，先回憶一個(gè)結(jié)

2、論：定理定理1：函數(shù)：函數(shù)的圖像關(guān)于點(diǎn)的圖像關(guān)于點(diǎn)對(duì)稱，則在對(duì)稱，則在)(xfy?)(baMbxafxf2)2()(???證明：設(shè)是圖像上任意一點(diǎn)，則A關(guān)于點(diǎn)的對(duì)稱點(diǎn))(yxA)(xfy?)(baM也在函數(shù)圖像上，)22(ybxaB??)(xfy?即又，所以)2(2xafyb???)(xfy?bxafxf2)2()(???定理定理2：三次函數(shù)：三次函數(shù)的對(duì)稱中心是的對(duì)稱中心是)0()(23?????adcxbxaxxf))3(3(abf

3、ab??證明1：設(shè)是圖像上任意一點(diǎn)，只要能證明點(diǎn))(yxA)(xfy?))3(232(yabfxabB???也在函數(shù)圖像上。cxbxaxdabcabdcxabcbxaxbabaxbxxababdxabcxabbxabaxabf?????????????????????????????23232223322232332274323494234278)32()32()32()32(dcxbxaxxfydabcabbabaabf???????

4、??????2323)()3()3()3()3(得??????????00323ndcmbmambam，。所以，函數(shù)的對(duì)稱中心是（）。定理定理3：若三次函數(shù)：若三次函數(shù)有極值，則它的對(duì)稱中有極值，則它的對(duì)稱中心是兩個(gè)極值點(diǎn)的中點(diǎn)心是兩個(gè)極值點(diǎn)的中點(diǎn)證明：不妨設(shè)為的導(dǎo)方程，判別式設(shè)0232???cbxax)(xf01242????acb兩極值點(diǎn)為)(xf))(())((2211xfxBxfxA????acxxabxxdxxcxxxxbxx

5、xxxxadxxcxxbxxadcxbxaxdcxbxaxxfxf3322)(2)(3)()(2)()()()()(2121212122121221212122213231222321213121????????????????????????????????又dabcabbabadabcacbabbacabaabaxfxf2)3(2)3(2)3(22)32(32323)32(332)()(232321?????????????????