第六節(jié)冪函數(shù)與二次函數(shù)【教材·知識(shí)梳理】1.冪函數(shù)的圖象與性質(zhì)(1)常見(jiàn)的5種冪函數(shù)的圖象(2)性質(zhì)①冪函數(shù)在(0,+∞)上都有定義.②當(dāng)α>0時(shí),冪函數(shù)的圖象都過(guò)點(diǎn)(1,1)和(0,0),且在(0,+∞)上單調(diào)遞增.③當(dāng)α<0時(shí),冪函數(shù)的圖象都過(guò)點(diǎn)(1,1),且在(0,+∞)上單調(diào)遞減.2.二次函數(shù)的圖形與性質(zhì)解析式f(x)=ax2+bx+c(a>0)f(x)=ax2+bx+c(a<0)圖象

定義域(-∞,+∞)(-∞,+∞)值域

解析式f(x)=ax2+bx+c(a>0)f(x)=ax2+bx+c(a<0)單調(diào)性在上單調(diào)遞增;在上單調(diào)遞減在上單調(diào)遞增;在上單調(diào)遞減奇偶性當(dāng)____時(shí)為偶函數(shù),當(dāng)b≠0時(shí)為非奇非偶函數(shù)頂點(diǎn)

對(duì)稱性圖象關(guān)于直線成軸對(duì)稱圖形b=0【常用結(jié)論】1.冪函數(shù)的圖象和性質(zhì)(1)冪函數(shù)的圖象一定會(huì)出現(xiàn)在第一象限內(nèi)，一定不會(huì)出現(xiàn)在第四象限，至于是否出現(xiàn)在第二、三象限內(nèi)，要看函數(shù)的奇偶性.(2)冪函數(shù)的圖象過(guò)定點(diǎn)(1，1)，如果冪函數(shù)的圖象與坐標(biāo)軸相交，則交點(diǎn)一定是原點(diǎn).(3)當(dāng)α>0時(shí)，y=xα在[0，+∞)上為增函數(shù)；當(dāng)α<0時(shí)，y=xα在(0，+∞)上為減函數(shù).2.若f(x)=ax2+bx+c(a≠0)，則當(dāng)時(shí)恒有f(x)>0，當(dāng)時(shí)，恒有f(x)<0.【知識(shí)點(diǎn)辨析】(正確的打“√”,錯(cuò)誤的打“×”)(1)函數(shù)y=2是冪函數(shù). (

)(2)當(dāng)n>0時(shí),冪函數(shù)y=xn在(0,+∞)上是增函數(shù). (

)(3)二次函數(shù)y=ax2+bx+c(x∈R)不可能是偶函數(shù). (

)(4)如果冪函數(shù)的圖象與坐標(biāo)軸相交,則交點(diǎn)一定是原點(diǎn). (

)(5)二次函數(shù)y=ax2+bx+c,x∈[a,b]的最值一定是 (

)提示:(1)×.不符合冪函數(shù)的形式.(2)√.根據(jù)5個(gè)基本冪函數(shù)知,n>0時(shí)為增函數(shù),n<0時(shí)為減函數(shù).(3)×.當(dāng)b=0時(shí),為偶函數(shù).(4)√.在y=xn中,令y=0,則x=0,令x=0,則y=0,所以正確.(5)×.只有當(dāng)對(duì)稱軸在區(qū)間內(nèi)時(shí)最值才是【易錯(cuò)點(diǎn)索引】序號(hào)易錯(cuò)警示典題索引1對(duì)冪函數(shù)概念理解錯(cuò)誤考點(diǎn)一、T12周期性、對(duì)稱性對(duì)應(yīng)代數(shù)式分辨不清考點(diǎn)二、T23二次函數(shù)最值點(diǎn)錯(cuò)誤考點(diǎn)三、角度34忽略最高次數(shù)項(xiàng)的系數(shù)是否為零考點(diǎn)三、角度2T1【教材·基礎(chǔ)自測(cè)】1.(必修1P119鞏固與提高T1(5)改編)已知冪函數(shù)f(x)=k·xα的圖象過(guò)點(diǎn)則k+α等于

(

)

【解析】選C.由冪函數(shù)的定義,知所以k=1,α=,所以k+α=.2.(必修1P63習(xí)題2-2AT8改編)已知函數(shù)f(x)=x2+2ax+3,若y=f(x)在區(qū)間[-4,6]上是單調(diào)函數(shù),則實(shí)數(shù)a的取值范圍為_(kāi)_______.

【解析】由于函數(shù)f(x)的圖象開(kāi)口向上,對(duì)稱軸是x