二次函數(shù)的圖形

一次函數(shù)與二次函數(shù)的意義一次函數(shù):y=f(x)=ax+b,其中a,b為常數(shù)且a=0.例:f(x)=500-x,h(x)=100x+600二次函數(shù):y=ax2+bx+c,其中a,b,c為常數(shù),且a=0.例:f(x)=100-4.9x2,g(x)=x2.一次函數(shù)的圖形一次函數(shù)的圖形是一直線一次函數(shù)的畫(huà)圖方法:畫(huà)一次函數(shù)圖形時(shí),只要描出相異兩點(diǎn),連接即得所求的直線.描繪f(x)=x+5的圖形列表描點(diǎn)xy0(2,7)(1,6)(0,5)(-1,4)(-2,3)描繪f(x)=x+5的圖形列表插入四個(gè)數(shù)對(duì),描點(diǎn)xy0(2,7)(1,6)(0,5)(-1,4)(-2,3)(-3,2)(-4,1)(3,8)(4,9)描繪f(x)=x+5的圖形列表再插入四個(gè)數(shù)對(duì),描點(diǎn)xy0(2,7)(1,6)(0,5)(-1,4)(-2,3)(3,8)(4,9)(-3,2)(-4,1)(5,10)(6,11)(-5,0)(-6,-1)描繪y=2x-3的圖形列表描點(diǎn),將兩點(diǎn)用直線連接起來(lái),即得y=2x-3的圖形xy0(,0)(0,-3)描繪y=x2

的圖形列表描點(diǎn)(2,4)(1,1)(-2,4)(-1,1)xy0描繪y=x2

的圖形列表插入四個(gè)數(shù)對(duì),描點(diǎn)(,),(-,)(2,4)(-2,4)(1,1)(-1,1)(,)(-,)xy0描繪y=x2

的圖形列表再插入四個(gè)數(shù)對(duì),描點(diǎn)(,)(,)(-,)(-,)(2,4)(-2,4)(1,1)(-1,1)(,)(-,)(,)(,)xy0描繪y=x2

的圖形列表插入(-(,)(,)(-,)(-,)(2,4)(-2,4)(1,1)(-1,1)(,)(-,)(,)(,)0xy,),(-,),(,),(,)四個(gè)數(shù)對(duì)觀察二次函數(shù)y=x2

圖形結(jié)論在描繪函數(shù)圖形時(shí),如描繪的點(diǎn)愈多,愈容易得到準(zhǔn)確的圖形.二次函數(shù)圖形是一拋物線畫(huà)二次函數(shù)圖形時(shí),須以頂點(diǎn)為準(zhǔn)左右兩邊各取兩個(gè)以上數(shù)對(duì)(即至少5個(gè)數(shù)對(duì))方可畫(huà)出拋物線圖形.二次函數(shù)的畫(huà)圖步驟首先列表:須以頂點(diǎn)為準(zhǔn)左右兩邊各取兩個(gè)以上數(shù)對(duì)(即至少5個(gè)數(shù)對(duì))其次在坐標(biāo)平面上將各數(shù)對(duì)描成點(diǎn).用平滑曲線把各點(diǎn)連接起來(lái),就得到二次函數(shù)圖形.描繪y=x2

的圖形列表描點(diǎn),並用平滑曲線將各點(diǎn)連接起來(lái)xy(2,4)(1,1)(-2,4)(-1,1)0描繪y=2x2

的圖形列表描點(diǎn)(1.5,4.5)(1,2)(-1.5,4.5)xy0(-1.2)描繪y=2x2

的圖形列表描點(diǎn)並用平滑曲線將各點(diǎn)連接起來(lái)xy0(1.5,4.5)(1,2)(-1.5,4.5)(-1,2)比較y=x2,y=2x2

兩圖形之異同表y=x2表y=2x2x0yy=x2y=2x2比較y=x2與y=2x2

兩圖形1.比較圖形大小時(shí),各圖形坐標(biāo)平面的單位長(zhǎng)度必須相同,方可比較.2.形狀相似,開(kāi)口都向上.3.y=x2

開(kāi)口較大,y=2x2

開(kāi)口較小.4.沿著y軸各把兩圖形對(duì)摺,我們發(fā)現(xiàn)y軸兩邊的圖形可以處處疊合.y=x2與y=2x2

的圖形都是以y軸為對(duì)稱軸的軸對(duì)稱圖形.描繪y=x2

的圖形列表描點(diǎn)xy0(3,4.5)(2,2)(-2,2)(-3,4.5)描繪y=x2

的圖形列表描點(diǎn),並用平滑曲線將各點(diǎn)連接起來(lái)xy0(3,4.5)(2,2)(-3,4.5)(-2,2)比較y=x2,y=2x2,y=x2各圖形形狀,開(kāi)口方向,開(kāi)口大小異同表y=x2表y=2x2表y=x2xy0y=x2y=2x2y=x2比較y=x2,y=x2,y=2x2異同1.形狀相似,頂點(diǎn)相同.2.開(kāi)口方向皆向上.3.開(kāi)口最大為y=x2,其次是y=x2,最小為y=2x2.描繪y=-x2

的圖形列表描點(diǎn)xy0(1,-1)(2,-4)(-1,-1)(-2,-4)描繪y=-x2

的圖形列表描點(diǎn),並用平滑曲線將各點(diǎn)連接起來(lái)(1,-1)(2,-4)(-1,-1)(-2,-4)xy0比較y=x2,y=-x2

兩圖形形狀,開(kāi)口方向,開(kāi)口大小異同表y=x2表y=-x2xy0y=x2y=-x2比較y=x2與y=-x2

兩圖形1.形狀,開(kāi)口大小,頂點(diǎn)都相同

.2.y=x2

開(kāi)口向上

,y=-x2

開(kāi)口向下

3.兩圖形對(duì)x

軸成對(duì)稱:把兩個(gè)圖形合在一起看成一個(gè)圖形時(shí),它是以x軸為對(duì)稱軸的軸對(duì)稱圖形.描繪y=-2x2

的圖形列表描點(diǎn)xy0(1,-2)(1.5,-4.5)(-1,-2)(-1.5,-4.5)描繪y=-2x2

的圖形列表描點(diǎn),並用平滑曲線將各點(diǎn)連接起來(lái)xy0(1,-2)(1.5,-4.5)(-1,-2)(-1.5,-4.5)比較

y=2x2,y=2x2各圖形形狀,開(kāi)口方向,開(kāi)口大小異同表y=2x2表y=-2x2xy0y=2x2y=-2x2比較y=-2x2與y=2x2

1.形狀,開(kāi)口大小,頂點(diǎn)都相同

.2.y=-2x2

開(kāi)口向下

,y=2x2

開(kāi)口向上

.3.兩圖形對(duì)x

軸成對(duì)稱:把兩個(gè)圖形合在一起看成一個(gè)圖形時(shí),它是以x軸為對(duì)稱軸的軸對(duì)稱圖形.

比較y=x2,y=-x2,y=2x2,y=-2x2,y=x2,y=-x2y=x2六圖形的形狀,開(kāi)口方向,開(kāi)口大小異同y=2x2y=

x2y=-x2y=-2x2y=-x2xy0比較y=x2,y=-x2,y=2x2,y=-2x2,y=x2,y=-x21.形狀相似,頂點(diǎn)相同.2.x2

項(xiàng)係數(shù)為正時(shí),開(kāi)口向上

;x2

項(xiàng)係數(shù)為負(fù)時(shí),開(kāi)口向下.3.x2

項(xiàng)係數(shù)之絕對(duì)值相同時(shí),開(kāi)口大小相同

,x2

項(xiàng)係數(shù)之絕對(duì)值愈小時(shí),開(kāi)口愈大.4.六個(gè)圖形皆是以y軸(方程式:x=0)

為對(duì)稱軸之軸對(duì)稱圖形.描繪y=(x-3)2

的圖形列表描點(diǎn)(5,4)(4,1)(1,4)(2,1)(3,0)xy0描繪y=(x-3)2

的圖形列表描點(diǎn),並用平滑曲線將各點(diǎn)連接起來(lái)xy0(5,4)(1,4)(4,1)(2,1)(3,0)比較y=x2,y=-(x-3)2

兩圖