公開課用函數(shù)觀點(diǎn)探究一元二次方程的根

課題:用函數(shù)觀點(diǎn)探究一元二次方程的根方程ax2+bx+c=0(a>0)的根函數(shù)y=ax2+bx+c(a>0)的圖象判別式△=b2－4ac△＞0△=0△＜0函數(shù)的圖象與x軸的交點(diǎn)有兩個(gè)相等的實(shí)數(shù)根x1=x2沒有實(shí)數(shù)根xyx1x20xy0x1xy0(x1,0),(x2,0)(x1,0)沒有交點(diǎn)兩個(gè)不相等的實(shí)數(shù)根x1、x2問題1：你能通過觀察二次方程的根及相應(yīng)的二次函數(shù)圖象，找出它們之間的關(guān)系嗎?函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20判別式△=b2－4ac方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20判別式△=b2－4ac方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20判別式△=b2－4ac方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20判別式△=b2－4ac方程ax2+bx+c=0(a>0)的根函數(shù)的圖象與x軸的交點(diǎn)xy0xy0x1函數(shù)y=ax2+bx+c(a>0)的圖象xyx1x20知識(shí)梳理⑶一正一負(fù)

⑵兩負(fù)根

1．設(shè)是方程的兩根典例剖析2.若根與其它實(shí)數(shù)進(jìn)行大小比較則利用二次函數(shù)的圖像數(shù)形結(jié)合加以分析.先作出符合根的分布的二次函數(shù)的圖象，由圖像可得到其開口方向，在區(qū)間端點(diǎn)的函數(shù)值與判別式的符號(hào)，對(duì)稱軸的位置等情況，從而找到參數(shù)滿足的條件。1.若根與零進(jìn)行大小比較利用韋達(dá)定理解決;例題小結(jié)2．關(guān)于x的二次方程有兩個(gè)小于1的實(shí)根，求實(shí)數(shù)m的取值范圍。3．已知方程在1≤x≤3上有解，求實(shí)數(shù)m的取值范圍。

【例2】、當(dāng)關(guān)于的方程的兩個(gè)根都在區(qū)間；求實(shí)數(shù)a的取值范圍.

三、回顧小結(jié)1、一元二次方程根的分布。（1）主要依靠Δ及韋達(dá)定理求解，但此法有時(shí)不大奏效.2、數(shù)形結(jié)合的思想；函數(shù)與方程思想；化歸與轉(zhuǎn)化思想.

（2）利用函數(shù)圖象及函數(shù)值來“控制”一元二次方程根的分布（函數(shù)觀