函数与方程教案_函数与方程公开课教案

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函数与方程教案

27.3实践与探索(第二课时)二次函数与一元二次方程的关系 晋城四中 李前进 【教学目标】

1、知识与技能:(1)体会函数与方程之间的联系，初步体会利用函数图象研究方程问题的方法;(2)理解二次函数图象与x轴(横轴)交点的个数与一元二次方程的根的个数之

间的关系，理解方程有两个不等的实根、两个相等的实根和没有实根的函数 图象特征;22(3)理解一元二次方程ax+bx+c=0的根就是二次函数与y=ax+bx+c图象与x轴交

点的横坐标。

2、过程与方法:(1)由一次函数与一元一次方程根的联系类比探求二次函数与一元二次方程之间的联系;(2)经历类比、观察、发现、归纳的探索过程，体会函数与方程相互转化的数学

思想和数形结合的数学思想。

3、情感、态度与价值观: 培养学生类比与猜想、不完全归纳、认识到事物之间的联系与转化、体验 探究的乐趣和学会用辨证的观点看问题的思维品质。【重点与难点】

重点:经历“类比--观察--发现--归纳”而得出二次函数与一元二次方程的关系的探

索过程。

难点:准确理解二次函数与一元二次方程的关系。【教法与学法】

教法:采用“发现式学习”的方式，注重“最近发展区”，寻根问源，以旧知识为

基础创设问题情境，引导学生经历“类比—猜想—观察—发现—归纳—应用” 的探究过程。学法:探究式学习。

appearance of the weld appearance quality technical requirements of the project must not have a molten metal stream does not melt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition.Width 2-3 mm from the edge of weld Groove.Surface reinforcement should be le than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm.Depth of undercut should be le than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous should not be greater than 100 mm.Wrong side should be le than or at 0.2T, and should not be greater than 2 mm(wall thickne mm t)incomplete or not allow 7.5 7.5.1 installation quality proce standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of convex, close to nature, close the door.7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color.7.5.3 uniform indirect gap from top to bottom, slot width

一、诗词导入

教师投影:我国著名数学家华罗庚曾经说过:“数缺形时少直观，形少数 时难入微，数形结合百般好，隔离分家万事休。”(学生齐读)师:数学家的寥寥数语就将数与形之间的内在联系表达的淋漓尽致。今天，我们通过研究二次函数中的数形结合来体会“数形结合百般好”的奥妙～ 设计思路:从学生熟悉的小诗入手，激发学生探究学习的积极性。

二、温故知新 y3那些年，我们一起做过的题: 2(1)解一元一次方程x+1=0;1(2)画一次函数y=x+1的图象，并指出函数y=x+1的图象 x –2–11O 与x轴的交点坐标。–1(3)你会不画图象求函数y=3x,3与x轴的交点坐标吗, 师生共同总结:一次函数y,kx，b的图象与x轴的交点的横坐标就是一元一次方程kx，b,0的根

设计思路:这一环节让学生通过对旧知识的回顾及对新知识的思考，梳理旧知识，起到承上启下之效，同时通过老师的引导，培养学生的形成解决一类问题的通用方法的思维品质

三、类比猜想

22你觉得一元二次方程ax+bx+c=0的根与二次函数y=ax+bx+c之间有联系吗,四、问题探究

教师分配研究的任务，然后小组合作完成，教师提问，学生展示研究成果。设计思路: 学生画函数图象比较慢，分配任务既可以节约时间，又可以使 每个学生都有事可做，能够很好地完成学习任务。

appearance of the weld appearance quality technical requirements of the project must not have a molten metal stream does not melt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition.Width 2-3 mm from the edge of weld Groove.Surface reinforcement should be le than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm.Depth of undercut should be le than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous should not be greater than 100 mm.Wrong side should be le than or at 0.2T, and should not be greater than 2 mm(wall thickne mm t)incomplete or not allow 7.5 7.5.1 installation quality proce standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of convex, close to nature, close the door.7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color.7.5.3 uniform indirect gap from top to bottom, slot width

五、归纳结论

2(1)从“数”的方面看，当二次函数y=ax+bx+c的函数值y=_0_ 时，二次函数 x2-2x+ 2 变为一元二次方程ax+bx+c=0,此时相应的_自变量的值即为二次方程 2ax+bx+c=0的_根_;2=0(2)从“形”的方面看，当二次函数的y值为0时，从图像看指的是二次函数图像与_x轴_的交点，此时二次函数y=ax+bx+c与x轴交点的_横坐标_即为二x2-2x+ 2次方程ax+bx+c=0的_根_。表格二: 2=0 2一元二次方程二次函数y=ax+bx+c的图象一元二次方程根的判别式 222b,4ac ax+bx+c=0的根的个数 与x轴交点的个数

x-2x+ 22=0 b,4ac>0 2 b,4ac=0 2 b,4ac

1六、基础练习 x–3–2–1123O2–1(1)已知二次函数y=x-x-6的图象如图所示: –2 –3图象与x轴有2个交点，交点的横坐标 –42 是______,则方程x-x-6=0有__个根，方程的根是________ 2(2)函数y= x-5x+6的图象与x轴有___个交点,其交点坐标为_________、__________。(3)自命题

每个小组按照教师的要求，小组内通过讨论写出一个一般式的二次函数关系式，用关系式出一道有关二次函数和一元二次方程的简单的题，(七个大组分三种情况布置有目的性的布置，各小组只知道自己小组的任务)。教师通过在教师内观察学生活动情况，选两个代表性题由其他小组来做。

设计思路:小组活动，激发学生的学习热情，巩固对上面总结结论的认识。

七、例题讲解 2 例1:已知二次函数y=ax+bx+c(a?0)的对称轴是x=2,它与x轴的一个交 2点坐标是(4，0)，则方程ax+bx+c=0的两个解是__________ 设计思路:鼓励学生自主思考，然后小组讨论，派代表上讲台讲解。

八、巩固练习

2(1)抛物线y=ax+bx+c(a?0)的图象全部在x轴下方的条件是()22(A)a,0 b,4ac?0(B)a,0 b , 4ac,0 22(C)a,0 b , 4ac,0(D)a,0 b , 4ac,0(2)下列函数中其图象与x轴有两个交点的是()11112222(A)y=()x23+155(B)y=()x+23+155(C)y=()x23155(D)y=()x+23+1554444 appearance of the weld appearance quality technical requirements of the project must not have a molten metal stream does not melt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition.Width 2-3 mm from the edge of weld Groove.Surface reinforcement should be le than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm.Depth of undercut should be le than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous should not be greater than 100 mm.Wrong side should be le than or at 0.2T, and should not be greater than 2 mm(wall thickne mm t)incomplete or not allow 7.5 7.5.1 installation quality proce standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of convex, close to nature, close the door.7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color.7.5.3 uniform indirect gap from top to bottom, slot width

七、拓展提高:

21、已知二次函数y=ax+bx+c(a?0)的图象 如图所示，根据图象回答下列问题: 2(1)方程ax+bx+c=0的两个解是__________ 2(2)方程ax+bx+c=4的两个解是__________ 设计思路:让学生对二次函数和一元二次方程的关系的认识上升高度。

22、你会利用二次函数的图象求出一元二次不等式x,x,2，0的解集吗,(看课堂时间情况决定是否出示)

八、课堂小结，提高认识

函数 方程 22ax+bx+c=0(a ?0)y=ax+bx+c(a?0)横坐标的值 图象与x轴交点 根 个数

一个关系:二次函数图象与一元二次方程根的关系: 两种思想:函数与方程互相转化的思想;数形结合思想(设计思路:用精炼的语言，使得学生记忆简便，而且印象加深，同时让学生在 总结中反思，完成升华。学生再次齐读华罗庚名言，下课。

九、布置作业，巩固提升

十、板书设计

课题:„„.课题:„„.方程与函数转化 例1: 方程与函数转化 例1: 函数 方程 22y=ax+bx+c(a?0)ax+bx+c=0(a ?0)横坐标的值 图象与x轴交点 根 个数 数形结合 数形结合appearance of the weld appearance quality technical requirements of the project must not have a molten metal stream does not melt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition.Width 2-3 mm from the edge of weld Groove.Surface reinforcement should be le than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm.Depth of undercut should be le than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous should not be greater than 100 mm.Wrong side should be le than or at 0.2T, and should not be greater than 2 mm(wall thickne mm t)incomplete or not allow 7.5 7.5.1 installation quality proce standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of convex, close to nature, close the door.7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color.7.5.3 uniform indirect gap from top to bottom, slot width