GeoGebra及其在几类二次问题中的应用

GeoGebra and Its Application to Some Kinds of Quadratic Problems

摘要: GeoGebra是全面且实用的教学软件，不仅能帮助学生理解抽象的数学概念，提高学生的听课效率、提升学生的数学思维能力，而且便于老师的讲解。二次问题是我们数学学习过程中一个十分重要的板块，它包括中学已经学习过的二次函数以及在大学中学习的二次曲面，并且在现实情境中二次问题的应用十分广泛。随着GeoGebra在国内的使用范围不断扩大，许多学者都对其应用做了一定的研究，并发表了自己的观点，他们中的许多人都涉及了GeoGebra在二次问题中的应用。本文通过对GeoGebra及其使用方法的简单介绍，发现使用GeoGebra能够方便快速地画出复杂的函数图像，这对于我们理解数学中抽象函数具有深刻的现实意义。因此本文首先介绍GeoGebra在中学中所学习的一元二次函数和一元二次方程的应用，再进一步地利用GeoGebra探讨大学中所学习的抽象难懂的二次曲线的类型，把一个抽象的二次函数的表达式输入GeoGebra的命令窗口，就可以在几何区得到直观的函数图像，进而判断二次曲线的类型。对于二次曲面来说，先利用二次型的知识将二次曲面的类型将二次曲面化为标准形，再通过输入函数的标准方程得到其图形，让我们快速直观地判断出二次曲面的类型并且会发现平时晦涩难懂的在二次型正定问题下的极值问题、不等式问题也可以利用GeoGebra画出其立体图形，让我们更形象地感受到二次问题的直观，另外在数学分析中的积分问题中也可以利用GeoGebra画出函数图像，明确积分区域，进而进行积分计算。研究表明GeoGebra在我们大学所学习到的诸多二次问题中的应用是十分广泛的。

Abstract: GeoGebra is a comprehensive and practical teaching software, which can not only help students understand abstract mathematical concepts, improve students’ listening efficiency, improve students’ mathematical thinking ability, but also facilitate teachers’ explanation. Quadratic problem is a very important part in our mathematics learning process. It includes the quadratic function we have learned in middle school and the quadric surface we have learned in university, and it is widely used in real situations. With the continuous expansion of GeoGebra’s application in China, many scholars have done some research on its application and published their own opinions, and many of them are involved in the application of GeoGebra in quadratic problems. Through a brief introduction of GeoGebra and its application method, this paper finds that it is convenient and fast to draw complex function images with GeoGebra, which has profound practical significance for us to understand abstract functions in mathematics. Therefore, this paper first introduces the application of quadratic functions of one variable and quadratic equations of one variable that GeoGebra learned in middle school, and then further discusses the abstract and difficult types of conic curves that GeoGebra learned in university by using GeoGebra. The expression of an abstract quadratic function is input into the command window of GeoGebra. You can get the intuitive function image in the geometric area, and then judge the type of conic curve. For quadric surface, firstly use the knowledge of quadric form to convert the type of quadric surface into standard form, and then get its graph by input function standard equation. Let us quickly and intuitively judge the type of quadric surface and find the extreme value problem and inequality problem under the quadratic positive definite problem which are difficult to understand at ordinary times. We can also use GeoGebra to draw its three-dimensional graph, so that we can feel the intuition of quadratic problem more vividly. In addition, in the integral problem of mathematical analysis, GeoGebra can also be used to draw the function image, clear the integral region, and then carry out the integral calculation. Research has shown that GeoGebra has a wide range of applications in quadratic problems that we learn in universities.