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含参变元的问题求解是中学数学教学中的难点,本文试图探讨一类通过构造函数、分析图象特征,可以利用二次函数图象作出简易解答的问题的解题规律。例1 已知方程asin~2x+sinx-a-2=0,其中a是不为0的可变常数,x∈[0,2π),试根据a的变化讨论方程的解。分析:按惯例,用求根公式 sinx=(-1±(4a~2+8a+1)~(1/2))/2a,然后根据|(-1±(4a~2+8a+1)~(1/2))/2a|≤1来讨论解的情况,这种解法比较麻烦。现试构造二次函数f(x)=ax~2+x-a-2f(x)=a(x~2-1)+(x-2),因为对于任意a,函数通过两个定点(1,-1),(-1,-3),作出图象,使原方程有解的函数图象必须如下所示,根据图象性质可得出有解
The problem-solving with parameter variables is a difficult point in middle school mathematics teaching. This paper attempts to explore the problem-solving law of a class of questions that can be easily solved by using quadratic function images by constructing functions and analyzing image features. Example 1 Known equations asin~2x+sinx-a-2=0, where a is a non-zero variable constant, x∈[0, 2π), and the solution to the equation is discussed in terms of a. Analysis: According to convention, use the root formula sinx=(-1±(4a~2+8a+1)~(1/2))/2a, then according to |(-1±(4a~2+8a+1) ~(1/2))/2a|≤1 to discuss the solution, this solution is more troublesome. Try to construct a quadratic function f(x)=ax~2+xa-2f(x)=a(x~2-1)+(x-2) because for any a, the function passes two fixed points (1, -1), (-1, -3), The image of the function to make the image so that the original equation has a solution must be as follows. According to the nature of the image, a solution can be obtained.