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数形结合是一种常用的解题方法,也是高考经常考查的一种数学思想。对于有些问题,若能抓住本质,以形辅数、数形结合,则可直观、快速地求解。本文以2005年高考题为例,谈谈数形结合在解题中的妙用。例1 函数f(x)=|sinx+cosx|的最小正周期是( )。(全国卷Ⅱ理科第1题) (A)π／4 (B)π／2 (C)π (D)2π常规解法用定义f(x+k)=f(x)进行验证。巧解先画出g(x)=sinx+cosx=2~(1/2)sin(x+π/4)的图像(见图1),然后将g(x)图像中x轴下方的部分沿x轴翻折上去,即得f(x)的图像,由图像可知f(x)的最小正周期是π,故选(C)。
The combination of number and shape is a commonly used method of solving problems, and it is also a kind of mathematics thought that is often examined in the college entrance examination. For some problems, if we can grasp the essence and combine the auxiliary numbers and numbers, we can solve them intuitively and quickly. This article takes the 2005 college entrance examination question as an example to talk about the magical use of the combination of numbers and shapes in solving problems. Example 1 The minimum positive period of the function f(x)=|sinx+cosx| is ( ). (National Volume II Science Question 1) (A) π / 4 (B) π / 2 (C) π (D) 2π conventional solution by the definition of f (x + k) = f (x) for verification. First, draw an image with g(x)=sinx+cosx=2~(1/2)sin(x+π/4) (see Figure 1) and then place the part below the x-axis in the g(x) image. Fold up along the x-axis to get the image of f(x). It can be known from the image that the minimum positive period of f(x) is π, so choose (C).