Point 1: properties and images of functions

函數的性質是研究初等函數的基石，也是高考考查的重點內容．在復習中要肯于在對定義的深入理解上下功夫．

The nature of function is the cornerstone of the study of elementary function and the key content of the college entrance examination. In the review, we should work hard on the in-depth understanding of the definition

復習函數的性質，可以從“數”和“形”兩個方面，從理解函數的單調性和奇偶性的定義入手，在判斷和證明函數的性質的問題中得以鞏固，在求復合函數的單調區間、函數的最值及應用問題的過程中得以深化．具體要求是：

Reviewing the properties of functions can start with understanding the definitions of monotonicity and parity of functions from two aspects of "number" and "shape", consolidate in judging and proving the properties of functions, and deepen in the process of finding the monotonic interval of composite functions, the maximum value of functions and application problems. The specific requirements are as follows:

1．正確理解函數單調性和奇偶性的定義，能準確判斷函數的奇偶性，以及函數在某一區間的單調性，能熟練運用定義證明函數的單調性和奇偶性．

1. Correctly understand the definition of monotonicity and parity of function, accurately judge the parity of function and the monotonicity of function in a certain interval, and skillfully use the definition to prove the monotonicity and parity of function

2．從數形結合的角度認識函數的單調性和奇偶性，深化對函數性質幾何特征的理解和運用，歸納總結求函數最大值和最小值的常用方法．

2. Understand the monotonicity and parity of function from the perspective of the combination of number and shape, deepen the understanding and application of the geometric characteristics of function properties, and summarize the common methods of finding the maximum and minimum of function

3．培養學生用運動變化的觀點分析問題，提高學生用換元、轉化、數形結合等數學思想方法解決問題的能力．

3. Train students to analyze problems from the viewpoint of movement change, and improve students' ability to solve problems with mathematical ideas and methods such as yuan exchange, transformation and combination of numbers and shapes

這部分內容的重點是對函數單調性和奇偶性定義的深入理解．

The focus of this part is the in - depth understanding of the definitions of monotonicity and parity of functions

函數的單調性只能在函數的定義域內來討論．函數y＝f(x)在給定區間上的單調性，反映了函數在區間上函數值的變化趨勢，是函數在區間上的整體性質，但不一定是函數在定義域上的整體性質．函數的單調性是對某個區間而言的，所以要受到區間的限制．

The monotonicity of function can only be discussed in the domain of function definition. The monotonicity of function y = f (x) in a given interval reflects the change trend of function value in the interval. It is the overall property of function in the interval, but not necessarily the overall property of function in the domain of definition. The monotonicity of function is for a certain interval, so it is limited by the interval

對函數奇偶性定義的理解，不能只停留在f(－x)＝f(x)和f(－x)＝－f(x)這兩個等式上，要明確對定義域內任意一個x，都有f(－x)＝f(x)，f(－x)＝－f(x)的實質是：函數的定義域關于原點對稱．這是函數具備奇偶性的必要條件．稍加推廣，可得函數f(x)的圖象關于直線x＝a對稱的充要條件是對定義域內的任意x，都有f(x＋a)＝f(a－x)成立．函數的奇偶性是其相應圖象的特殊的對稱性的反映．

To understand the definition of function parity, we should not only stay in the two equations of F (- x) = f (x) and f (- x) = f (x). It should be clear that for any X in the definition field, there is f (- x) = f (x), and the essence of F (- x) = f (x) is that the definition field of function is symmetrical about the origin. This is a necessary condition for the function to have parity. A little generalization, The necessary and sufficient condition for the symmetry of the image of function f (x) with respect to straight line x = a is that f (x + a) = f (A-X) holds for any X in the definition domain. The parity of function is the reflection of the special symmetry of its corresponding image

這部分的難點是函數的單調性和奇偶性的綜合運用．根據已知條件，調動相關知識，選擇恰當的方法解決問題，是對學生能力的較高要求．

The difficulty in this part is the comprehensive application of the monotonicity and parity of the function. According to the known conditions, mobilizing relevant knowledge and selecting appropriate methods to solve problems are high requirements for students' ability

函數的圖象是函數性質的直觀載體，函數的性質可以通過函數的圖像直觀地表現出來。

The image of function is the visual carrier of function properties, and the properties of function can be intuitively expressed through the image of function.

因此，掌握函數的圖像是學好函數性質的關鍵，這也正是“數形結合思想”的體現。復習函數圖像要注意以下方面。

Therefore, mastering the image of function is the key to learn the nature of function, which is the embodiment of "the combination of number and shape". When reviewing function images, pay attention to the following aspects.

1．掌握描繪函數圖象的兩種基本方法——描點法和圖象變換法．

1. Master the two basic methods of drawing function images - point drawing method and image transformation method

2．會利用函數圖象，進一步研究函數的性質，解決方程、不等式中的問題．

2. Be able to use function images to further study the properties of functions and solve problems in equations and inequalities

3．用數形結合的思想、分類討論的思想和轉化變換的思想分析解決數學問題．

3. Analyze and solve mathematical problems with the idea of combination of number and shape, classified discussion and transformation

4．掌握知識之間的聯系，進一步培養觀察、分析、歸納、概括和綜合分析能力．

4. Master the connection between knowledge and further cultivate the ability of observation, analysis, induction, generalization and comprehensive analysis

以解析式表示的函數作圖象的方法有兩種，即列表描點法和圖象變換法，掌握這兩種方法是本節的重點．

There are two methods of making images with functions expressed analytically, namely, list point drawing method and image transformation method. Mastering these two methods is the focus of this section

運用描點法作圖象應避免描點前的盲目性，也應避免盲目地連點成線．要把表列在關鍵處，要把線連在恰當處．這就要求對所要畫圖象的存在范圍、大致特征、變化趨勢等作一個大概的研究．而這個研究要借助于函數性質、方程、不等式等理論和手段，是一個難點．用圖象變換法作函數圖象要確定以哪一種函數的圖象為基礎進行變換，以及確定怎樣的變換．這也是個難點．

When using the point drawing method to make an image, we should avoid the blindness before the point drawing, and we should also avoid blindly connecting points into lines. We should list the table at the key place and connect the lines at the appropriate place. This requires a general study on the existing range, general characteristics and change trend of the image to be drawn. This study should rely on the theories and means of function properties, equations, inequalities and so on, It is a difficult point. When using image transformation method to make function image, it is necessary to determine which kind of function image is the basis for transformation and what kind of transformation is determined. This is also a difficult point