1, the functional relationship between line segments:

Because the main element of this kind of test questions is geometric figures, when solving this kind of test questions, we should first observe the characteristics of geometric figures, and then find out the relationship between geometric elements according to the related graphic properties (such as right triangle properties, special quadrilateral properties, parallel line proportion theorem and its inference, similar triangles properties, basic properties of circles, proportional line segments in circles, etc.). ). Finally, we can express their relationship with mathematical expressions, sort it into a functional relationship, and then solve it on the basis of this functional relationship. When solving this kind of problems, we should pay special attention to the range of independent variables.

2. Establish the functional relationship between region and line segment;

In order to solve this kind of problem, besides mastering the first kind of knowledge, we should also pay attention to the following two points: (1) common graphic area formula; (2) Learn to flexibly convert the area of non-special graphics into the area of special graphics; Convert the area ratio of two triangles with the same base (or height) into their height (or base) ratio; And the ratio of similar triangles area is converted into a similar ratio (or the ratio of perimeter to corresponding edge).

(2) The "signal" problem:

Looking at the mid-term exam questions over the years, functions are almost all geometric questions. From the difficulty point of view, most of the questions are difficult, and a few are intermediate. Judging from the types of questions, most of them are exploration questions, and a few are calculation questions. They all pay attention to the innovation of design methods, the proposition of the intersection of junior high school mathematics knowledge, and the mathematical thinking method and ability (especially the comprehensive application of thinking ability, inquiry ability, innovation ability and knowledge). Therefore, when solving this kind of problems, we should flexibly use functional knowledge, pay attention to the hidden conditions in the topic, and pay attention to the application of mathematical ideas such as the combination of numbers and shapes, mathematical modeling and classified discussion. Let's talk about the classification of such problems first.

The problem of graphic area in 1 and three basic elementary functions;

When solving this kind of problems, the graphics in the coordinate system are usually divided into triangles (or trapezoid or rectangle) with two sides (or three sides) on the coordinate axis or two sides (or three sides) parallel to the coordinate axis. Pay attention to the difference between the distance from the point to the coordinate axis and the coordinates of the point, and use the coordinates of the point to represent the length of the line segment.

2. The problems of triangle, quadrilateral and circle in three basic elementary functions;

This kind of topic generally consists of 1~3 questions. The first problem is often to find the resolution function. On this basis, it is combined with triangles in geometry (whether congruent, similar or special triangles exist, etc.). ), quadrilateral (whether there is an area function relationship, whether there is a special quadrilateral), circle (judging the position relationship between straight line and circle, and whether the proportion formula in the circle is established), using the main knowledge of junior high school to examine students' comprehensive problem-solving ability. When solving this kind of problems, we should pay attention to several problems: (1) Pay attention to the concepts involved in the topic and be familiar with relevant theorems, formulas, skills and methods; (2) Pay attention to analyzing the structure of comprehensive questions, find out the connection between knowledge points, and be good at decomposing a comprehensive question into several basic problems. The intersection of various knowledge points is often the key to transforming one basic problem into another. (3) Pay attention to exploring ways to solve problems from different angles, and use comprehensive methods and analytical methods such as "knowing from the known" and "knowing from the conclusion" to communicate known conditions and conclusions.

"The Problem of Several Letters" and "The Problem of Several Letters" involve a wide range of knowledge, a large span of knowledge, strong comprehensiveness, many applied mathematical methods, complex vertical and horizontal relations, novel and flexible structure, and emphasize basic ability, exploration and innovation, and mathematical thinking methods. It requires students to have good psychological quality and excellent mathematical basic skills, and be able to extract mathematical problems from known information, so as to flexibly use what they have learned and master the basic skills of creative problem solving.

1, comprehensive application analysis method, comprehensive method. It is from the conditions and conclusions, "from the known to the known" and "from the requirement to the demand" that the problem is "attacked from both sides" so that they can be linked in a certain link in the middle, and the problem can be solved.

2. Apply the idea of equation. It is to find the equivalent relationship between quantity and quantity in the problem to be solved, establish the equation of known quantity and unknown quantity, and solve the equation to solve the problem; When applying this idea, we should pay attention to fully excavating the hidden conditions of the problem, looking for equivalence relations and establishing equations or equations; For example, the solution of problem (2) in Example 2 of this paper uses this idea;

3. Pay attention to the idea of classified discussion. In the comprehensive problem of combining function and geometry, we often pay attention to the mathematical thought of students' classified discussion. Therefore, when solving such problems, we must think carefully from the side and explore all the possibilities of the conclusion with the thinking of classified discussion, so as to make the answer complete.

5. Use transformation ideas. The mathematical thought of reduction is the core idea of solving mathematical problems. Because the combination of function and geometry is comprehensive, boldly speaking, it is difficult to solve the comprehensive problem of the combination of function and geometry correctly and comprehensively without mastering the mathematical thought of reduction.

4. Use the idea of combining numbers with shapes. In middle school mathematics, "number" and "shape" are not isolated, and their dialectical unity is as follows: "number" can accurately clarify the ambiguity of "shape", and "shape" can intuitively inspire the calculation of "number"; When solving problems with the idea of combining numbers and shapes, we should pay attention to linking their properties with figures, and linking the corresponding figures with properties to simplify the problem.