In junior high school mathematics learning, many students are at a loss when they see the function and the finale. Although the learning of functions is gradual, from simple to difficult, functions are first proportional, then inversely proportional, and then quadratic. However, there are still many students who don't know where to start when facing the function problem, and quadratic function is an insurmountable hurdle for junior high school students to learn the function well. Function problems and finale problems in junior high school are obstacles for most students to learn math well. How to solve the problem of students' function finale is worth pondering by every junior high school math teacher.

Functions and their images play a very important role in junior high school mathematics. In the senior high school entrance examination, the score of function knowledge points accounts for about 25%, while the function synthesis questions involve more knowledge and skills. There are many mathematical ideas and methods in solving problems, with hidden conditions and complex structures. The comprehensive questions related to functions are the key and difficult points in the senior high school entrance examination. Only by breaking through this difficulty can we get high marks. But the function synthesis problem is a headache for many of our students, and many students give up or get few marks. So, how can we improve our ability to solve the problem of function synthesis? Personally, I think that students should first strengthen the "two basics" training. Only when they have solid basic skills can they have "inspiration" when solving problems. Secondly, they should improve their reading ability and their ability to analyze and solve problems. Solving comprehensive problems can be roughly divided into three steps:

First, carefully examine the questions and tap the implied conditions;

Second, explore ways to solve problems;

Third, write the solution process correctly.

It is necessary to master basic knowledge and skills and their internal relations, and flexibly use mathematical thinking methods (including transforming ideas, combining numbers with shapes, discussing ideas and equations in categories, etc. ). In the face of problems such as function synthesis, we should first be calm, because the first and second sub-questions of such problems can be scored, and the difficult part is the third sub-question, so we can't lose the points we deserve. Generally speaking, the third problem is a comprehensive problem of quadratic function and geometry, so when doing the problem, we should list the known conditions given in the problem, find out the relationship between the conditions and the problem, and at the same time combine the properties of geometric figures and algebraic knowledge and methods to analyze and solve the problem. There are three ways to solve the problem. One: Assuming that the point P on the parabola is consistent with the problem, replace the coordinates with (x, y), and then solve the equation according to the analysis of the conditions listed above; 2. Set the required quantity as X, find out the related quantity in the topic, and then list another quadratic function and turn it into a vertex to find the maximum and minimum value of X or Y; Thirdly, combined with geometric knowledge, the relationship between conditions and problems is comprehensively analyzed.

Quadratic function problem is basically a comprehensive problem, as the name implies, it is to examine all aspects of knowledge. Generally speaking, we should think from the following aspects (1): the basic knowledge of quadratic function (definition, graphics, properties, etc. of quadratic function. ); (2) The most common knowledge in the coordinate system is: "If a point is on a straight line, the coordinate of the point is suitable for the analytical formula of the straight line, so the equation is obtained"; "The coordinates of the intersection of two lines apply to the analytical formula of two lines,"; "To find the coordinates of the intersection of two lines, it is necessary to solve the equations formed by the analytical expressions of the two lines", "Take a point in the coordinate system as the vertical line segment of the two axes, and its length is equal to the absolute value of the abscissa and ordinate"; (3) Geometric knowledge, such as Pythagorean theorem, right triangle, triangle (similar triangles is the difficulty), quadrilateral and circle; (4): Equation knowledge: the most commonly used equations (groups), the discriminant of roots, the relationship between roots and coefficients, etc. (5): inequality (group) and so on.

In the senior high school entrance examination, the final question has always been a hot topic in the senior high school entrance examination paper, and it is also an evergreen tree. It is the finale that can fully reflect the characteristics of the subject, the finale that can really consider the level of candidates, and the finale that can open the fractional distance between candidates. The finale has the characteristics of novelty, typicality, pertinence and practicality, which can be roughly divided into the following eight questions.

First, the function problem usually examines the ability of candidates to solve problems by combining several functions. Candidates are not only required to correctly understand the basic knowledge (definition, image and nature) of linear function (proportional function), inverse proportional function and quadratic function, but also to master several common basic problems and conduct corresponding training.

Secondly, the straight line problems in space and graphics mainly include triangle problem, quadrilateral problem, congruence and similarity problem of graphics, solving right triangle and so on. This kind of problem needs students to observe, compare, summarize, guess and operate.

Third, the circle is one of the key contents of the senior high school entrance examination, and the open and exploratory topics tend to increase compared with the past. The comprehensive problem of circle often combines many knowledge points such as triangle, quadrilateral, similar shape, equation, function and so on. Many questions are original, innovative and difficult. Therefore, it needs to be comprehensively mastered and comprehensively applied.

Fourth, the problem of existence is a hot topic in the recent senior high school entrance examination. This type of question has a wide range of knowledge, strong comprehensiveness, exquisite topic conception and flexible problem-solving methods, which requires candidates' ability to analyze and solve problems. Its characteristic is to explore and discover whether some mathematical conclusions or laws exist under certain conditions, and it is open because there are two possibilities for conclusions to exist and not to exist.

5. Graphics and transformation, mainly including the axis symmetry, translation and rotation of graphics, changed the specific position relationship between the front and back graphics, and ran through the exploration of the properties of geometric graphics. This kind of questions not only examines students' understanding of the essence of basic graphics, but also cultivates students' hands-on and operation ability, and forms the idea of spatial concept and motion change. Therefore, this kind of question has become the new favorite of the final exam. The strategies to solve this kind of problems are: (1) mastering the properties of various graphs; (2) To master the relationship before and after graphic changes; (3) Transformation solution.

Sixth, the moving point problem that frequently appears in the finale is characterized by the fact that the main elements in the conditional point are moving. There is a little bit of motion problem. Recently, the double moving point problem with parabola as the carrier has become a beautiful scenery in the finale. This kind of problem requires students to use the ideas of combination of numbers and shapes, classified discussion and mathematical modeling. Through observation, guessing, reasoning, calculation and other processes. , use the method of equation or function to describe the process of change, and then solve it.

Seven, the maximum problem is also a comprehensive problem, which runs through junior high school mathematics learning and is a hot issue in the senior high school entrance examination. Mainly use important geometric conclusions (such as the shortest line segment between two points, the sum of two sides of a triangle is greater than the third side, the difference between the two sides is less than the third side, the shortest vertical line segment, etc.). ) to find the maximum value, using the properties of linear function and quadratic function to find the maximum value.

Third, the importance of mathematics reading

Eighth, practical application, the important purpose of learning mathematics is to be able to use what you have learned to solve problems in real life. Therefore, practical application is a compulsory part of the senior high school entrance examination. Recently, there have been many new changes and new features in this kind of questions. Many questions are rich in background and close to life, which examine students' mathematical modeling ability and application innovation ability. It covers all the knowledge points of junior high school mathematics, among which equations, inequalities, functions, solving right triangles and statistics are the key points.

In view of the above eight questions, students should first learn to interpret the meaning of the questions, instead of skipping and browsing the stem of the questions, they should chew slowly, seize every keyword, find out the hidden conditions in the information, and be good at comprehensive analysis combining conditions and conclusions. Secondly, analyze the core content of function, get familiar with the basic properties of geometric figures and the basic methods and skills of algebraic problem solving, find out the internal relationship between geometry and function, and solve the problem comprehensively through thinking methods such as combination of numbers and shapes, equation thinking, reduction thinking, transformation thinking and classified discussion. At the same time, we should master the methods and skills of solving some basic problems, so that the basic knowledge can be systematized, the basic problems can be modeled, the problem-solving methods can be shortened and the training methods can be scientific. After the exercises are modeled, we can sum up the methods and skills of solving problems in the back, and truly know everything at once; Do a problem and solve a lesson.