One knowledge point of compulsory mathematics in senior one 1. Parity of functions.
(1) If f(x) is an even function, then f(x)=f(-x).
(2) If f(x) is odd function and 0 is in its domain, then f(0)=0 (which can be used to find parameters).
(3) The parity of the judgment function can be defined in equivalent form: f (x) f (-x) = 0 or (f(x)≠0.
(4) If the analytic formula of a given function is complex, it should be simplified first, and then its parity should be judged.
(5) odd function has the same monotonicity in the symmetric monotone interval; Even functions have opposite monotonicity in symmetric monotone intervals.
2. Some questions about compound function.
Solution of the domain of (1) composite function: If the domain is known as [a, b], the domain of the composite function f[g(x)] can be solved by the inequality a≤g(x)≤b; If the domain of f[g(x)] is known as [a, b], find the domain of f(x), which is equivalent to x∈[a, b], and find the domain of g(x) (that is, the domain of f(x)); When learning functions, we must pay attention to the principle of domain priority.
(2) The monotonicity of the composite function is determined by "same increase but different decrease".
3. Function image (or symmetry of equation curve).
(1) Prove the symmetry of the function image, that is, prove that the symmetry point of any point on the image about the symmetry center (symmetry axis) is still on the image.
(2) Prove the symmetry between the image C 1 and C2, that is, prove that the symmetry point of any point on C 1 about the symmetry center (symmetry axis) is still on C2, and vice versa.
(3) curve C 1: f (x, y)=0, and the equation of symmetry curve C2 about y=x+a(y=-x+a) is f(y-a, x+a)=0 (or f(-y+a,-x+a) =
(4) Curve C 1: f (x, y)=0 The C2 equation of the symmetrical curve about point (a, b) is: f(2a-x, 2b-y)=0.
(5) If the function y=f(x) is constant for x∈R and f(a+x)=f(a-x), then the image y=f(x) is symmetrical about the straight line x = a. ..
4. The periodicity of the function.
(1)y=f(x) for x∈R, f(x+a)=f(x-a) or f (x-2a) = f (x) (a >: 0) is a constant, then y=f(x) is a period of 2a.
(2) If y=f(x) is an even function whose image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 2 ~ a ~.
(3) If y=f(x) odd function, whose image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 4 ~ a.
(4) If y=f(x) is symmetric about points (a, 0) and (b, 0), then f(x) is a periodic function with a period of 2.
5. When judging whether the corresponding relationship is a mapping, grasp two points.
The elements in (1)A must all have images and be unique.
(2) All elements in B may not have the original image, and different elements in A may have the same image in B. ..
6. Prove the monotonicity of the function by using the definition skillfully, find the inverse function and judge the parity of the function.
7. For the inverse function, we should grasp the following conclusions.
A monotone function in (1) field must have an inverse function.
(2) odd function's inverse function is also odd function.
(3) Even functions whose domain is not a single element set have no inverse function.
(4) The periodic function has no inverse function.
(5) The two inverse functions have the same monotonicity.
(6)y=f(x) and y=f- 1(x) are reciprocal functions. Let the domain of f (x) be a and the domain of f(x) be b, then there is f [f- 1 (x)] = x (x ∈ b).
8. When dealing with quadratic functions, don't forget the combination of numbers and shapes.
Quadratic function must have a maximum in the closed interval, and the problem of finding the maximum is "two views": look at the opening direction; Second, look at the relative position relationship between the symmetry axis and a given interval.
9. According to monotonicity, we can solve the range problem of a class of parameters by using the sign-preserving property of linear functions on intervals.
10. Work out the solution to the invariant problem.
(1) separation parameter method.
(2) The solution of transforming the inequality (group) of distribution table into the root of quadratic equation.
Expanding reading: the method of learning mathematics 1. Establish confidence in learning high school mathematics well.
When you enter high school, you must set up correct learning goals and lofty ideals. Encourage yourself to think positively, be enterprising, cultivate interest in learning mathematics, and establish confidence in learning mathematics well.
Read your notes first, then do your homework.
Some high school students think. I heard what the teacher said clearly. But why is it so difficult to do the problem by yourself? The reason is that students' understanding of what the teacher said did not reach the level required by the teacher. Therefore, before you do your homework every day, you must take a look at the relevant contents of the textbook and the class notes of that day. Whether you can stick to this point is often the biggest difference between good students and poor students. Especially when the exercises are not matched, there are often no questions in the homework that the teacher just talked about, so it is impossible to compare and digest them. If we don't pay attention to this realization, it will cause great losses over time.
3. Strengthen reflection after doing the questions.
Students must make it clear that the topic they are doing now is definitely not the topic of the exam. But to use ideas and methods to solve the problems we are doing now. Therefore, we should reflect on every question we have done. Sum up your gains. To sum up, this is a question of what content and how to use it. Make knowledge into pieces, problems into strings, accumulate over time, and build a scientific network content and method system.
4. Take the initiative to review, summarize and improve.
It is very important to summarize the chapters. In junior high school, it is the teacher who gives the students a summary, which is meticulous, profound and complete. In the third year of senior high school, I made my own summary, but the teacher not only refused to do it, but also said where to take the exam, where to take the exam, leaving no review time, and did not clearly point out the time to make the summary.
5. Accumulate information and organize it at any time.
Pay attention to accumulating review materials. Organize class notes, exercises, unit tests and various papers in chronological order. Every time you read it, mark the key content of the next reading on it. This way, the review materials can be more refined and clear at a glance.
6. Jump out of the endless ocean of problems.
Save time and focus on the research of high-quality projects. Make maximum use of two kinds of excellent questions: one is the mother question covering multiple test sites, and the other is the wrong question with high frequency in the same question type.
7. Summarize the laws of mathematics.
Mathematics is not difficult, in fact, it is just doing problems according to the law. The reason is very simple, because the person who gave the question gave the question according to law. So as long as you master the rules, don't be afraid. The key is to find the pattern. The same type of topic, this time wrong, summed up the law, will do it next time. More and more rules, just like having more keys, are not afraid of all kinds of locks. If someone else sums it up for you, you should sum it up again, so that it can become yours. Our mathematics is based on the laws summarized by mathematicians before.