What quadratic functions are introduced in the knowledge points of quadratic functions in junior high school mathematics?
The basic expression of quadratic function is y=ax? The highest degree of the quadratic function of +bx+c(a≠0) must be quadratic, and the image of the quadratic function is a parabola whose symmetry axis is parallel or coincident with the Y axis. Its definition is quadratic polynomial (or monomial).
If the value of y is equal to zero, a quadratic equation can be obtained. The solution of this equation is called the root of the equation or the zero of the function.
What is a quadratic function expression?
(1) vertex type
y=a(x-h)？ +k(a≠0, a, h, k are constants), the vertex coordinates are (h, k), the symmetry axis is a straight line x=h, and the position characteristics of the vertex and the opening direction of the image are related to the function y=ax? When x=h, y = the maximum (minimum) value of k.
(2) Intersection point
y=a(x-x？ )(x-x？ ) [limited to the parabola intersecting with the x axis, that is, y=0, that is, b? -4ac & gt； 0]
Function and image intersect at (x? 0) and (x? ,0)
(3) general formula
y=aX？ +bX+c=0(a≠0)(a, B and C are constants)
Symmetry relation of quadratic function image
(1) For the general formula:
①y=ax2+bx+c and y=ax2-bx+c are symmetrical about Y ..
② The two images Y = AX2+BX+C and y=-ax2-bx-c are symmetrical about X axis.
③y=ax2+bx+c and y=-ax2-bx+c-b2/2a are symmetrical about the vertex.
④y=ax2+bx+c and y=-ax2+bx-c are symmetrical about the center of the origin. (that is, the graph obtained after rotating 180 degrees around the origin).
(2) For vertices:
① two images, y = a (x -h) 2+k and y=a(x+h)2+k, are symmetrical about y axis, that is, vertices (h, k) and (-h, k) are symmetrical about y axis, and the abscissas are opposite, but the ordinate is the same.
② two images, y = a (x-h) 2+k and y=-a(x-h)2-k, are axisymmetrical about x, that is, vertices (h, k) and (h, -k) are axisymmetrical about x, with the same abscissa and opposite ordinate.
③y=a(x-h)2+k and y=-a(x-h)2+k are symmetrical about the vertices, that is, the vertices (h, k) and (h, k) are the same and the opening directions are opposite.
④y=a(x-h)2+k and y=-a(x+h)2-k are symmetrical about the origin, that is, the vertices (h, k) and (-h, -k) are symmetrical about the origin, and the abscissa and ordinate are opposite.
The Method of Finding Quadratic Resolution Function
(1) If the parabola passes through three known points, use the general formula: y=ax? +bx+c are substituted into the ternary linear equations respectively, and the values of a, b and c are obtained, thus the analytical expressions are obtained.
(2) Given vertex coordinates and another point, use vertex: y=a(x-h)? +k, after the point coordinates are substituted, it becomes a one-dimensional linear equation about a, and the value of a is obtained, thus the analytical formula is obtained.
(3) Among the three points that the parabola passes through, there are two points on the X axis, and the intersection point can be used (two types): y=a(x-x? )(x-x？ ), substitute the coordinates of the third point to find a, and get the parabolic analytical formula.
Properties of quadratic function
The image of (1) quadratic function is a parabola, and parabola is an axisymmetric figure. The symmetry axis is a straight line x=-b/2a.
(2) Quadratic coefficient A determines the opening direction and size of parabola. When a>0, the parabolic opening is upward; When a<0, the parabolic opening is downward. The larger |a|, the smaller the opening of parabola; The smaller the |a|, the larger the opening of the parabola.
(3) Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
Both linear coefficient b and quadratic coefficient a*** determine the position of the axis of symmetry. When A and B have the same number (ab>0), the symmetry axis is on the left side of Y axis; When a and b have different numbers (i.e. AB
The constant term c determines the intersection of the parabola and the y axis. The parabola intersects the y axis at (0, c).
Extended reading: What are the preparation methods for the senior high school entrance examination? 1. The novelty and flexibility of the senior high school entrance examination questions are getting stronger and stronger.
Many teachers and students focus on the difficult comprehensive problems, thinking that only by solving the difficult problems can they cultivate their abilities, thus relatively ignoring the review of basic knowledge, basic skills and basic methods. In the review, concepts, formulas and theorems are given first, and then several examples are given to train through a large number of topics. In fact, the process of deducing theorems and formulas contains important problem-solving methods and laws. Teachers do not fully expose the thinking process and discover its internal laws, and try to "realize" some truths by doing a lot of questions. As a result, they can't "understand" methods and laws, have a superficial understanding and weak memory, only imitate mechanically, have a low level of thinking, and sometimes even copy things mechanically, which complicates simple problems and leads to losing points.
2. Look for the "shadow" of the exam questions from the textbook.
Many test questions are formed by analogy, processing, transformation, strengthening or weakening conditions, extension or expansion on the basis of examples and exercises in textbooks. Therefore, in the first stage of review, we should review the basic knowledge based on the new curriculum standards and textbooks.
3. Highlight the characteristics of review.
Judging from the review arrangement, the review of basic knowledge mainly depends on systematic review. In the review of each chapter, in order to make students understand the structure of knowledge effectively, students should be allowed to check and fill in the gaps according to their own reality and review freely with a purpose. Then through appropriate training, students can strengthen their understanding of concepts, grasp conclusions, apply methods and improve their ability. Then we can cultivate students' abstract thinking ability.
4. Combing knowledge and strengthening variant training.
The proposition of the senior high school entrance examination is "based on the curriculum standard and closely related to the teaching materials", which is in the test paper. Many topics are corrected by taking examples and exercises in textbooks as examples. Therefore, no matter what review materials can replace textbooks, only by carefully reviewing the basic knowledge in textbooks, mastering basic skills and doing some variant exercises on typical topics in textbooks can we master the double basics flexibly and correctly answer the questions in the senior high school entrance examination. In the process of double-base review, we should sort out the textbook knowledge, strengthen variant training in the combing of key knowledge, and comb it with auxiliary teaching methods and auxiliary lines, so as to master it skillfully.
5. Clear the context and grasp the foundation.
In the review, we should stick to the textbook, lay a solid foundation, focus on the review of basic questions and the training of basic mathematical thinking methods, interspersed with a small amount of comprehensive review, and at the same time pay attention to new knowledge, systematically sort out the textbook knowledge, form a knowledge network, and carry out variant training on typical problems, so as to achieve the purpose of drawing inferences from others and improving the ability to take exams.
6. Different treatments have different emphases.
In review, students should review their knowledge in a targeted way. If you are an ordinary student, you must be strict with yourself and solve problems strictly and seriously; Top students may wish to strengthen exercise training in review and pay attention to logical relations in solving problems. In addition, we should focus on the difficulty and proportion of knowledge points in the senior high school entrance examination, and review them differently and emphatically. At the same time, error correction training is carried out purposefully to analyze the problems that are easy to make mistakes.
What are the answering skills of multiple-choice questions in mathematics for senior high school entrance examination? 1. exclusion method.
Because the answer to multiple-choice questions is one of the four options, there must be only one correct answer, so we can use the exclusion method to exclude the answer that is easy to judge as wrong from the four options, so the remaining one is naturally the correct answer.
Second, the method of giving special value.
That is, according to the conditions in the topic, select special values that meet the conditions or make special graphics for calculation and reasoning. When solving problems with special value method, we should pay attention to selecting values that meet the conditions and are easy to calculate.
Third, directly observe or get the results by guessing and measuring.
This method is often used to explore the regularity in junior high school problems in recent years. The main solution to this kind of problem is to use incomplete induction to solve the problem through experiment, guess, trial and error verification and summary.
Fourth, direct solution method.
Some multiple-choice questions are adapted from some fill-in-the-blank questions, true-false questions and solution questions, so we can often adopt direct methods, directly proceed from the conditions of the questions, draw conclusions directly through correct operation or reasoning, and then compare with the options to determine the options. Most of us will use this method when solving problems.
Verb (verb abbreviation) undetermined coefficient method
To find a certain functional relationship, we can first assume the undetermined coefficients, then list the equations (groups) according to the meaning of the question, and then get the undetermined coefficients by solving the equations (groups), thus determining the functional relationship. This method is called undetermined coefficient method.
Six, incomplete induction
When a mathematical problem involves many or even infinite situations and it is difficult to start with a chaotic clue, the effective method is to find out the general law and solve the problem by investigating some simple situations.