(1) We will look for quantitative relations and changing rules from specific problems.
⑵ Understand the meaning of constants and variables, understand the concept and three representations of functions, and give practical examples of functions.
⑶ You can determine the range of independent variables in simple algebraic expressions, fractions and simple practical problems, and you can get the function values.
(4) Understand the concept of plane rectangular coordinate system and know the coordinate characteristics of each quadrant point and coordinate axis; You will find the coordinates of the point symmetrical about the X axis or Y axis or the origin.
5] Understand the concepts of linear function (including proportional function) and inverse proportional function in combination with specific conditions.
[6] Understand the images and properties of linear functions and inverse proportional functions and apply them.
Once analytical expressions of linear functions (including proportional functions) and inverse proportional functions can be determined according to practical problems.
⑻ Using the image of linear function to find the approximate solution of binary linear equations.
⑼ Combined with the analysis of function image, this paper tries to make a preliminary prediction of the changing law of variables and solve practical problems.
Second, the preparation points
1. Plane Cartesian coordinate system
A figure composed of two mutually perpendicular number axes on the (1) plane is called a plane rectangular coordinate system.
(2) The ordered real number pairs corresponding to a point on the coordinate plane are called the coordinates of the point. By establishing rectangular coordinate system on the plane, shape (point on the plane) and number (ordered real number pair) can be closely combined.
(3) The coordinate features of the first, second, third and fourth quadrant points are (+,+), (-,+), (-,-) and (+,-) respectively.
(4) If point (a, b) is on the horizontal axis, then b = 0；; If the point (a, b) is on the vertical axis, then a=0.
(5) The distance from point P(a, b) to origin O is equal to the distance from |b| to X axis and the distance from |a| to Y axis.
(6) The point where point (a, b) is symmetrical about X axis is (a,-b); The point about the Y axis symmetry is (-a, b); The point symmetrical about the origin O is (-a,-b);
2. The concept of function
(1) Suppose there are two variables X and Y in a certain change process. If every fixed value of x has a unique fixed value correspondence within a certain range, then y is said to be a function of x, and x is called an independent variable.
(2) There are three representations of functions, namely, image method, list method and analytical method.
(3) In a certain change process, the quantity that remains constant is called a constant, and the quantity that can take different values is called a variable.
(4) The range of independent variables of the function. For practical problems, independent variables must make practical problems meaningful. For pure mathematical problems, independent variables should make mathematical expressions meaningful.
3. Linear function and its properties
The form of (1) is y=kx(k is a constant, k≠0), then y is called the proportional function of X. 。
(2) The image with the proportional function y=kx is a straight line passing through (0,0) and (1, k); When k>0, the straight line passes through the first and third quadrants, when k
(3) the nature of the proportional function y=kx
(1) when k >; 0, y increases with the increase of x.
② when k
(4) If y = kx+b, k and b are constants, and the value of k is not 0, then y is called a linear function of x, and the proportional function is a special linear function when b=0.
(5) The image of linear function (k≠0) is a straight line passing through two points (0, b) and (0); When k>0 is a straight line passing through the first and third quadrants, when k
(6) Properties of linear functions
At that time, y increased with the increase of X.
② when k
4. Inverse proportional function and its properties
The form of (1) is y=k/x (k is a constant, k≠0), then y is called the inverse proportional function of x, and the inverse proportional function has three different expressions: ① y = k/x2y = kx-1; ③ xy=k
(2) The image of inverse proportional function y=k/x(k≠0) consists of two curves, usually called hyperbola.
Description: ① The two branches of hyperbola cannot be connected;
② The two branches are infinitely close to the X axis and the Y axis, but never intersect;
③ The image is symmetrical both in axis and center;
④ When drawing an inverse proportional function image, one branch is usually drawn first, and then another branch is drawn according to symmetry.
(3) The properties of inverse proportional function:
(1) when k >; At 0k, y decreases with the increase of x in each quadrant;
② when k
Third, the preparation suggestions
1. Plane rectangular coordinate system is the high-frequency test center of the senior high school entrance examination and the basic content of each exam. This paper mainly investigates the thinking method of combination of number and shape and movement change. Generally in the form of fill-in-the-blank questions and multiple-choice questions. In recent years, some provinces and cities have designed this part with the knowledge of probability, equation and circle, forming a novel finale. When reviewing, we should make clear the one-to-one correspondence between a point on the coordinate plane and an ordered real number pair. Understand the coordinate characteristics of points in the coordinate plane; According to the structural characteristics of the function, the range of independent variables of the function can be determined, so as to get the function value; Can accurately analyze the function relationship and predict the changing law of variables.
2. Linear function and inverse proportional function are widely used in real life. Using linear function and inverse proportional function to solve practical problems has become the highlight of the senior high school entrance examination in recent years. The function image information questions in the examination papers of many provinces and cities are novel in design, close to life, reflect the characteristics of the times, and comprehensively examine the mathematical quality of candidates. So review this section to master the images and properties of linear function and inverse proportional function. Can understand the meaning of linear function and inverse proportional function in combination with specific conditions; We can use the image information of linear function and inverse proportional function to solve practical problems. When reviewing, design some training questions about the interpenetration and interrelation of linear function, linear equation, linear inequality and linear equations, and strengthen training, so as to master the related properties of function, understand its laws and improve comprehensive ability.