Discuss the function model and solve practical problems, and realize that function is an important mathematical model to describe the changing law of the real world.
2. With examples, understand related concepts such as constants, variables and functions, understand the idea of "variable correspondence" and understand three representations of functions.
Methods (list method, analytical formula method, image method) By combining the number and shape of images, simple functional relationships can be analyzed.
3. Through certain inquiry activities, explore and understand the concepts of proportional function and linear function, draw their images and discuss them with images.
Regarding the basic properties of these functions, we can use these functions to analyze and solve simple practical problems.
Key points:
Understand the concepts of function and linear function, and master the images and properties of linear function.
Difficulties:
Understanding the concept of function and its application in the thought of function model.
Second, sort out the main points of knowledge
Knowledge point 1: the concept of function
Important point: There are two variables X and Y in a change process. If for each value of X, Y has a unique and definite value corresponding to it, then X is an independent variable and Y is a function of X. (To understand the concept of function, grasp three points: ① a change process, ② two variables and ③ a corresponding relationship. Judging whether two quantities have a functional relationship is also based on these three points)
Knowledge point 2: Definition of linear function and proportional function
Key point explanation:
1. Generally, if it is in the form of y=kx and ten b(k, b are constants,), it is said that y is a linear function of x;
2. Especially when b=0, that is, a function in the form of y=kx () is called a proportional function, where k is called a proportional coefficient;
3. Obviously, the proportional function is a linear function, which is not necessarily a proportional function, but it is a special case of linear function.
Key point explanation:
1, the nature of the proportional function: the image of the proportional function y=kx () is a straight line passing through the origin, which can also be called straight line y = kx; When k>0 and y increase with the increase of x, the straight line y=kx passes through the first and third quadrants, and the image rises; When k < 0, y decreases with the increase of x, and the straight line y=kx passes through the second and fourth quadrants, and the image decreases.
2. Properties of linear function: linear function y=kx ten b(k, b is constant,), which can also be called straight line y = kxten b; It has the same properties as the proportional function, that is, when k >; 0, y increases with the increase of x, and when k
Knowledge point 5: Determination of linear function expression
Focus on explanation: the determination of a linear function expression usually has the following situations:
(1) Using the undetermined coefficient, according to the coordinates of two points on a straight line, k and b are determined by the listed equations, and the linear function expression is obtained.
(2) Find the linear function expression according to the image.
Knowledge point 6: Three representations of functions: list method, image method and analytical method.
Key point explanation:
1, analytical formula method-the mathematical formula used to express functional relationship is called resolution function or functional relationship;
2. List method-the corresponding relationship between function y and independent variable x is given by list;
3. Mirror image method —— Take the independent variable X as the abscissa of the point and the corresponding function value Y as the ordinate of the point, and draw the corresponding point in the rectangular coordinate system. The set of all these points is called the image of this function. The corresponding relationship between function y and independent variable x is represented by images.
Note: The three representations of functions have their own advantages, and one or more of them are often selected according to the needs of solving problems. The resolution function is given. Through calculation, a numerical table reflecting the corresponding relationship between two variables can be listed. According to the listed lattice, we can get the ordered number pairs representing the functional relationship, and we can draw an image by tracing. However, functions expressed by list or mirror image method do not necessarily have analytical expressions, such as the function of temperature changing with time on a certain day. We can't use analytical expressions to express functional relations, but we can use three methods to express linear functions, but we usually use analytical expressions and image methods. To find the analytical expression of linear function, the undetermined coefficient method is usually used. As long as two points on a straight line are found, the image of linear function can be drawn.
Knowledge point 7: Determine the conditions that the resolution function should have.
Key point explanation:
1, because there is only one undetermined coefficient k in the proportional function y=kx(k is a constant), as long as there is a pair of values of x and y or a point outside the origin, the value of k can be found;
2. There are two undetermined coefficients k and b in the linear function y=kx. Determining two equations about K and B requires two independent conditions, which are usually two points or two pairs of values of X and Y. ..
Third, the guidance of legal methods.
(A summary of thinking methods
1. undetermined coefficient method:
It means that the unknown coefficient in the formula is set first, and then the unknown number is obtained according to the conditions, so as to write the solution method of this formula. This chapter is mainly used to find the expression of a linear function.
2. The idea of combining numbers and shapes:
It refers to a way of thinking that combines quantity and graphics to comprehensively analyze, study and solve problems. The combination of numbers and shapes is an important channel to develop thinking. Function diagram can intuitively show the dependence of two variables, which is convenient for observing the changing trend of two variables. The meaning embodied in the function diagram should be related to equations, equations and inequalities.
3. Functional concept:
Function reflects the relationship between the movement of the objective world and the quantity of real things, and is a powerful weapon to solve problems. In addition, functions are closely related to equations and inequalities.
4. Turn to think of:
It refers to a way of thinking that the problems to be solved or unsolved are reduced to problems that have been solved or are relatively easy to solve through transformation, and finally solve the problems. The idea of transformation runs through mathematics.
5. The idea of classified discussion:
This chapter fully embodies the idea of classified discussion by studying the images and properties of linear functions.
(2) Problems that should be paid attention to
1. Understand the concept of function
The definition of (1) function includes three elements:
(1) the range of independent variables;
② Correspondence between two variables;
③ The range of function value (or dependent variable).
(2) The function is not a number, but its essence is a correspondence. The function relation is a correspondence between variables, which is a special correspondence. It must be "for every value of x, y has a unique value corresponding to it".
(3) It doesn't matter what letter the independent variable and the function are represented by. The independent variable can be represented by X, or any letter of T, U, P, …, and the function can be represented by Y, or any letter of S, V, Q, ….
Uniqueness means that for every value of X, Y has a unique value corresponding to it, so Y is a function of X. For example, Y is a function of X in the two variables reflected in Figure A and Y is not a function of X in the two variables reflected in Figure B. 。
2. Judge whether two functions are the same function.
Consider two aspects from the definition of function:
First, look at the range of independent variables;
The second is to see whether the corresponding way between two variables is the same. When the range of independent variables is the same and the independent variables are the same, the corresponding function values are also the same, that is, the two variables correspond in the same way.
3. Observe the image of the function and get relevant information.
(1) maximum;
(2) increase or decrease;
(3) The practical significance of the coordinates of points.
From the coordinates of the highest or lowest point in the image, it can be explained that when the independent variable takes any value, the function value is maximum or minimum.
From left to right, we can see that the function value increases or decreases with the increase of the independent variable value.
Dialysis of classical examples
The first category: the range of independent variables.
1. Xiao Qiang wants to make an isosceles triangle with a circumference of 80cm. Please write the functional relationship between base length y(cm) and waist length x(cm) and find the range of independent variable X.
Thinking: The waist length x and the bottom length y should be greater than 0, and at the same time, it should be noted that the sum of the two waist lengths 2x should be greater than the bottom length y. 。
Analysis: y=80-2x
∫x+x = 2x > y，
∴0