Mathematical definition of function: given a set of numbers A, apply the corresponding rule F to A, and record it as f(A), and get another set of numbers B, that is, B=f(A). Then this relationship is called functional relationship, or function for short. The following is my summary of the second grade function knowledge, welcome to refer to!
First, the main points of knowledge
1. function concept: there are two variables x and y in a changing process. If y has a unique value for each value of x, then x is an independent variable and y is a function of X. 。
2. The concepts of linear function and proportional function.
If the relationship between two variables x and y can be expressed as y=kx+b(k, b is a constant, k? 0), y is said to be a linear function of x (x is an independent variable), especially when b=0, y is said to be a proportional function of X. 。
Note: The range of independent variables of (1) linear function is all real numbers, but it should be determined according to the actual meaning of the function in practical problems.
(2) linear function y=kx+b(k, b is a constant, b? 0)? Once? There are also unary linear equations and unary linear inequalities? Once? The meaning is the same, that is, the degree of the independent variable x is 1, the coefficient k of the first term must be non-zero constant, and b can be any constant.
(3) When b=0, k? 0, y=b or a linear function.
(4) When b=0 and k=0, it is not a linear function.
3. Linear function image (three-step drawing image)
Because the linear function y=kx+b(k, b is a constant, k? 0) is a straight line, so the image of linear function y=kx+b is also called straight line y = kx+B.
Because two points determine a straight line, when making a function diagram in the future, you only need to trace two points that are suitable for the relationship and then connect them into a straight line. Generally, two special points are selected: the intersection of a straight line and the Y axis (0, b) and the intersection of a straight line and the X axis (-0). However, it is not necessary to choose these two special points. When drawing an image with the proportional function y=kx, you only need to trace the point (0,0).
4, linear function y=kx+b(k, b is a constant, k? 0) (the attribute of the proportional function is omitted)
The sign of (1)k determines the inclination direction of the straight line; ①k & gt； The values of 0 and y increase with the increase of x value;
②k & lt； The values of o and y decrease with the increase of x value. P = "">& lt/o, the value of y decreases with the increase of x value. >;
(2) The size of |k| determines the inclination of the straight line, that is, the greater the | k |, the greater the acute angle of the straight line intersecting the X axis (steep line), while the smaller the | k |, the smaller the acute angle of the straight line intersecting the X axis (slow line);
(3) The sign of B determines the position where the straight line intersects the Y axis;
(1) when b >; 0, the straight line intersects the Y axis on the positive semi-axis;
② when b
③ When b=0, the straight line passes through the origin, which is a proportional function.
(4) Because the symbols of K and B are different, the quadrants that the straight line passes through are also different;
5. Determine the conditions of proportional function and linear function expression.
(1) Because the proportional function y=kx(k? 0) has only one undetermined coefficient k, so the value of k can be obtained by only one condition (such as a pair of x, y values or a point).
(2) because the linear function y=kx+b(k? There are two undetermined coefficients k and b in 0). Two independent conditions are needed to determine two equations about k and b, and then the ` value of k and b can be obtained. These two conditions are usually two points or two pairs of x and y values.
6, undetermined coefficient method
Set the relationship between the functions to be solved (including unknown constant coefficients), and then list the equations (or equations) according to the conditions to find the unknown coefficients, so as to get the results. The method is called undetermined coefficient method, in which the unknown coefficients are also called undetermined coefficients. For example, in the function y=kx+b, k and b are undetermined coefficients.
7. The general steps to determine the expression of linear function by undetermined coefficient method.
(1) Let the function expression be y = kx+b;
(2) Substituting the coordinates of known points into the function expression to solve the equation (group);
(3) Find the values of k and b to get the function expression.
8, this chapter thinking method
(1) function method. Function method is to analyze the quantitative relationship in the problem from the viewpoint of motion change, and the essence of function is to study the corresponding relationship between two variables.
(2) Number-shape combination method. The combination of numbers and shapes refers to a way of thinking that combines numbers and shapes to analyze, study and solve problems.
Second, typical cases
Example 1. Is the function y=-(m-2)x+(m-4) a linear function when m is a value?
Example 2: The length of the spring is 15cm, and the mass of the object it hangs cannot exceed 18kg. For every 1kg object, the spring will extend by 0.5cm. Write the functional relationship between the length y(cm) of the spring and the mass x(kg) of the suspended object, write the range of the independent variable x, and judge y.
Example 3, (2003? Xiamen) The temperature M(℃) of an object from 7 am to 4 pm is a function of time t (hours): M=t2-5t+ 100 (where t=0 means noon 12 and t= 1 means afternoon 1), so the morning is.
Example 4: It is known that y+m is proportional to x-n (where m and n are constants).
Is (1)y a linear function of x? Please explain the reasons; Under what conditions is y a direct proportional function of x?
(2) If x=- 1 and y =-15; When x=7 and y= 1, find the analytical expression of this linear function. Find the area of the triangle surrounded by this straight line and the coordinate axis.
Example 5. (Harbin) If the image of the proportional function y=( 1-2m)x passes through point A (x 1, y 1) and point B (x2, y2), when x 1 y2, the range of m is _ _ _ _ _ _.
Example 6. The range of the independent variable x of the linear function y=kx+b is -3? x？ 6. The range of the corresponding function value is -5? y？ -2, then the analytical expression of this function is.
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