The concepts of 1, proportional function and linear function

Basic knowledge induction: generally speaking, if y = kx+b (k, b is a constant, k≠0), it is called a linear function of x, especially when b in the linear function y = kx+b is 0, y=kx(k is a constant, k≠0). At this time, y is called the proportional function of X.

The basic methods are summarized as follows: the key to judge whether a function is linear is whether its k is not 0 and the independent variable index is1; In order to judge whether it is a proportional function, the condition of b=0 should be added to the linear function.

2. Linear function image

Basic knowledge induction: the images of all linear functions are a straight line; The image with linear function y = kx+b is a straight line passing through point (0, b).

The image with the proportional function y=k/x is a straight line passing through the origin (0,0).

K>0, b>0, the image passes through the first, second and third quadrants, and Y increases with the increase of X.

K>0, b<0, the image passes through the first, third and fourth quadrants, and Y increases with the increase of X.

K<0, b>0, the image passes through the first, second and fourth quadrants, and Y decreases with the increase of X.

K<0, b<0, the image passes through two, three and four quadrants, and Y decreases with the increase of X.

When b=0, a linear function becomes a proportional function, which is a special case of linear function.

The basic method is summarized as follows: use the proportional function y=kx to translate the linear function y=kx+b up and down. It is necessary to judge the quadrant that the linear function passes through, and then shift up or down with the sign of B to get the quadrant that it passes through. The increase or decrease is only determined by the positive or negative of k, and has nothing to do with the value of B.

3. Determination of the proportional function and the first resolution function.

Basic knowledge induction: to determine a proportional function is to determine the constant k in the definition formula y=kx(k≠0) of the proportional function. To determine a linear function, we need to determine the constants k and b in the definition of linear function Y = KX+B (k ≠ 0). The general method to solve this kind of problem is the undetermined coefficient method.

4. Triangle area surrounded by linear function image and coordinate axis.

Basic knowledge induction: the coordinates of the intersection of straight line y=kx+b with X axis and Y axis; You can find the area of a triangle surrounded by a straight line and two coordinate axes.

5. The application of linear function

Basic knowledge induction: it mainly involves the application of economic decision-making and market economy. Using linear function and combining equation (group) and inequality (group) to solve the design problems of interest rate, profit, rent and production plan in real life.

Basic method induction: the general steps to solve application problems by using functional knowledge;

(1) Set variables in practical problems.

(2) Establish the functional relationship between variables, such as linear function, quadratic function or other compound functions.

(3) Determine the range of independent variables to ensure that the independent variables have practical significance.

(4) Using the properties of functions to solve problems.

(5) write the answer.

Attention should be paid to problem induction: when reading a picture, we should first understand the actual meaning expressed by abscissa and ordinate, and also convert the coordinates of each point on the image into quantities representing the actual meaning; The range of independent variables should be accurate and in line with practical significance.