Examples of linear functions
1. Definition of linear function and proportional function:
(1) linear function: in general, if y=kx+b (where k and b are constants, k ≠ 0), then y is called a linear function of X. 。
(2) proportional function: when b = 0 b=0, k ≠ 0, y=kx, then y is the proportional function of X. 。
2. The difference and connection between linear function and proportional function:
(1) From the analytical formula, y=kx+b (k ≠ 0, b ≠ 0) is a linear function, and y=kx (k ≠ 0, b=0) is a proportional function. Obviously, proportional function is a special case of linear function, and linear function is a generalization of proportional function. They all are.
(2) From the image, y=kx (k ≠ 0) is a straight line passing through point (0,0), while y=kx+b (k ≠ 0) is a straight line passing through point (0,b) and parallel to y=kx.
3. The positional relationship between the symbols of k and b and the image of linear function y=kx+b (k ≠ 0);
4. Determine the conditions of linear function and proportional function:
? The undetermined coefficient in the proportional function y=kx (k 0) is k, so only one condition is needed to determine the proportional function; The undetermined coefficients in the linear function y=kx+b(k ≠ 0) are k and b, so two conditions are needed to determine the linear function. From the geometric point of view, the image of the proportional function passes through the (0,0) point, and "two points determine a straight line", so we only need to know the other point, and the linear function must know two points.
Problem solving skills of first-order function
Linear function: a function with the shape y=kx+b(k≠0, k and b are constants) is called a linear function, and the image is a straight line. Usually, the specific problems are to find the analytical formula, the area of the figure surrounded by the coordinate axis, the coordinates of the intersection of the two left axes, and the practical application problems. The most difficult point is to find the law.
Problem solving skills:
First find the known conditions, such as symmetry, coordinate points, xy axis intersection, etc.
The analytical formula is obtained by using conditions.
List the problem meaning equation, such as the intersection problem, that is, two sets of analytical expressions constitute the equation.
The area problem is usually a regular number. If it is irregular, it is often solved by digging and filling, and using ""instead of ""regular graphics.
Note: In practical application, there is often a range of values. For example, the unit price of a commodity is -500 yuan, which is obviously unrealistic.