Linear functions, also known as linear functions, can be represented by straight lines on the x and y axes. When the value of one variable in a linear function is determined, the value of another variable can be determined by a linear equation. Basic concept of function: In a changing process, there are two variables X and Y, and for each definite value of X, there is a unique definite value in Y, so we say that Y is a function of X, or that X is an independent variable and Y is a dependent variable. Expressed as y = kx+b (k ≠ 0, k and b are constants). When b = 0, y is the proportional function of x, and the proportional function is a special case of linear function. It can be expressed as y=kx. Variable: variable (with different values) constant: constant (fixed) The independent variable k has the following relationship with the linear function y of x: Y = KX+B (where k is an arbitrary non-zero constant and b is an arbitrary constant). When x takes a value, y has one and only one value corresponding to x, and if there are two or more values corresponding to x, it is not a linear function. X is an independent variable, Y is a dependent variable, K is a constant, and Y is a linear function of X, especially when b=0, Y is X. That is, the image of the proportional function of y=kx (k is a constant, but K≠0) passes through the origin. Domain: the range of independent variables should make the function meaningful; It should be realistic. Related properties, practices and figures: List (2) trace points through the following three steps (1); [Generally, two points are taken and a straight line is determined by two points]; (3) The connection can be the image of a function-a straight line. So the image of a function only needs to know two points and connect them into a straight line. (Usually, the intersections of the function image with the X axis and the Y axis are -k points B and 0, 0 and B, respectively. B) 2 .. Property: any point P(x, y) on the linear function (1) satisfies the equation: y=kx+b(k≠0). (2) The coordinates of the linear function intersecting with the Y axis are always (0, b), and the images of the proportional function intersecting with the X axis at (-b/k, 0) are all at the origin. 3. Function is not a number, it refers to the relationship between two variables in a certain change process. 4. When K, B and the quadrant where the function image is located: y=kx (that is, B is equal to 0, and Y is proportional to X): When k > 0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X; When k 0, the straight line must pass through the first and second quadrants; When b < 0, the straight line must pass through the third and fourth quadrants. Particularly, when b=0, the image of the proportional function is represented by a straight line of the origin o (0 0,0). At this time, when k > 0, the straight line only passes through the first and third quadrants, but not the second and fourth quadrants. When k < 0, the straight line only passes through the second and fourth quadrants, but not through the first and third quadrants. 4. Special positional relationship When two straight lines in the plane rectangular coordinate system are parallel, the k value of the resolution function (that is, the coefficient of the first term) is equal. When two straight lines in the plane rectangular coordinate system are perpendicular, the k values of the resolution function are reciprocal (that is, the product of the two k values is-1). Expression analytic type ① General formula ax+by+c=0 ② Oblique y=kx+b (k is the slope of a straight line. Where the proportional function b=0) ③ Point inclination y-y 1=k(x-x 1) (k is the slope of a straight line, (x 1, Y 1) is the point where the straight line passes) ④ Two-point formula (y-y1)/(y2-y1) = (x-x1)/(x2-x1) (known because of the undetermined coefficient method, Note that the expression "a straight line without slope is parallel to the Y axis" is inaccurate, because x=0 coincides with the Y axis) ④ There are many parameters and the calculation is too complicated; ⑤ Cannot represent a straight line parallel to the coordinate axis and a straight line passing through the origin. The concept of inclination angle The angle between the X axis and the straight line (the angle formed by the straight line and the positive direction of the X axis) is called the inclination angle of the straight line. Let the inclination of the straight line be α, and the slope of the straight line is k=tanα. The range of tilt angle is [0, π]. Relationship with binary linear equation 1. (1) The image composed of points whose coordinates are the solution of the binary linear equation ax+by=c is the same as that of the linear function Y =-A/BX+C/D. (2) The solution of the binary linear equation {A1X+B65438+Y = C.A2X+B2Y = C2 can be regarded as. Summary of the intersection method of x+C 1/d 1: rewrite two binary linear equations into linear functions. The solution of the equation can be known. First, the difference and connection difference: the binary linear equation has two unknowns, while the linear function only says that the number of unknowns is one, without limiting several variables, so the binary linear equation is only one of the linear functions. Connection: (1) In the plane rectangular coordinate system, draw points whose coordinates are the solutions of binary linear equations, and these points are all on the images of corresponding linear functions. For example, the equation 2x+y = 5 has countless solutions, such as x= 1 and y = 3；; x=2，y = 1； The points (1, 3) (2, 1) whose coordinates are these solutions are all on the image of linear function y =-2x+5. (2) The coordinates of any point on the linear function image are suitable for the corresponding binary linear equation. For example, the linear function y =-x+2 (.2). The relationship between the intersection of two images of this function and the solution of the equations is in the same plane rectangular coordinate system, and the coordinates of the intersection of two images of the linear function are the solutions of the corresponding binary linear equations. On the contrary, the point whose coordinates are the solution of binary linear equations must be the intersection of the corresponding images of two linear functions. Third, the relationship between the corresponding function images when the equations have no solution. When the binary linear equations have no solution, the images of the corresponding two linear functions in the plane rectangular coordinate system do not intersect, that is, the images of the two linear functions are parallel. Conversely, when two linear function images are parallel, the corresponding binary linear equations have no solution. If the binary linear equations 3x-y=5 and 3x-y=- 1 have no solution, then the linear function y = 3x-5 is parallel to the image of y = 3x+ 1, and vice versa. 5. Solving the Binary Linear Equation System by Graph Method To solve the Binary Linear Equation System by Graph Method, there are generally the following steps: (1) Rewrite the corresponding Binary Linear Equation System into the analytical formula of linear function; (2) Make the images of these two linear functions in the same plane rectangular coordinate system; (3) Find the coordinates of the image intersection point and get the solution of the binary linear equations. Common formula 1. Find the k value of the function image: (y 1-y2)/(x 1-x2) 2. Find the midpoint of the line segment parallel to the X axis: | x1-x2 |/2 3. Find the midpoint of the line segment parallel to the Y axis: | Y65438+. 2+(y 1-y2) 2 under the radical sign (note: (x 1-x2) and (y 1-y2). 5. Find the coordinates of the intersection point of two linear function images: Solve two linear functions: y1= k1x+b1y2 = k2x+b2 to make y 1=y2 get k1x+b1. 0 y2=k2x+b2, if y=y0, then (x0, y0) is the coordinate of the intersection of y 1=k 1x+b 1 and y2=k2x+b2. 6. Find the midpoint coordinates of a line segment connected by any two points: [(x 1+x2)/. (y 1+y2)/2] 7。 Find the first resolution function of any two points: (x-x1)/(x1-x2) = (y-y1)/(y1-y2). Negative) in the third quadrant+,-(positive and negative) in the fourth quadrant 8. If two straight lines y1= k1x+b1∑ y2 = k2x+B2, then k 1=k2, b1.Then k1× k2 =-/kloc-. Y = K (x-n)+B means to translate n units to the right, and y=k(x+n)+b means to translate n units to the left: right minus left plus (for y=kx+b, only change b) y=kx+b+n means to translate n units up, and y=kx+b-n means to translate n units down: When the time t is constant, the distance s is a linear function of the speed v .. s = vt.2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t, and the original water quantity s in the pool is s=vt .2 G = S- ft. 3. When the original length b of the spring (the length when the weight is not hung) is fixed, the length y of the spring after the weight is hung is a linear function of the weight X, that is, y=kx+b(k is an arbitrary positive number).