1. Key points of knowledge: 1. Linear function: If the relationship between two variables X and Y is Y = KX+B (k ≠ 0, k and b are constants), then Y is said to be a function of X. Note: (1)k≠0, otherwise the coefficient of the highest term of the independent variable X is not1; (2) when b=0, y=kx, and y is a proportional function of x.2. Image: The image of a linear function is a straight line with two common special points (1): it intersects the Y axis at (0, b); Intersect with the x axis at (-,0). (2) The image of the proportional function y=kx(k≠0) is a straight line passing through (0,0) and (1, k); The image of linear function y=kx+b(k≠0) is a straight line passing through (-,0) and (0, b). (3) According to the image, the straight line y=kx+b is parallel to the straight line y=kx, for example, the straight lines y=2x+3 and y=2x-5 are both parallel to the straight line y=2x. 3. Properties of linear function image: (1) position of image in plane rectangular coordinate system: (2) increase or decrease: k >；; 0, y increases with the increase of x; K<0, y decreases with the increase of x. There are three main ways to find the resolution function: one is derived from a known function, such as1; The second is to list two unknown equations from practical problems and then convert them into resolution functions, as shown in the first question of Example 4. The third is to use the undetermined coefficient method to find the resolution function, such as the second small question in Example 2 and Example 7. The steps are as follows: according to the given conditions, write the analytical formula of the undetermined coefficient; (2) Substituting several pairs of values of X and Y or the coordinates of several points on the image into the above analytical formula to obtain an equation or set of equations with undetermined coefficients as unknowns; (3) solving the equation to obtain the specific value of the undetermined coefficient; ④ Substitute the determined undetermined coefficients into the required resolution function. 2. Example: for example 1, it is known that the relationship between variable Y and y 1 is y=2y 1, and the relationship between variable y 1 and X is y 1=3x+2. Find the functional relationship analysis of variables Y and X: Two groups of functional relationships are known, in which the same variable is y 1, so the relationship between Y and X can be found through y 1. Solution: ∫y = 2y 1y 1 = 3x+2, ∴ Y = 2 (3x+2) = 6x+4, that is, the relationship between variable y and x is: y=6x+4. Example 2, solve the following problem (1) (Gansu Province senior high school entrance examination) It is known that the straight line and the Y axis intersect at point A, then the coordinate of point A is (). (A)(0,–3) (b) (c) (d) (0, 3) (2) (Hangzhou senior high school entrance examination) The proportional function is known. When x =–3, y = 6. Then the proportional function should be (). (A) (B) (C) (D) (3) (Fuzhou senior high school entrance examination) The image of the linear function y=x+ 1, the quadrant that fails is (). (A) First Quadrant (B) Second Quadrant (C) Third Quadrant (D) Fourth Quadrant Analysis and Solution: (1) The coordinate feature of the intersection of the straight line and the Y axis is that the abscissa is 0, and the ordinate can be obtained by substituting it into the functional relationship. Or directly use the intersection of a straight line and the Y axis as (0, b) to get the intersection point (0, 3), and the answer is d(2). The key to finding the analytical formula is to determine the coefficient k. When x=-3 is known, substitute y=6 into y=kx, and the analytical formula can be determined. Answer D: y=-2x. (3) According to the image properties of the linear function y=kx+b, the following conclusions are obtained: y=x+ 1, k =1> 0, the function image must pass through one or three quadrants; b = 1 & gt； 0, then the straight line and the Y axis intersect the positive semi-axis, so you can judge the position of the straight line, draw a sketch, or take two stippling sketches to judge that the image is exactly the fourth quadrant. Answer: D. Example 3: (Liaoning Provincial Senior High School Entrance Examination) A unit urgently needs a car; But they are not going to buy a car. They will sign monthly rental contracts with car owners or state-owned taxi companies. Assuming that the car travels x kilometers per month, the monthly fee payable to the individual owner is y 1 yuan, and the monthly fee payable to the taxi company is y2 yuan. The functional relationship between y 1 and y2 and x (two rays) is shown in the figure. Observe the image and answer the following questions: (1) When the monthly driving distance is within what range, is it economical to rent a car from a state-owned enterprise? (2) When the distance traveled each month is equal, the cost of renting two cars is the same? (3) If the unit is expected to travel 2300 kilometers per month, which car is more economical for the unit to rent? Analysis: Because the images of two functions are given, we can know that one is a linear function and the other is a special form of proportional function of linear function. The abscissa of the intersection of two straight lines is 1500, which means that when x= 1500, the function values of the two straight lines are equal, and we can know from the image that x >. At 1500, y2 is above y 1; 0<x< is in 1500, and y2 is lower than y 1. Using images, three questions are easy to answer. Answer: (1) When the monthly driving distance is less than 1500 km, it is more economical to rent a car from a state-owned enterprise. [or answer: when 0 ≤ x < 1500 (km), it is more economical to rent a car from a state-owned enterprise]. (2) When the driving distance in the current month is equal to1500km, the cost of renting two cars is the same. (3) If the distance traveled every month is 2300 kilometers, then it is cost-effective for this unit to rent a car owned by an individual owner. Example 4 (Senior High School Entrance Examination in Hebei Province) A factory has two production lines, A and B, which have been put into production successively. Before production line B was put into production, production line A had produced 200 tons of finished products. Since production line B was put into production, production lines A and B have produced 20 tons and 30 tons of finished products every day. (1) Calculate the functional relationship between the total output y (tons) of the two production lines A and B and the time x (days) since B started production, and calculate that the total output of the two production lines A and B is the same at the end of the previous days; (2) In the rectangular coordinate system as shown in the figure, make the image of the above two functions in the first quadrant; Observe the image and point out which production line has high total output at the end of day 15 and day 25 respectively. Analysis: (1) According to the given conditions, first list the functions of y and x, =20x+200, = =30x, when =, find x ... (2) Draw the images of the two functions in the given rectangular coordinate system. According to the coordinates of the point, we can see the total output of the two production lines A and B at the end of day 15 and day 25. Solution: (1) According to the meaning of the question, the functional relationship between production line A and production line B is: y=20x+200, and the functional relationship between production line B is: y=30x, so 20x+200=30x, and the solution is x=20, that is, at the end of the 20th day, the output of the two production lines is the same. (2) According to (1), the production function image corresponding to production line A must pass through two points: A (0 0,200) and B (20 20,600); The production function image corresponding to production line B must pass through two points: O (0 0,0) and B (20 20,600). Therefore, the image is displayed on the right. As can be seen from the image, at the end of 15, the total output of production line A was higher; By the end of the 25th day, the total output of production line B was very high. Example 5. The straight line y=kx+b is parallel to the straight line y=5-4x, intersects with the straight line y=-3(x-6), and the intersection point is on the y axis. Find the analytical expression of this straight line. Analysis: the position of the straight line y=kx+b is determined by the coefficients k and b: the direction is determined by k, and the intersection with the y axis is determined by b. If the two straight lines are parallel, the linear term coefficient k of the analytical formula is equal. For example, images with y=2x and y=2x+3 are parallel. Solution: ∫y = kx+b and y=5-4x are parallel, ∴ k=-4, ∫y = kx+b and y=-3(x-6)=-3x+ 18 intersect on the y axis, ∴ b = Note: the position of the linear function y=kx+b image is determined by the coefficients k and b: the direction is determined by k, and the fixed point is determined by b, that is, the function image is parallel to the straight line y=kx and passes through the (0, b) point, and vice versa, that is, the direction of the function image is determined by k, and the intersection point with the y axis is determined by b ... Example 6. The straight line intersects with the X axis at point A (-4,0) and with the Y axis at point B. If the distance from point B to the X axis is 2, find the analytical formula of the straight line. Solution: The distance from point B to X axis is 2, and the coordinate of point B is (0,2). Let the analytical formula of the straight line be y = kx 2, and the ÷ straight line passes through point A (-4,0) and ∴ 0 =-4k 2, and the solution is: k = Note: This example looks simple, but it actually implies a lot of reasoning processes, which is necessary to find a resolution function. (1) The function that the image is a straight line is a linear function; (2) If the straight line intersects the Y axis at point B, then point B (0, Yb); (3) If the distance from point B to the X axis is 2, then | Yb | = 2；; (4) The ordinate of point B is equal to the constant term of the linear analytical formula, that is, b = yB(5) Given the ordinate Yb of the intersection of a straight line and the Y axis, we can set Y = KX+Yb; We just need to make sure. Third, improve and think about the example 1. Knowing that the vertical coordinate of the intersection of the image of the linear function y 1=(n-2)x+n and the Y axis is-1, judge what function y2=(3- )xn+2 is, write the analytical expressions of the two functions, and point out the positions and increases and decreases of the two functions in the rectangular coordinate system. Solution: According to the meaning of the question, it is found that n=- 1, ∴ y 1 =-3x- 1, Y2 = (3-) x, and Y2 is a proportional function; The image of y 1=-3x- 1 passes through the second, third and fourth quadrants, and y 1 decreases with the increase of x; The image of y2=(3- )x passes through the first and third quadrants, and y2 increases with the increase of X. Note: Because the analytical formula of the linear function contains undetermined coefficient n, the key to solving the analytical formula is to construct the equation about n. In this topic, the equation is constructed by using "the constant term of the linear resolution function is the ordinate of the intersection of the image and the Y axis". Example 2. Given the image of a linear function, the X axis is at a (-6,0), the image of a proportional function is at point B, and point B is in the third quadrant. Its abscissa is -2, and the area of delta △AOB is 6 square units. Find the analytic expressions of proportional function and linear function. Analysis: The sketch is as follows: Solution: Let the proportional function y=kx, the linear function y=ax+b, the point b is in the third quadrant, the abscissa is -2, and let B(-2, yB), where Yb.

Satisfied, please adopt.