1. Review content: constants and variables; The concept of function; Determination of the range of independent variables; Function value; Function image and drawing method; Application of function image; Three representations of functions; Positive proportional function image and its properties: linear function image and its properties; Determination of resolution function; Application of linear function; Look at equations, equations and inequalities from the functional point of view.
2. Review key points: the concept of function; Application of function image; Determination of the range of independent variables; Linear function image and its properties; Determination of resolution function; Application of linear function.
3. Review difficulties: comprehensive application of linear function; Look at equations, equations and inequalities from the functional point of view.
Four. On determining the type of resolution function.
① Define the type.
Example 1. Assuming that the function is a linear function, find its analytical expression.
② Point oblique type
Example 2. Given the image intersection point (2,-1) of a linear function, find the analytical expression of this function.
Variant method: when the linear function y =- 1 is known, find the analytical formula of this function.
③ Two-point formula
Example 3. Given that the coordinates of the intersection of the image of a linear function with the X axis and the Y axis are (-2,0) and (0,4) respectively, the analytical formula of the function is _ _ _ _ _ _ _ _.
④ Image type
Example 4. Given the image of a linear function as shown in the figure, the analytical formula of the function is _ _ _ _ _.
⑤ Diagonal 4-question map
Example 5. It is known that the straight line is parallel to the straight line, and the distance from its intersection with the Y axis to the origin is 2.
Then the analytical formula of this straight line is _ _ _ _ _.
⑥ Translation type
Example 6. The analytical formula of the image obtained by translating the straight line down by 2 units is _ _ _ _ _ _ _ _.
⑦ Practical application type
Example 7. There is 20 liters of oil in a fuel tank, and the oil flows out of the pipeline at a uniform speed with a flow rate of 0.2 liters/minute. Then the functional relationship between the remaining oil in the tank (liter) and the outflow time t (minute) is _ _ _ _ _ _ _ _.
8 regional types
Example 8. Given that the area of a triangle surrounded by a straight line and two coordinate axes is equal to 4, the analytical formula of the straight line is _ _ _ _ _ _ _.
Pet-name ruby symmetric type
If a straight line is related to a straight line
(1)x is axisymmetric, so the analytical formula of the straight line L is _ _ _ _ _ _ _ ().
(2) The Y axis is symmetrical, then the analytical formula of the straight line L is _ _ _ _ _ _ _ _ _ ()
(3) If the straight line Y = X is symmetrical, the analytical formula of the straight line L is _ _ _ _ _ _ _ ().
(4) If the straight line is symmetrical, the analytical formula of the straight line L is _ _ _ _ _ _ _ _ _ ()
(5) If the origin is symmetrical, the analytical formula of the straight line L is _ _ _ _ _ _ _ _ ()
Example 9. If the straight line L and the straight line are symmetrical about y, the analytical formula of the straight line L is _ _ _ _ _.
Attending the open.
Example 10. Given that the image of the function passes through point A (1, 4), please write the resolution function that meets the conditions.
Example11(unified examination in 2009). If a function has the following two properties:
(1) Its image is a straight line passing through the origin; (2) Value increases with the increase of value.
Please write the analytical expression of the function that satisfies the above two conditions.
Several problems needing attention in verb (abbreviation of verb);
1. Pay attention to the background of the actual problem and find out the relationship between the related variables in the problem.
2. Solve practical problems with function analysis, and find the relationship between variables with the help of function images, tables and formulas.
3. For piecewise functions, special attention should be paid to the range of the corresponding independent variables.
4. Pay attention to the idea of combining numbers and shapes, pay attention to the internal relationship between knowledge, and unify binary linear equations, linear inequalities and linear equations with linear functions.
Consolidation exercises of intransitive verbs
First, the basic knowledge review
(a) variables and functions
1. Functional concept
Generally speaking, in a process, if there are two variables x and y, and for
So let's assume that x is the independent variable and y is.
2. Three representations of functions
(1) The method of expressing the functional relationship with mathematical expressions is called;
(2) The method of expressing the function relationship by listing the values of independent variables and corresponding function values is called;
(3) Generally speaking, for a function, if the independent variable and the function are taken as points in the plane rectangular coordinate system, these points are called the image of the function. This method of expressing functional relationships is called.
(2) Linear function
1. Concept of linear function: Generally speaking, functions with shapes are called linear functions.
Especially when y is said to be a function of X.
2. Images and properties of linear functions
The image of (1) proportional function is: the image of linear function is a straight line passing through point (0,0) and point (0,0). An image of a linear function is also called.
(2) For linear functions and their images:
linear function
( )
Graphical functions and attributes of images
The image passes through the fourth quadrant, and y increases with the increase of x;
The image passes through the fourth quadrant, and y increases with the increase of x;
The image passes through the first, second and fourth quadrants, and y increases with the increase of x;
The image passes through the first, third and fourth quadrants, and y increases with the increase of x;
The image passes through the fourth quadrant, and y increases with X.
The image passes through the fourth quadrant, and y increases with X.
(3) Translation relationship:
When a unit length is translated along a straight line, a straight line can be obtained;
When, a straight line can be obtained by translating a unit length in the direction of a straight line.
Straight time,,; When a straight line intersects the y axis at the same point.
3. Linear functions and linear equations (groups), linear inequalities
(1) Solving a linear equation can be transformed into finding the coordinates of the intersection of a straight line and the X axis (straight line).
(2) Solving binary linear equations can be transformed into finding the coordinates of straight lines and intersections.
(3) Solving inequality can be transformed into observing the range of values corresponding to the square part of a straight line; Or observe the value range corresponding to the upper part of the straight line.
Second, the classification of supplementary exercises.
(A) the concept of function
1. According to the program in the figure on the right, when the input value x is -2, the output value y is ().
A.4 B.6 C.8 D. 10
2. According to the program calculation as shown in the figure, if the value of x entered at the beginning is 48, we find that the first result is 24, and the second result is 12, ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
3. With the elevation, the atmospheric pressure decreases, so does the oxygen content in the air, that is, the oxygen content is directly proportional to the atmospheric pressure. When it is a functional relationship.
4. The waist length of an isosceles triangle with a circumference of 18 is x, and the base length is y. Find the functional relationship between y and x and the range of x. 。
5. Among the following functions: ①, ②, ③, ④ and ⑤, one of them is a linear function.
6(20 1 1). A match with a length of 4cm can be made into a parallelogram with a length of 4cm as shown in Figure 1 or a parallelogram with a length of 4cm as shown in Figure 2. Then, it is represented by an algebraic expression containing, and _ _ _ _ _ _ _ _.
(b) Find the range of independent variables of the function
7 (district unified examination in 2009). In the function, the value range of the independent variable x is.
8. In the function, the range of independent variables is.
9. The range of the independent variable x in the function is.
(C) the application of functional images
10. As shown in the figure, in the quadrilateral A-B-C-D, the moving point P starts from point A and proceeds at a constant speed along the path of A-B-C-D until D. In the process, the relationship between the area S of △APD and time t is correctly represented by images ().
1 1. The image (dotted line ABCDE) in the diagram describes the driving of the car in a straight line.
In this process, the distance between the car and the starting point is between S (kilometers) and the driving time T (hours).
According to the information provided in the figure, give the following statement:
(1) the car * * * driving120km; ② The car stays for 0.5 hours while driving;
③ The average speed of the car during the whole driving process is km/h;
(4) The speed gradually decreases from 3 hours to 4.5 hours after departure.
The correct statement * * * has ().
1。
12. A school organized members to hold publicity activities for the successful Olympic bid. Ride from school, go uphill to station A first, and then publicize for 8 minutes. Then go downhill to B for publicity and return for 8 minutes. The itinerary is as shown. If the speed of going uphill and downhill is the same when they return, and it takes 8 minutes to publicize in place A, then the time it takes them to return to school from place B is ().
45.2 minutes 48 minutes 46 minutes 33 minutes
13(20 1 1 district unified examination). Wang Peng and Li Ming started from school along the same road at the same time.
Go to the library to check the information. The distance from the school to the library is 4 kilometers. Peng Wang
Li Ming rides a bike and walks. Li Ming happened to be in when Wang Peng came back to school the same way.
Arrive at the library. The dotted line O-A-B-C and the line OD in the figure respectively represent the separation of two people.
The functional relationship between the distance s (km) of the school and the elapsed time t (min),
Please answer the following questions according to the pictures:
(1) Wang Peng spent _ _ _ _ minutes in the library consulting materials, and Wang Peng returned to school at the speed of _ _ _ _ _ _ _ km/minute;
(2) Find the functional relationship between the distance S (km) from Li Ming to school and the elapsed time T (min);
(3) When Wang Peng and Li Ming met head-on, how many kilometers were they from the school?
(4) Images and properties of linear functions
14. If the point m is on a straight line, the coordinate of the point m can be ().
A.(- 1，0) B.(0， 1) C.( 1，0) D.( 1，- 1)
15. In the linear function, the value of decreases with the increase of, so the value range of is ().
A.B. C. D。
16. In the plane rectangular coordinate system, a straight line passes through ().
A. First, second and third quadrants B. First, second and fourth quadrants C. First, third and fourth quadrants D. Second, third and fourth quadrants
17. If the image of the linear function passes through the first quadrant and intersects the negative semi-axis of the shaft, then ().
A.、b、c、d、
18(20 1 1 regional unified examination). When is, the image of the function does not pass ().
A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant
19. In the linear function, the function value y decreases with the increase of x, and its image does not pass through the first quadrant, so the value range of k is ().
A.B. C. D。
20(20 1 1 district unified examination). Both points A () and B () are on a straight line, so the relationship with is ().
A.B. C. D。
2 1(20 1 1 regional unified examination). The image of the known linear function is shown in the figure. When,
The value range of is ().
A.B. C. D。
22(20 1 1 regional unified examination). It is known that the straight line is parallel to the straight line and passes through the point (1, 1), then the straight line can be regarded as the translation of the straight line to _ _ _ _ _ _ _.
(5) Determine the resolution function according to the known conditions.
23. If the image of the proportional function passes through the point (,), then its analytical formula is _ _ _ _ _ _ _.
24. As shown in the figure, the image of linear function passes through this point and intersects the image of proportional function at this point.
The expression of linear function is ().
A.B. C. D。
25. As shown in the figure, translate the straight line upward by 1 unit to obtain an image of a linear function.
Then the analytical expression of this linear function is.
26. The analytical formula of the straight line obtained by translating the straight line y = 2x by 2 units to the right is ().
a . y = 2x+2 b . y = 2x-2 c . y = 2(x-2)d . y = 2(x+2)
27. Given that the area of the triangle surrounded by the image of the linear function and the X and Y axes is 8, find the analytical formula of the linear function.
28. It is known that the straight line intersects the X axis at point A, the Y axis at point B, and the straight line L intersects the line segment AB at point C. The area of △ABO is divided into two parts: 1∶2, and the analytical formula of the straight line L is obtained.
29 (district unified examination in 2009). As shown in the figure, cut out a square piece of paper with a side length of 14cm to get a "Japanese" pattern. It is known that the sum of the perimeters of the two cut rectangles is 40, and the width of each stroke in the Japanese pattern is not less than 2cm. Let the length and width of each small rectangle be 10 cm.
A.B. C. D。
(6) Look at equations (groups) and inequalities from the functional point of view.
30. The image of the linear function is shown in the figure. When is, the value range of is ().
A.B. C. D。
3 1. The image of the known function is shown in the figure. When is, the value range of is ().
A.B. C. D。
32. The image of a function is as shown in the figure, and the following conclusions are drawn: ② ; (3) The correct number for when, when and when is ().
a . 0b . 1c . 2d . 3
33. The images of straight lines and straight lines in the same plane rectangular coordinate system are shown in the figure, and the solution set of inequality is.
32 questions 33 questions 34 questions 35 questions
34. As shown in the figure, if a straight line passes through two points A (-2,-1) and B (-3,0), the solution set of the inequality group is.
35. As shown in the figure, get the image of function sum. The coordinates of the point symmetrical about the x axis are.
36 (district unified examination in 2009). As shown in the figure, the abscissa of the intersection of known straight lines is 1. According to the image, the following four conclusions are drawn: ①; ② ; ③ For any two points on a straight line, if, then
④ is the solution set of inequality. The correct conclusion is ().
A.①② B.①③ C.①④ D.③④
(7) Problems related to linear functions and geometry.
37. In the plane rectangular coordinate system, the acute angle between the straight line AB and the positive direction of the X axis is 60 degrees, and the A coordinate is (? 2,0), and point B is above the X axis. Let AB=a, then the abscissa of point B is ().
A.B. C. D。
38. The straight line y=x+ 1 intersects the coordinate axis at point A and point B, point C is on the coordinate axis, and △ABC is an isosceles triangle, so point C that meets the conditions has at most ().
A.4 B.5 C.7 D.8
39. As shown, points A, B and C are on the image of linear function, and their abscissas are as follows.
-1, 1, 2, respectively, indicate that when these points are perpendicular to the X axis and the Y axis, the sum of shadow areas in the figure is ().
A.B. C. D。
40. As shown in figure 1, in a right-angled trapezoid, a moving point starts from one point, moves along that point and ends at one point. Let the distance of point movement be, and the area be. If the correlation function image is shown in Figure 2, the area is ().
a . 3b . 4c . 5d . 6
4 1. As shown in the figure, the straight line AB: Y = X+ 1 intersects the X axis and the Y axis respectively.
Point A, point B and straight line CD: y = x+b intersect with X axis and Y axis respectively.
Points C and D. Lines AB and CD intersect at point P, which is called =4.
Then the coordinate of point P is ().
A.(3，)B.(8，5) C.(4，3) D .)
42. It is known that the plane rectangular coordinate line () intersects the straight line () at the point ().
(1) Find the analytical formula of a straight line ();
(2) If the straight line () intersects another straight line at point B, and the abscissa of point B is, find the area of △ABO.
(8) the application of linear function
43. As the picture shows, two piles of rice bowls with the same specifications are neatly stacked on the desktop. Please answer the following questions according to the data information given in the picture:
(1) Find an analytic function between the height y(cm) and the number x (rice bowls) of rice bowls arranged neatly on the table;
(2) What is the height of these two piles of rice bowls when they are neatly arranged in a pile?
44. Xiao Qiang participated in a social practice activity on Sunday. He bought several kilograms of strawberries from the fruit farm at the price of one kilogram of 3 yuan and sold them in the market. When he sold 65,438+00 kilograms, he earned 50 yuan. The remaining price per kilogram was reduced by 1 yuan, and all of them were sold out, earning 70 yuan twice. It is known that the sales income (yuan) before the price reduction is between the sales weight (kg).
(1) Find the functional relationship between the sales income (yuan) before price reduction and the weight (kg) of strawberries sold; And draw its functional diagram;
How many kilograms of strawberries are wholesale in Xiao Qiang? Xiao Qiang decided to donate all the money earned from selling strawberries to the Wenchuan earthquake-stricken area, so how much did Xiao Qiang donate?
45. China is one of the countries with severe water shortage in the world. In order to enhance residents' awareness of water conservation, a water supply company in a city adopts the method of charging by households. That is, users who lack water 10 ton (including 10 ton) in January will be charged RMB per ton; Users who consume more than 10 tons of water in January are still charged at RMB per ton, and users who consume more than 10 tons are charged at RMB per ton (). Suppose a household uses tons of water every month, and the water fee receivable is RMB. The functional relationship between them is shown in the figure.
( 1); A family used 8 tons of water last month. How much is the water charge receivable?
(2) Find the value of and write the functional relationship between and when;
(3) It is known that Resident A used 4 tons more water than Resident B last month, and 46 yuan charged the water fee. How many tons of water did they use last month?
46. An express train goes from A to B, and a local train goes from B to A. Both cars start at the same time. Let the local train drive, and the distance between the two cars is. The dotted line in the figure indicates the functional relationship between and.
According to the pictures, make the following inquiries:
Information reading: (1) The distance between A and B is km;
(2) Please explain the actual meaning of the dots in the picture;
Image understanding: (3) Find the speed of the local train and the express train;
(4) Find the relationship between the line segment and the function it represents, and write the range of independent variables;
Problem solved: (5) If the second express also leaves from A to B, the speed is the same as that of the first express. Thirty minutes after the first express train meets the local train, the second express train meets the local train. How many hours does the second express leave after the first one?
47. A vegetable processing factory undertakes the task of processing vegetables for export, and a batch of vegetable products need to be packed into cartons with certain specifications. There are two options for supplying this kind of cartons:
Option 1: Customized purchase from carton factory, with the price of 4 yuan per carton;
Scheme 2: The vegetable processing factory rents machines to process and manufacture this kind of cartons, and the machine rental fee is charged according to the number of cartons produced. The factory needs a one-time investment of 16000 yuan for machine installation, and the processing cost of each box is 2.4 yuan.
(1) If cartons of this specification are needed, please write down the functional relationship between the cost (yuan) of buying cartons from carton factory and the cost (yuan) of processing and manufacturing cartons in vegetable processing factory.
(2) Suppose you are the decision maker, which scheme do you think should be chosen? And explain why.
48. A company wants to print a batch of publicity materials for the Beijing Olympic Games. On the premise of paying 600 yuan plate-making fee and 0.3 yuan printing fee for each material, two printing plants, A and B, respectively put forward different preferential terms. The first printing factory proposed that the printing fee for more than 2000 copies could be charged at 10%, and the second printing factory proposed that the printing fee for more than 3000 copies could be charged at 20%.
(1) If the company wants to print 2400 copies, the cost of printing plant A is, and the cost of printing plant B is.
(2) According to the printing quantity, please discuss which printing factory in our unit can get more preferential treatment for printing materials.
49. A school plans to rent six buses to send a group of teachers and students to the annual Harbin Ice Sculpture Festival to feel the charm of ice sculpture art. There are two types of buses, A and B, and their passenger capacity and rent are shown in the following table. If you rent a bus, the total rental fee is RMB.
Class a bus, class b bus
Passenger volume (person/car) 45 30
Rent (RMB/vehicle) 280 200
(1) Find the functional relationship between (yuan) and (car), and point out the range of independent variables;
(2) If there are 240 teachers and students in the school, and the leader teacher advances the car rental fee of 1650 yuan to the school, can the prepaid car rental fee be balanced? If so, what is the maximum balance?
50. In earthquake relief, in order to ensure the safety of grain storage, a county grain bureau decided to transfer all the grain from warehouses A and B to warehouses A and B with strong earthquake resistance. It is known that the grain in warehouse A is 100 tons, the grain in warehouse B is 80 tons, the grain in warehouse A is 70 tons and the grain in warehouse B is 1 10 tons. "Kilometers" refers to the RMB required for transportation of grain per ton 1 kilometer).
(1) If warehouse A transports tons of grain to warehouse A, please write the functional relationship between the total freight (yuan) and (tons) for transporting grain to warehouse A and warehouse B;
(2) When warehouse A and warehouse B transport several tons of grain to warehouse A and warehouse B, what is the most economical total freight?
5 1. It is known that in the plane rectangular coordinate system xoy, points A (0 0,4), B, C are on the X axis (point B is on the left of point C), point C is on the right of the origin, and the vertical foot E (point E is on the line segment AC, and point E does not coincide with point A), and the straight line BE intersects with the Y axis.
(1) Find the coordinates of point B; (2) Let the length of OC be m and the area of △BOD be s, find the functional relationship between S and M, and write the range of the independent variable m. 。
52. As shown in the figure, in the equilateral triangle ABC, AB = 2, point P is the moving point on the side of AB (point P can coincide with point A, but not with point B), crossing point P is PE⊥BC, vertical foot is E, crossing point E is EF⊥AC, vertical foot is F, and crossing point F is FQ⊥AB.
(1) Write the functional relationship between y and x; (2) When the length of BP is equal to what, point P coincides with point Q;
53. In the plane rectangular coordinates, the two vertices of a square with a side length of 2 are on the axis and the positive semi-axis of the axis respectively, and the point is at the origin. Now the square rotates clockwise around the point and stops rotating when the point falls on a straight line for the first time. In the process of rotation, an edge intersects a straight line, and an edge intersects an axis at a point.
(1) Find the area swept by the edge when rotating;
(2) In the process of rotation, when parallel, find the square rotation.
Degree;
(3) Suppose that in the process of rotating a square,
Has the value changed? Please prove your conclusion.
-_-。 Excuse me! It's a bit messy to copy. You can give me your email address and I will send it to you.