What is the topic of the function test?

linear function

First, knowledge points:

1. The meaning of linear function (the meaning of proportional function); 2. Linear function image;

3. Properties of linear functions; 4. Application of linear function: undetermined coefficient method, the positional relationship between two straight lines.

Second, the requirements of the curriculum standards for senior high school entrance examination

Test center/site

Curriculum standard requirements

Knowledge and skills

Understand; Understanding

grasp

Flexible application

one

time

believe

count

Understand the concept of linear function (including proportional function)

∨

The image of the function (including the proportional function) will be drawn once.

∨

Understand the properties of linear functions and apply them.

∨

Linear functions can be listed according to practical problems, and the analytical formula of linear functions can be determined by undetermined coefficient method.

∨

Approximate solution of binary linear equations by using images of linear functions

∨

Third, sort out the knowledge of the senior high school entrance examination

1. Relationship between proportional function and linear function

When Y = KX+B = 0, the proportional function is a special linear function.

2. Determine the analytical expressions of proportional function and linear function by undetermined coefficient method.

Usually, when one point is known, the analytic expression of the proportional function can be determined by the undetermined coefficient method, and the resolution function can be determined once when two points are known.

3. Linear function image

The proportional function y=kx(k≠0) is a straight line passing through (0,0) and (1, k); The linear function y=kx+b(k≠0) is a straight line passing through (0, b) and (0).

4. the relationship between the position of the straight line y=kx+b(k≠0) and the symbols of k and b.

When k>0 is a straight line passing through the first and third quadrants y=kx+b, when k

5. The positional relationship between the straight line L 1 and L2 is determined by K and B.

When the straight line l 1∑L2, k is the same, but b is different; When the straight line L 1 coincides with L2, k and b are the same; When the straight line L 1 intersects L2 at the same point on the Y axis, k is different from B. 。

6. Linear functions are often associated with linear equations and linear inequalities.

Fourth, the analysis of examples of senior high school entrance examination questions

1. Linear function image

Example 1 (Fuzhou, 2003) if the straight line y=ax+b passes through the first, second and third quadrants, then ab____0 _ 0 _ 0 (fill in ">", "<," = ").

Analysis: When the straight line y=ax+b passes through the first, second and third quadrants, you can draw a sketch first, from which you can see a>0, b>0 or press the straight line Y = KX+B when K >; 0 straight through the first and third quadrants, b >；; When 0, the Y axis passes through the positive half axis to judge.

Answer: You can draw a sketch according to the meaning of the question, a>0, B>0, ∴ AB > 0, so the answer is >.

Answer: >.

Comments: The key to solve this problem is to find out the relationship between the symbols of K and B in the linear function y=kx+b and learn to use the mathematical thinking method of combining numbers and shapes.

Example 2 (Qingzhou, 2003) In the figure below, the image representing the linear function y=mx+n and the proportional function y=mnx(m and n are constants and mn≠0) is ().

Analysis: For the problem that two images with different functions exist in the same coordinate system, it is often assumed that one image is correct, and then the other image is judged according to the actual meaning expressed by the letter coefficient to solve the problem. For example, if the straight line y=mx+n in option B is correct, then M

Answer: a.

2. Properties of linear functions

Example 3 (Gansu, 2003) The image of a linear function passes (1, 2), and if y increases with the increase of x, the resolution function is _ _ _ _ _.

Analysis: Starting from the point (1, 2) of the linear function image, we can first set the analytical formula to y=kx+b (or y=kx) and substitute the point (1, 2) into its analytical formula. But the function y increases with the increase of the independent variable x, and this condition cannot be lost.

Solution: (1) let the primary resolution function be y=kx+b (or y=kx)(k≠0)y increases with the increase of independent variable x, then k >；; 0, substitute (1, 2) into y=kx+b to get 2=k+b, that is, k = 2-B. 。

Let's take k= 1 and get b= 1.

∴ The analytical formula is y = x+1;

If k=2 and b = 0, the analytical formula is y = 2x.

Take k=3 and get b =- 1. The analytical formula is y=3x- 1.

…

There are countless analytical formulas that meet the conditions, so the answers are: y=x+ 1 or y=2x or y=3x- 1 and so on.

Comments: This question is to determine the openness of analytical formula. The key to solve this kind of problem is to think about the properties of functions under known conditions.

3. The application of linear function

Example 4 (Harbin, 2003) gives the images of a ship and a speedboat from Port A to Port B along the same route, which are the images of proportional function and linear function respectively. According to the picture, answer the following questions:

(1) Please find out the resolution function representing the running process of the ship and the speedboat respectively (the range of independent variables is not required;

(2) What is the speed of ships and speedboats on the way (excluding the starting point and the end point)?

(3) How long does it take for the speedboat to catch up with the ship?

Analysis: According to the known conditions, we can set two straight lines as y=k 1x(k 1≠0) or y=k2x+b(k2≠0), and then determine (1) according to the coordinates of the points given in the figure. (2) As can be seen from the figure, the speed of the ship is 8h 160km, and the speed of the speedboat is 4h 160km. (3) It can be solved according to "fastest distance-distance = slow distance" in the catching-up problem.

Solution: (1) Let the resolution function representing the ship operation process be y=kx. We know from the figure that when x=8, y= 160.

∴8k= 160, the answer is k=20.

∴ The resolution function representing the ship's running process is y=20x.

Let the resolution function representing the speed of speedboat be y = ax+B.

According to the image, when x is 2, y is 0; When x=6, y= 160.

Get a solution

The resolution function of speedboat is y=40x-80.

(2) According to the image, the ship travels 8h 160km and the speedboat travels 4h 160km, so the speed of the ship on the way is 20 (km/h) and the speed of the speedboat on the way is 40 (km/h).

(3) Set a boat and let Xia Houyuan's speedboat catch up with it.

20x=40x-80，x=4，∴x-2=4-2=2.

A: The speedboat leaves for two hours to catch the boat.

Comments: This question mainly solves the practical problem through the meaning of the intersection point between the function image and the coordinate axis, so it is the key to find out the meaning of the intersection point and then solve the resolution function by using the undetermined coefficient method.

Basic standard acceptance quantity

First, multiple choice questions

1. (Hangzhou, 2003) Images with linear function y=x- 1 fail ().

A. the first quadrant B. the second quadrant; The third quadrant and the fourth quadrant

2. (Fuzhou, 2004) It is known that the image of the proportional function y=kx(k≠0) passes through the second quadrant and the fourth quadrant, then ()

A.y decreases with the increase of x; B.y increases with the increase of x.

C. when x

D. no matter how x changes, y remains the same.

3. (Harbin, 2003) If the image of the proportional function y=( 1-2m)x passes through point A (x 1, y 1) and point B (x2, y2), when x 1

Morning & lt0b.m > 0 cubic centimeter <

4. (Gansu, 2003) Combined with the image of the proportional function y=4x, the answer is: when x >；; When 1, the value range of y is ()

a . y = 1 b . 1≤y & lt； 4c . y = 4d . y & gt； four

5. (Harbin, 2004) The straight line y=x- 1 intersects the coordinate axis at points A and B, point C is on the coordinate axis, and △ABC is an isosceles triangle, so the most eligible point C is ().

A.4 B.5 C.7 D.8

6. (Qinghai, 2003) When the tractor started to work, there was 40L oil in the fuel tank. If the fuel consumption is 5L/ hour, the functional relationship between the remaining fuel quantity Q(L) in the fuel tank and the working time t(h) can be expressed as ().

Second, fill in the blanks

1. (Guangzhou, 2003) If the image of the proportional function passes through the point (2, 1), then the resolution function is _ _ _ _ _ _ _ _.

2. (Weifang, 2002) If the image of a linear function passes through the first, third and fourth quadrants, the analytical formula of the linear function is _ _ _ _ _ (only one is filled in).

3. (Sichuan, 2004) In the plane rectangular coordinate system, the straight line y=kx+b(k and b are constants, k≠0, b>0) can be considered as the straight line y=kx moves b units in parallel along the y axis, then the straight line y=kx moves m units in parallel to the right along the x axis (M >；); 0) The equation of the straight line obtained is _ _ _ _ _.

4. (Tianjin, 2004) It is known that the side length of square ABCD is 1, e is the point on the side of CD, and p is the moving point on the side of square ABCD. The moving point P starts from point A and moves along A→B→C→E to reach point E. If the distance traveled by point P is an independent variable X and the area of △ Ape is a function Y, then when y=

Third, answer questions.

1. (Zhenjiang, 2002) It is known that y is proportional to x+2, and when x= 1, y=-6. (1) Find the functional relationship between y and x; (2) If point (a, 2) is on the function image, find the value of a. 。

2. (Jilin, 2004) As shown in the figure, when the thumb and little finger are opened as far as possible, the distance between the two fingertips is called the finger distance. A study shows that, generally speaking, people's height h is a linear function of distance d. The following table is a set of measured finger distance and height data:

(1) Find the functional relationship between h and d (the range of independent variable d is not required).

(2) The height of a person is 196cm. What is the distance between his fingers?

3.(2003, Shaanxi) For the sake of students' health, the height of school desks and stools is scientifically designed according to a certain relationship. Xiao Ming observed and studied a batch of desks and stools bought by the school and found that they can adjust their height according to people's height. So he measured the four heights corresponding to a set of tables and chairs and got the following data:

first gear

second gear

third gear

Fourth gear

Step height x (cm)

37.0

40.0

42.0

45.0

Table height y (cm)

70.0

74.8

78.0

82.8

(1) After exploring the data, Xiao Ming found that the table height y is a linear function of the stool height x, please find out the relationship of this linear function (it is not required to write the value range of x);

(2) After Xiao Ming came home, he measured the desk and stool at home. The height of the desk is 77cm and the height of the stool is 43.5cm. Please judge whether they match and explain the reasons.

(2003. Liaoning) A museum attracts a large number of Chinese and foreign tourists every week. If there are too many tourists, it will have a negative impact on the precious cultural relics in the museum. However, considering the cost of cultural relics restoration and preservation, it is necessary to ensure a certain ticket income. Therefore, museums control the number of visitors by raising and lowering the ticket prices. During the implementation of this method, it is found that there is a linear functional relationship between the number of weekly visitors and the ticket price, as shown in figure 1- 13-9. In this case, if the ticket income of 40 thousand yuan per week is guaranteed, what should be the number of visitors per week? What should the fare be?

Ability improvement exercise

First, interdisciplinary application.

1.(2003. Enshi Autonomous Prefecture) In a certain circuit, if the voltage remains unchanged, the image of the functional relationship between current intensity I and resistance R is roughly ().

2. (Hangzhou, 2003) The brownish red flue gas produced by converter steelmaking will pollute the atmosphere. A device can reduce pollution by recovering iron oxide from brown-red smoke. The iron oxide recovery rate of this device is related to the current it passes through. The following data are obtained through experiments:

Current intensity through (unit a)

1.7

1.9

2. 1

2.4

Iron oxide recovery rate (%)

Seventy-eight

Establish a rectangular coordinate system as shown in the figure, with the abscissa indicating the current intensity and the ordinate indicating the iron oxide recovery rate.

(1) The data obtained from the test are represented by points in the rectangular coordinate system, as shown in the figure; (Note: The intersection of coordinate axes in this drawing represents the point (1, 70).

(2) Connect the points (1) drawn in the question with line segments from left to right. If this image is used to simulate the functional relationship between iron oxide recovery y and current x, try to write the expression of this function when 1.7≤x≤2.4;

(3) Using the function relation obtained in question (2), when the iron oxide recovery rate is greater than 85%, the current range that the device should control (accurate to 0. 1A).

Second, the practical application problems

3. (Fuzhou, 2004) As shown in the figure, L 1 and L2 respectively represent the function images of the cost y (cost = lamp price+electricity fee, unit: yuan) and the lighting time x(h) of incandescent lamps and energy-saving lamps. Assuming that the service life of both lamps is 2 000h, the lighting effect is the same.

(1) According to the image, the functional relationships of L 1 and L2 are obtained respectively;

(2) When the lighting time is what, the cost of two lamps is equal?

(3) Liang Xiao's room is planned to be illuminated for 2 500 h. He bought an incandescent lamp and an energy-saving lamp. Please help him design the most economical way to use the lamp (give the answer directly without writing the solution process).

4. (Shenyang, 2004) A county and B county in a city are in urgent need of 90 tons and 60 tons of chemical fertilizer respectively, and C county and D county in this city store 0/00 tons and 50 tons of chemical fertilizer respectively, all of which are distributed to A county and B county. The freight (yuan/ton) of known chemical fertilizer from county C and county D to county A and county B is shown in the following table.

point of departure

transportation charge

destination

(1) Let the chemical fertilizer transported from county C to county A be x tons, find the functional relationship between the total cost W (yuan) and X (ton), and write the range of independent variable X;

(2) Find the lowest total freight, and explain the delivery plan when the total freight is the lowest.

Fourth, open to explore questions.

5. (Jilin, 2003) As shown in Figure (1), in right-angle ABCD, AB= 10cm, BC=8cm, point P starts from A, moves along the route A→B→C→D, and ends at D; Point Q starts from point D, moves along the route of D→C→B→A, and ends at point A. If point P and point Q start at the same time, the speed of point P is 1cm/s, the speed of point Q is 2 cm/s, the speeds of point P and point Q change with as at the same time, the speed of point P becomes bcm/s, and the speed of point Q becomes dcm/s (.

(1) Refer to Figure (2) and find the values of A, B and C in Figure (2);

(2) find the value of d;

(3) Assume that the distance from point P to point A is y 1(cm), and the distance from point Q to point A is y2(cm). Please write the function expressions of y 1 and y2 and the movement time x(s) of moving points P and Q after speed change respectively, and find the value of x when P and Q meet;

(4) When point Q leaves _ _ _ _ s, the distance between point P and point Q on the movement route is 25cm.

Fourth, the issue of innovation.

6.(200 1 Hebei) Two cars, A and B, are driving at a constant speed on an expressway. In order to determine the position of the car, we use the number axis Ox to represent this road, and the origin O is a zero-kilometer road sign (as shown in the figure), and make the following agreement:

① Speed v>0 means that the car is driving in the positive direction of several axes; Speed c < 0 means that the car is driving in the negative direction of several axes; Speed v=0 means that the car is stationary.

② The coordinate s>0 of the car position on the number axis indicates that the car is located on the right side of the zero kilometer road sign; The coordinate position of the car on the number axis s

According to the above agreement, two cars driving at a constant speed on the expressway are drawn in the same rectangular coordinate system in the form of a function image, as shown in the figure.

Please answer the following questions:

(1) Fill in the following table about the driving situation of two cars on this expressway reflected by these two linear function images.

Direction of travel

Speed (km) h

Position before departure

Jiache

B car

(2) Can A and B meet? If you can meet, find the time of meeting and your position on the highway; If you can't meet, please explain why.

Answer:

Basic standard acceptance quantity

I. 1. B 2。 A 3。 D 4。 D 5。 C 6。 C

Second,1.y = x 2. y = x-1(just make k >；; 0，b & lt0) 3.y=k(x-m) 4。

3. 1.( 1)y =-2x-4； (2)a=-3。

2. Solution: (1) Let h=kd+b(k≠0), depending on the meaning of the problem.

solve

The functional relationship between h and d is h=9d-20.

(2) when h= 196, 9d-20= 196,

∴d=24cm.

The finger distance of a person with height 196cm is 24cm.

3.( 1)y = 1.6x+ 10.8； (2) Do not match.

4. Solution: Let the linear functional relationship between the number of visitors per week and the ticket price be y = KX+b. 。

Get a solution from the meaning of the problem

∴y=-500x+ 12 000。

According to the meaning of the question, xy=40 000,

That is x(-500x+ 12 000)=40 000,

X2-24x+80=0, then the solution is x 1=20, x2=4.

Substitute X 1 = 20 and X2 = 4 into y=-500x+ 12 000 respectively.

Y 1=2 000，y2= 10 000。

Because the number of tourists is controlled, x=20, y=2 000.

A: The number of visitors per week should be controlled within 2,000, and the ticket price is 20 yuan.

Ability improvement exercise

1.D

2. Solution: (1) As shown in Figure (2) Connect as shown in Figure.

(3) when 1.7≤x≤ 1.9, it is changed from 45x+2.5 >: 85, 1.8.

When 2. 1≤x≤2.4, from -30x+ 150 >: 85, we get 2.1≤ x < 2.2;

And when 1.9 ≤ x

It can be seen from the above that the current of the device should be controlled between 1.8A ~ 2.2A when the requirements are met.

3.( 1) Let the analytical formula of the straight line L 1 be y 1=k 1x+2, and get k 1=0.03 from the image.

∴y 1=0.03x+2(0≤x≤2 000)。

Let the analytical formula of the straight line L2 be y2=k2x+20,

From the image of 26=500k2+20, we can get k2=0.0 12.

y=0.0 12x+20(0≤x≤2 000)。

(2) When y 1=y2, the cost of the two lamps is equal.

0.03x+2=0.0 12x+20, the solution is x = 1 0,000.

When the lighting time is 1 000 hours, the cost of the two lamps is equal.

(3) 2 000 hours for energy-saving lamps and 500 hours for incandescent lamps.

4.( 1) The transportation of chemical fertilizer from county C to county A is x tons, while that from county C to county B is (100-x) tons, that from county D to county A is (90-x) tons, and that from county D to county B is (x-40) tons.

w = 35x+40(90-x)+30( 100-x)+45(x-40)= 10x+4800，40 ≤ x ≤ 90。

(2)∵W decreases with the decrease of x,

When x=40, the minimum w = 10×40+4 800=5 200 (yuan).

When the freight rate is the lowest, x=40, so 100-X = 60, 90-X = 50, and X-40 = 0.

Transportation scheme: 40 tons of chemical fertilizer from county C 100 to county A, 60 tons to county B, and all 50 tons of chemical fertilizer from county D to county A. 。

5. Solution: (1) Observation chart (2)

S△APD=PA PD=× 1×a×8=24，

∴a=6(s),b==2(cm/s)，

c=8+= 17(s)。

(2) According to the meaning of the question (22-6)d=28- 12,

The solution is d= 1 (cm/s).

(3)y 1=2x-6，y2=22-x。

2x-6=22-x according to the meaning of the question.

∴x=。

(4) 1, 19.

6. Solution: (1) A car: negative direction of X axis (left), 40, right side of zero kilometer road sign 190 kilometers;

B train: positive direction of X axis (to the right), 80km to the left of 50000km road sign.