1, main concept: variable is; The independent variable is; The dependent variable is.

2. Three manifestations of the relationship between variables.

Its characteristics are: list: for each value of the independent variable in the table, the value can be found directly without calculation, which is convenient for query; But it can't reflect the whole picture of change, and it's not easy to see the corresponding law between variables.

Relationship: concise, standardized and accurate; However, the relationship between some variables is difficult or cannot be expressed by relational expressions. Image: The image is intuitive. It can vividly reflect the changing process, changing trend and some characteristics of things; But the image is approximate and partial, so it is inaccurate to determine the value of the dependent variable from the image.

3. Main mathematical thinking methods: analogy and comparison (illustrate with examples); The idea of combining numbers with shapes and mathematical modeling (for example).

Second, learn to navigate.

1, application of related concepts

Which of the following questions are changing? What are independent variables and dependent variables?

① A rectangular field with a length of L(m) and an area of S(m2) is enclosed by a fence with a total length of 60;

② The side length of a square is 3. If the side length increases x, the area increases y.

2, using the table to find the law of change

The study of Example 2 shows that the fixed application amount of potassium fertilizer and phosphorus fertilizer, the yield of potato and the application amount of nitrogen fertilizer have the following relations:

fertilizing amount

(kg/ha) 0 346710135 202 259 336 404 471

Potato yield

(ton/hectare)15.1821.36 25.72 32.29 30.03 39.45 3438+05 43.46 40.83 30.75.

Which two variables are reflected in the above table? Which is the independent variable? Which is the dependent variable? According to the data in the table, what do you think is the appropriate amount of nitrogen fertilizer?

Variant (Hunan) After 10 seconds, the speed of the car from standstill to start on the expressway is measured as follows:

Time/sec 0 1 2 3 4 5 6 7 8 9 10

Speed/m/s 00.31.32.8 4.9 7.61.01.61.8.4 24.2 28.9

① Which two variables are reflected in the above table? Which is the dependent variable?

(2) If T represents time and V represents speed, what is the change trend of V with T?

③ When t increases 1 sec, will the change of V be the same? At 1 sec, v increases?

(4) If the upper speed limit on the expressway is 120km/h, how many seconds does it take to reach this upper speed limit?

3. Use relational expressions to express the relationship between two variables.

Example 3. ① Set a rectangular box with a height of 10 and a square bottom area, and find the relationship between the volume v of this rectangle and the side length a of the bottom. (2) Let the ground temperature be 20℃, and the temperature will drop by 6℃ every time it rises 1km, and find the relationship between the temperature and the height h of t..

Variant (Jiangxi) as shown in the figure, a rectangular sliding window with a window height of 1.5m, the relationship between the ventilation area a (m2) of the movable sash and the opening length b (m) is as follows:

4. Use images to represent the relationship between two variables.

Example 4. (Guilin) This year, an outbreak of atypical pneumonia occurred in the mainland of China. Under the correct leadership of the party and the government, the epidemic situation has been effectively controlled. The following figure shows the data trend of newly confirmed cases in the Mainland from May 1 to May 14 this year (data source: Daily Epidemic Bulletin of the Ministry of Health). From the picture, we can know that:

(1) On May 6th, the number of newly diagnosed people was;

(2) During the three days from May 9th to May 1 1 day, the number of newly confirmed cases was * * *;

(3) As can be seen from the figure, the newly confirmed cases showed a general trend in the first half of May.

Example 5. Xiaohong came out for a walk from home after dinner on Sunday. The following figure describes the functional relationship between the distance from home s (meters) and the walking time t (minutes). According to the image, the following description conforms to Xiaohong's walking scene ().

A. Starting from home, I went to a public newspaper column, read the newspaper for a while, and then went home.

B. From home, I went to a public newspaper column. After reading the newspaper for a while,

Walk for a while, then go home.

C. Start from home, go straight (without stopping) and then go home.

D. I started from home and walked for a while to find my classmates. I didn't start to return until 18 minutes later.

Variant (Chengdu) The picture on the right shows A riding an electric bicycle and B driving a car along the same route for 45 km. From A to B, the relationship between distance y (km) and elapsed time X (hours), please fill in the blanks according to the image during driving: the car meets the electric bicycle when it leaves; The speed of electric bicycle is km/h; The speed is km/h; The car arrived at B a few hours earlier than the electric bicycle.

Third, try your hand.

1, (Guiyang) Xiaoming wrote a poem according to the story of his neighbor: "When my son returned from school, my father arrived at the station early. After the son arrived, he studied hard and the father and son returned to China happily. " If the distance between father and son on the journey is represented by the vertical axis Y, and the time when father left home is represented by the horizontal axis, then the meaning of the following image is roughly the same as that of the above poem ().

2. In a candle burning experiment, two candles, A and B, remain after burning.

Part height y (cm) and combustion time x (hours)

The relationship between them is shown in the figure.

Please answer the following questions according to the information provided in the picture:

(1) The heights of the two candles before burning are,

The time from ignition to burnout is:

(2) How long does it take to burn, and the height of two candles A and B is equal (regardless of the situation of all burning out)? In which period, candle A was taller than candle B? In what period, candle A was lower than candle B?

(Suqian Curriculum Reform in 2006) Xiaoming goes to school by bike from home. He went up the mountain first, and then went down to school. The time and distance used are shown in the figure. If the speed of uphill and downhill is the same when he returns, then the time for him to get home from school is ().

8.6 minutes

C. 12 minutes

4. A motor vehicle has 42l oil in the fuel tank before it leaves, and a few liters of oil is added at the gas station after driving for several hours. The relationship between the remaining fuel quantity Q(L) in the fuel tank and the driving time t(L) is shown in fig. 8.

Answer the question: (1) How many hours after the motor vehicle runs?

(2) refueling on the _ _ _ _ _ _ _ _ _ L；;

(3) It is known that the gas station is far from the destination and the vehicle speed is,

Is there enough oil in the tank to reach the destination? And explain why.

In an experiment, Xiao Ming fixed the upper end of a spring and hung an object at its lower end. The following is a set of corresponding values of the measured spring length y and the mass x of the suspended object.

Hanging mass

0 1 2 3 4 5

Spring length

18 20 22 24 26 28

(1) Which two variables are reflected in the table above? Which is the independent variable? Which is the dependent variable?

(2) How long is the spring when the weight of the suspended object is 0? What about when you don't hang heavy objects?

(3) If the hanging weight is too long (within the allowable range), can you tell the length of the spring at this time?

6. Xiaoming bought several kilograms of melons from the wholesale market in the summer social distance activity, and sold them at the price of one kilogram of 0.8 yuan in the market. After selling 40 kilograms of watermelons, the rest were sold out at the reduced price of 0.4 yuan per kilogram. The relationship between the sales volume and the kilograms of watermelons sold is shown in Figure 9. Please complete the following questions according to the information provided in the picture:

(1) Find the relationship between the sales amount Y (yuan) before the price reduction and the sold watermelon (kg);

(2) How many Jin of watermelon did Xiao Ming buy from the wholesale market?

(3) How much did Xiao Ming earn by selling melons this time?

7. The dotted line ABC in the figure is a visual representation of the relationship between the telephone bill Y (yuan) that Party A needs to pay for making a long-distance call to Party B and the call time T (minutes).

(1) 1 min. How much do I have to pay? How much is the phone bill for a five-minute call?

(2) How many minutes does the phone bill remain unchanged?

(3) If the call lasts more than 3 minutes, the relationship between the phone bill y (yuan) and the time t (minutes) is, what is the phone bill for 4 minutes?

8. As shown in the figure, it shows the relationship between the reservoir capacity V (10/0,000m3) and the drought duration t (days). Answer the following questions:

(1) What was the original capacity of this reservoir? What is the storage capacity of the reservoir after the drought lasts 10 days?

(2) If the reservoir capacity is less than 4 million m3, a severe drought alarm will occur. How many days after the continuous drought, will there be a severe drought warning?

(3) According to this rule, how many days will the reservoir dry up after continuous drought?

9. (Chengdu) A mobile communication company has set up two kinds of "GSM" communication services: one is to pay the 50 yuan fee every month, and then pay the 0.4 yuan call fee every 1 minute; "M-Zone": no monthly fee. Pay 0.6 yuan per call 1 minute (all calls in this topic refer to local calls). If you talk for x minutes a month, the fees of the two methods are RMB and RMB respectively.

(1) Write the relationship with X;

(2) How many minutes a month is the call, and the cost of the two mobile communications is the same?

(3) It is estimated that people talk for 300 minutes a month, which mobile communication should be more economical?