"Linear function and proportional function" linear function PPT

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"Linear function and proportional function" linear function PPT

Part One: Learning Objectives

1. Master the concepts of linear functions and proportional functions. (Key points)

2. Be able to find the relational expression of a linear function based on conditions. (difficulty)

PPT on linear functions and proportional functions, part 2: importing new lessons

Observe and think

In ancient times, many peoples and regions used water clocks to keep time, as shown in the picture. People at that time judged the time by the flow of water released from the container. So do you know why water flow can be used to tell time?

Assuming that the amount of water leakage is uniform, the float in the kettle will rise uniformly, that is to say, the float will rise to a height h = kt (k is a constant)

PPT on linear functions and proportional functions, part 3: teaching new lessons

linear function and proportional function

In real life, many problems can be attributed to function problems. Can you give some examples?

Scenario 1: The natural length of a spring is 3 cm, within the elastic limit, so

For every 1 kilogram increase in the mass x of the hanging object, the length y of the spring increases by 0.5 cm.

(1) Calculate the length of the hanging object when its mass is 1 kg, 2 kg, 3 kg, 4 kg and 5 kg respectively, and fill in the following table:

(2) Can you write the relationship between y and x?

y=3+0.5x

Scenario 2: The original oil in the fuel tank of a certain car is 60 L, and the car consumes 6 L of fuel every 50 kilometers traveled.

(1) Complete the following table:

(2) Can you write the relationship between y and x?

y=60-0.12x

The above two functional relationships:

(1)y=3+0.5x

(2) y=60-0.12x

If the relationship between two variables x and y can be expressed in the form of y=kx+b (k, b are constants, k is not equal to 0), then y is said to be a linear function of x. (x is the independent variable, y is the dependent variable.)

When b=0, y is said to be a proportional function of x.

Method summary

1. Conditions for judging whether a function is a linear function:

The independent variable is a linear integer, and the coefficient of the linear term is not zero;

2. Conditions for judging that a function is a proportional function:

The independent variable is a linear integer, the coefficient of the linear term is not zero, and the constant term is zero.

Analysis of classic examples

Example 1: Write the relationship between y and x in each of the following questions, and judge: Is y a linear function of x? Is it a proportional function?

(1) The car is traveling at a constant speed of 60km/h, and the relationship between the driving distance is y (km) and the driving time x (h);

Solution: From distance = speed × time, we get y=60x. y is a linear function of x and a proportional function of x.

(2) The relationship between the area y (cm2) of a circle and its radius x (cm).

Solution: From the area formula of a circle, we get y=πx2,

y is not a directly proportional function of x, nor is it a linear function of x.

PPT on linear functions and proportional functions, part 4: practice in class

1. Judgment:

(1)y=2.2x, y is a linear function of x and a proportional function of x. ( )

(2)y=80x+100, y is a linear function of x. ( )

2. In the function y=(m-2)x+(m2-4), when m____, y is a linear function of x; when m____, y is a proportional function of x.

3. It is known that the function y=(m-1)x|m�+1 is a linear function, find the value of m;

Solution: According to the meaning of the question, we get �m�=1,

The solution is m=±1,

But m-1≠0, that is, m≠1,

So m=-1.

4. If the function y=(m-3)x+m2-9 is a proportional function, find the value of m;

Solution: According to the meaning of the question, m2-9=0,

The solution is m=±3,

But m-3≠0, that is, m≠3,

So m=-3.

5. A bookstore offers two ways to rent books: one is to rent books sporadically, charging 1 yuan per book, and the other is to charge a membership card, the card fee is 12 yuan per month, and the book rental is 0.4 yuan per book. Xiaobin comes often This store rents books, if the number of books rented per month is x.

(1) Write the functional relationship between the amount y1 (yuan) payable for sporadic book rentals and the number of books x (books) rented.

(2) Write the functional relationship between the amount y2 (yuan) payable by the membership card for renting books and the number of rented books x (books).

(3) Which method of renting books should Xiaobin choose to be more cost-effective? Why?

PPT on linear functions and proportional functions, Part 5: Class summary

The concept of a linear function

The concept of proportional function

Determination of functional relationship expressions

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Update Time: 2024-10-04

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