"Linear function and its image" PPT courseware

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"Linear function and its image" PPT courseware

Can you list it?

1. Today, Xiao Li brings 50 yuan to buy a notebook. It is known that each notebook sells for 3 yuan. The functional relationship between the money Y (yuan) left by Xiao Li and the number of notebooks bought (X) is y =50-3x.

2. Xiao Zhang is going to save some of his usual pocket money and save it. He has saved 60 yuan and will save 12 yuan every month from now on. Try to write down the functional relationship between Xiao Zhang’s deposit Y (yuan) and the number of months from now X, y=60+12x.

3. A classmate’s home is 2,000 meters away from the school. When he returns to school by bicycle, he travels 200 meters per minute. The functional relationship between the time X (minutes) he spent riding the bicycle and the remaining distance Y (meters) is y=2000-200x.

Observe and think

Observe the six functional relationships and complete the following fill-in-the-blanks:

(1)S=10+300t (2)y=50-3x

(3)y=60+12x (4)y=2000-200x

(5)y=-5x+2 (6)y=-3/2x

1. These functions have ____ variables, the degree of the independent variable is ___, and the coefficient of the independent variable is _____0 (fill in =, ≠).

2. These function expressions are all _______ expressions of independent variables, and their general form is ___________.

Summary and summary

A function of the form y=kx+b (k, b are constants, k≠0) is called a linear function of x.

Thinking: When b=0, what will happen if you observe the function y=kx+b once?

y=kx+b(k≠0)--b=0--y=kx(k≠0)

When b=0, the linear function y=kx (k≠0) is also called a proportional function. k is called the proportional coefficient.

Example 1. The mass m (unit: g) of copper is directly proportional to its volume v (unit: cm3). When the volume of copper is v=3cm3, its mass is measured to be m=26.7g

(1) Find the functional expression between the mass m and the volume v of copper;

(2) When the volume of the copper block is 2.5cm3, find its mass.

Solution: (1) Because m and v are directly proportional quantities,

So let m=kv, where k is the proportional coefficient.

Substitute v=3, m=26.7 into ,

We get 26.7=3k, and the solution is k=8.9.

Therefore, the functional expression between mass m and volume v is m=8.9v (v>0)

(2) When v=2.5, m=8.9×2.5=22.25.

Therefore, when the volume of the copper block is 2.5cm3, the mass of the copper block is 22.25g.

Expand and improve

1. It is known that the function y=(2-m)x+2m-3. Find the value of m,

(1) This function is a proportional function

(2) This function is a linear function

2. If the function y=(m-1)x|m|+m is a linear function about x, try to find the value of m.

Solution: (1) According to the question, 2m-3=0, m=3/2, so when m=3/2, the function is a proportional function y=1/2x

(2) According to the meaning of the question, 2-m≠0, m≠2, so when m≠2, this function is a linear function

summary

1.Definition of a linear function

2. The proportional function is a special linear function

3. For practical problems in daily life, the key to problem-solving is to transform the problem into a mathematical problem, that is, to construct a corresponding mathematical model, establish a functional relationship, and make an answer based on the conditions in the problem.

1. Write down whether the following functional relationship is a linear function? Is it a directly proportional function? If it is a directly proportional function, indicate the value of the proportional coefficient k.

(1) The relationship between the circumference C of a circle and its radius r;

C=2πr C is a linear function of r and a proportional function, k=2π

(2) The relationship between the area s of a circle and its radius r

S=πr² is not a linear function, nor is it a proportional function

(3) The functional relationship between the perimeter l of the square and the side length a:

l=4a is a linear function of a, which is also a proportional function, k=4

(4) The upper base of the trapezoid has a length of 2, a height of 3, and the relationship between the area of ​​the trapezoid s and the lower base b.

S=3/2b+3 S is a linear function of b, but it is not a proportional function

2. Fill in the blanks:

(1) Proportional function y=kx(k≠0)

①If the proportion coefficient is -1/3, then the functional relationship is y=-1/3x;

②If x=5, y=1, then the functional relationship y=1/5x.

(2) The known function y=(m-3)xm-1,

When m=2, y is a proportional function of x;

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"Linear function and its image" PPT courseware 2:

"Linear function and its image" PPT courseware 2 Review knowledge 1. The general form of a linear function y=kx+b (kb is a constant, k0) The general form of a proportional function is y=kx (k is a constant, k0) There are two What connection? The proportional function is a special linear function 2. Use...

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Update Time: 2024-07-22

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