"Proportional Function" linear function PPT courseware 6

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"Proportional Function" linear function PPT courseware 6

think:

What kind of functions can be used to represent the variable correspondence patterns in the following questions? What do these functions have in common?

(1) The circumference L of the circle changes with the change of the radius r;

(2) The density of iron is 7.8g/cm³, and the mass m (unit: g) of the iron block changes with the size of its volume V (unit: cm³);

What kind of functions can be used to represent the variable correspondence patterns in the following questions? What do these functions have in common?

(3) The thickness of each exercise book is 0.5cm. The total thickness h (unit: cm) of some exercise books stacked together changes with the change of the number n of these exercise books;

What kind of functions can be used to represent the variable correspondence patterns in the following questions? What do these functions have in common?

(4) Freeze an object at 0°C so that it drops by 2°C per minute. The temperature T (unit:°C) of the object changes with the freezing time t (unit: minutes).

Proportional function:

Generally, a function of the form y=kx (k is a constant, k≠0) is called a proportional function, where k is called the proportional coefficient.

Notice:

(1) Analytical formula:

The function is a proportional function and its analytical formula can be reduced to the form y=kx (k is a constant, k≠0);

(2) Characteristics of analytical expressions:

Characteristics of the analytical formula of proportional function y=kx (k is a constant, k≠0):

①k≠0,

②The index of the independent variable x is 1;

(3) The value range of the independent variable:

Generally speaking, the value range of the independent variable of a proportional function is all real numbers; in practical problems or under the premise that the value range is specifically specified, the value range of the independent variable of a proportional function is not all real numbers.

Example 3 Determine whether the following statements are correct?

(1) Formula for the circumference of a circle

Where C is the perimeter, R is the radius, and the perimeter C is a proportional function of the radius R;

(2) The formula for the area of ​​a square is

Where S is the area, a is the side length of the square, and the area S is a proportional function of the side length a.

According to the image discovery rules:

Both images pass through the origin.

The graph of function y=2x _________ from left to right,

Passing through the ________quadrant;

The graph of the function y=-2x ______ from left to right, passing through the _______ quadrant.

Properties of the graph of proportional function:

Generally, the graph of the proportional function y=kx (k is a constant, k ≠0) is a straight line passing through the origin, and we call it the straight line y=kx.

When k > 0, the straight line y=kx passes through the first and third quadrants and rises from left to right, that is, as x increases, y also increases;

When k <0, the straight line y=kx passes through the second and fourth quadrants and decreases from left to right, that is, as x increases, y decreases.

practise:

(1) If the function y=(m-2)x+5-m is a proportional function, then the value of m is ______, and the analytical formula of this function is _______.

(3) When the independent variable x=____, the function value of the proportional function y=8x is 4.

(4) If in the proportional function y=(2m-1)x, y decreases as x increases, then the value range of m is ______.

Class exercises:

1. The image of the function y=0.3x passes through the point (0, ____) and the point (1, ____), and y increases with the increase of x;

2. If the function y=mxm+5 is a proportional function, then m=____, the graph of this function must pass through the ____ quadrant;

3. If the graph of function y=kx(k≠0) passes through point (5,-4), then k=____;

4. Point A (1, m) is on the graph of function y=2x, then m=____;

5. When a____, the straight line y=(1-a)x descends from left to right

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