Positive proportional function is a kind of function proposed by Jack Louny in 1911Mathematical terminology, mainly forfunction。The positive proportional function is essentially a function of one degree.
Generally, twovariablex. The relationship between y can be expressed as y=kxfunction(k isconstantIf the degree of x is 1 and k ≠ 0), then y=kx is called a positive proportional function.[1]
The positive proportional function belongs to the primary function, but the primary function is not necessarily the positive proportional function, it is a special form of the primary function.
That is, in the form of a linear function, for example, y=kx+b (k is a constant, and k ≠ 0), when b=0, that is, the so-called "intercept on the y-axis" is zero, it is called the proportional case functionNumber.[1]
Relational expression
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In general, an image in the form of y=kx (k is a constant, k ≠ 0) is a pass throughoriginWe call it the line y=kx.[1]
The relationship of the positive proportional function is expressed as: y=kx (k is the proportional coefficient).
When k>0 (I, IIIquadrant）, k'sabsolute valueThe larger the image is, the closer the distance between the image and the y-axis is;The function value y followsindependent variableX increases with the increase of x;
When k<0 (two four quadrants), the smaller the absolute value of k, the farther away the image is from the y-axis.independent variableAs the value of x increases, the value of y gradually decreases.[2]
nature
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Monotonicity
When k>0, the image passes through the first and thirdquadrant, rising from left to right, y increases with the increase of x (monotonic increase)Increasing function；[1]
When k<0, the image passes through the second and fourthquadrant, decreases from left to right, y decreases with the increase of x (monotonically decreasing)Subtractive function。
Symmetry
Symmetry point: central symmetry about the origin.[1]
Symmetry axis: the straight line of itself;The bisector of the line in which it is located.
image
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Image description
The image of a positive scale function is passed through coordinatesorigin(0, 0) and the fixed point (1, k)SlopeIs k (k represents the angle between the positive proportional function and the x-axis), horizontal and verticalinterceptBoth are 0, and the image of a positive scale function is a straight line across the origin.[1]
The positive proportional function y=kx (k ≠ 0), the larger the absolute value of k, the steeper the straight line;The smaller the absolute value of k, the flatter the straight line.
1. Given the coordinates of a point, use the undetermined coefficient method to find the analytic formula of the function.Let the analytical formula be y=kx, and then substitute in the coordinates of the known points to solve the value of k.
2. After solving the value of k, mark the points on the number axis and connect the points
Image method
(I)[2]
Picture of a positive scale function
1. Take a value within the allowable range of x, and calculate the value of y according to the analytic formula;
2. Draw the point according to the value of x and y calculated in the first step;
3. Make a straight line between the point traced in the second step and the origin (because two points determine a straight line).
(II)
1. Given the coordinates of a point, use the undetermined coefficient method to find the analytic formula of the function.Let the analytical formula be y=kx, and then substitute the coordinates of the known points to solve the value of k;
2. After solving the value of k, mark each point on the number axis and connect points.
Image Properties
Positive proportional functionlinear programmingThe power embodied in the problem is also infinite.[2]
such asSlopeThe problem depends on the value of kFunction imageThe greater the angle with the x-axis, and vice versa.
Also, y=kx is the axis of symmetry of the image of y=k/x.
Positive proportion
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①Positive proportion: Two related quantities. One quantity changes, and the other also changes. If the ratio (i.e. quotient) of the two numbers corresponding to these two quantities is fixed, these two quantities are calledProportionalAnd their relationship is calledPositive proportional relationship。[1]
② Represented by letters: if the letters x and y are used to represent two related quantities, k is used to represent their ratio, and (certain) positive proportional relationship can be expressed by the following relationship:
Where k is a constant.
③ The change rule of two related quantities in a positive proportional relationship: for a positive ratio, that is, y=kx (k is a constant, k ≠ 0), at this time, y and x expand and shrink simultaneously, and the ratio remains unchanged.For example, a car travels at a certain speed per hour, and the distance traveled is proportional to the time spent.If all the above quotients are fixed, then the divisor and the two associated quantities represented by the divisor are in direct proportion.
Note: When judging whether the two related quantities are in positive proportion, we should pay attention to the two related quantities. Although one kind of quantity changes with the other, the ratio of their corresponding two numbers is not certain, so they cannot be in positive proportion.For example, a person's age cannot be in direct proportion to his weight, nor can the side length of a square be in direct proportion to his area.The quantity of unit price is directly proportional to the total price (the unit price remains unchanged, and the total price increases or decreases with the increase or decrease of quantity).
Example
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First, five functions are obtained through five questions. By observing these five functions, the concept of positive proportional function can be derived.[2]
According to the above five practical problems, we get five functions.Let's observe the common points of these five functions in order to summarize the concept of positive proportional function.
①h=2t ；② m=7.8n； ③s=0.5t； ④T=t/3 ；⑤y=200x。
What are the common characteristics of these five functions?
What are the forms of constants and independent variables on the right side of these five functions?
These five functions are the product of constant and independent variable, which can be expressed in the form of y=kx (k is not equal to 0).
Here are four functions. Which are positive proportional functions?
①y=3； ②y=2x； ③y=1/x； ④y=x^2。
answer:
② Is a positive proportional function.Because it conforms to the definition of positive proportional function. ①, ③, ④Is not a positive proportional function. ①:It isConstant function, no independent variable. ③:It isInverse proportional function。 ④：It isQuadratic function.
