On the properties of y=ax+x/b

Team leader: Xia

Members: Xu, Liao Kefei and Zhang.

Ruan Nianshou Yang Longkun Chen Xiupeng

Instructor: Dou Chunhong

Date: 20 10 year120 February.

Research papers on the properties of y=ax+x/b

This paper discusses the general properties and characteristics of the function y=ax+b/x (mainly in the case of A B > 0), and investigates the simple application of the function. By means of group cooperation, network investigation and literature research. It is concluded that the symbolic function is a special hyperbola with focus, asymptote and eccentricity.

Keywords: special hyperbola, application of function properties

First, the background of the project

Properties of function y=ax+b/x and its application in mathematics and real life.

Second, the purpose of the project

This study mainly explores the general properties and characteristics of the function y=ax+b/x (mainly in the case of A B > 0) through group cooperation, and investigates the simple application of the function. This paper focuses on the properties of the function y=ax+b/x, and then uses the Internet and other multimedia means to understand the problems solved by the function y=ax+b/x in daily life.

The purpose of this team cooperation study is to improve the cooperation ability and communication ability among members; At the same time, it will also cultivate our ability to understand and solve mathematical problems and enhance our logical abstract thinking ability.

Third, research methods.

This study mainly explores the general properties and characteristics of the function y=ax+b/x (mainly in the case of A B > 0) through group cooperation, and investigates the simple application of this function. This paper focuses on the properties of the function y=ax+b/x, and then uses the Internet and other multimedia means to understand the problems solved by the function y=ax+b/x in daily life.

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Fourth, the research process

Referring to the teacher's inquiry ideas about functions in the teaching process, we decided to discuss A and B first.

When a = 0 and b = 0.

The function y=ax+b/x is the x axis.

When a = 0 and b = 0.

The function y=ax+b/x is a hyperbola.

When a≠0 and b=0

The function y=ax+b/x is a straight line.

When a≠0 and b≠0

The function y=ax+b/x is a hyperbola with y=ax and y axis as asymptotes.

Draw the images of functions y=x+ 1/x and y=x+3/x by geometric drawing. Observe the monotonicity and symmetry of the function from the function image, as well as the approximate value domain and definition domain of the function. In order to get the precise range and definition of the function, we use the knowledge of basic inequalities.

Take y=x+ 1/x as an example, its monotonicity is [- 1, 0] and (0, 1), and the function decreases; In the interval of (-∞,-1) and (1,+∞), the function decreases.

Symmetry: the function image is a central symmetric figure with the origin as symmetry neutrality.

Range: (-∞, -2]∞[2,+∞]

Domain: (-∞, 0)∩(0,+∞).

After mastering the general properties of the function with special values, we searched the related contents of the function y=ax+b/x on the Internet, and we know that the function y = ax+b/x is called the sign check function, which is also called Nike function.

Fifth, the research results of the subject.

A summary of the properties of y = ax+b/x (mainly the properties when a > 0 and b > 0)

Rough image

Domain of definition

(-∞,0)∪(0,+∞)

range

(-∞，-2〖ab〗∩[2〗ab，+∞)

symmetrical

On the symmetry of origin o

Monotonicity:

(1) (0, "b/a" ∨(-" b/a, 0), the function decreases.

(2) (-∞, --" b/a "∨(+" b/a+"b/a,+∞), the function is increasing.

Most valuable

(1) x < 0, when x =-"b/a and ymax =-2" ab.

② x > 0, when x = "b/a and ymin = 2" ab.

Special properties:

The function image is infinitely close to the straight lines x=0 and y=ax.

From particularity to generality. Referring to the information obtained from the Internet, we summarize some features in the following table.

Special properties:

① Symbolic function is obtained by hyperbolic rotation. Like double lines, asymptotes, vertices and so on.

(Take y=x+ 1/x as an example: its equation is rsin α = rcos α+ 1/rcos α, and it is rsin (α-π/8) = rcos (α-π/8)+1/rcos after rotating 22.5 degrees counterclockwise.

Based on the above properties of symbolic function, it is often used to study the establishment of maximum value and constant of function. For example, for the function f(x)= 12/x+3x, the maximum value is taken when x < 0, and the minimum value is taken when x > 0. We can easily know that ymax=-6 when x < 0. When x > 0, Ymin=6. Of course, this is only a simple and basic application in mathematics. A slightly more complicated application will find the maximum value of two variables, for example, the known positive number x, y satisfies the minimum value of 8/x+ 1/y= 1 and x+2y.

Using the above properties of symbolic function will be very simple in solving mathematical problems. In solving the problems in production, scientific research and daily life, the role of symbols is also promising. For example: ① A food factory regularly purchases flour. It is known that this factory needs 6 tons of flour every day, and the price per ton of flour is 1.800 yuan. Other expenses such as flour storage are 3 yuan per ton per day, and transportation fee is 900 yuan per time. How many days does the factory buy flour, and the average daily total cost is the least? (1) It is very difficult to buy flour per x, and its purchase volume is 6x tons. According to the meaning of the question, other expenses such as flour storage are 3 [6x+6 (x-1)+…+6 * 2 = 6 *1] = 9x (x+65438).

Let the average daily total cost be y yuan, then y =1/x [9x (x+1)+900]+6 *1800 = 900/x.

+9x+ 10809 According to the properties of the sign function, when x= 10, the minimum value is 10989. That is, the factory should buy flour every 10 day, so that the average daily total cost is the least.

When solving the reagent problem, it is nothing more than establishing a symbolic function model and then solving it by using the function properties. Another example is:

(2) Through observation, there is a functional relationship between the traffic volume Y( 1000 vehicles/hour) and the average speed v (km/hour) in a certain period: y=920v/v? +3v= 1600(v>0)

(1) At this time, when the average speed of the car is V, what is the maximum traffic flow Y? What is the maximum traffic volume?

(2) In order to ensure that the traffic flow at this time is at least 10000 vehicles/hour, what range should the average speed of vehicles be controlled?

The solution to the problem is similar.

Sixth, research experience

Through this study of mathematics, we deeply realize that mathematics is everywhere, and dare not imagine what our world would be like without it. The team's cooperation spirit has been improved, and each of us has experienced the ability to find and solve problems; At the same time, it also cultivated good communication and expression skills.

In a word, the success of this research study is the result of teamwork.

Seven. refer to

Dialogue master: Weimin Popular Literature and Art Publishing House.

Editor-in-Chief: Yan Chuanyu Jilin Education Publishing House.