The methodical material is for reference purposes and covers a wide range of topics. The article provides an overview of the graphs of the main elementary functions and considers the most important question – how to correctly and FAST build a graph. During the study higher mathematics without knowledge of basic charts elementary functions it will be hard, so it is very important to remember how the graphs of a parabola, hyperbola, sine, cosine, etc. look like, remember some function values. Also we will talk on some properties of basic functions.

I do not pretend to completeness and scientific thoroughness of the materials, the emphasis will be placed, first of all, on practice - those things with which one has to face literally at every step, in any topic of higher mathematics. Charts for dummies? You can say so.

By popular demand from readers clickable table of contents:

In addition, there is an ultra-short abstract on the topic
– master 16 types of charts by studying SIX pages!

Seriously, six, even I myself was surprised. This abstract contains improved graphics and is available for a nominal fee, a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!

And we start right away:

How to build coordinate axes correctly?

In practice, tests are almost always drawn up by students in separate notebooks, lined in a cage. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for the high-quality and accurate design of the drawings.

Any drawing of a function graph starts with coordinate axes.

Drawings are two-dimensional and three-dimensional.

Let us first consider the two-dimensional case Cartesian rectangular system coordinates:

1) We draw coordinate axes. The axis is called x-axis , and the axis y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo's beard.

2) We sign the axes with capital letters "x" and "y". Don't forget to sign the axes.

3) Set the scale along the axes: draw zero and two ones. When making a drawing, the most convenient and common scale is: 1 unit = 2 cells (drawing on the left) - stick to it if possible. However, from time to time it happens that the drawing does not fit on a notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). Rarely, but it happens that the scale of the drawing has to be reduced (or increased) even more

DO NOT scribble from a machine gun ... -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, .... For coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero and two units along the axes. Sometimes instead of units, it is convenient to “detect” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely set the coordinate grid.

It is better to estimate the estimated dimensions of the drawing BEFORE the drawing is drawn.. So, for example, if the task requires drawing a triangle with vertices , , , then it is quite clear that the popular scale 1 unit = 2 cells will not work. Why? Let's look at the point - here you have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale 1 unit = 1 cell.

By the way, about centimeters and notebook cells. Is it true that there are 15 centimeters in 30 notebook cells? Measure in a notebook for interest 15 centimeters with a ruler. In the USSR, perhaps this was true ... It is interesting to note that if you measure these same centimeters horizontally and vertically, then the results (in cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. It may seem like nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automotive industry, falling planes or exploding power plants.

Speaking of quality, or a brief recommendation on stationery. To date, most of the notebooks on sale, without saying bad words, are complete goblin. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! Save on paper. For clearance control works I recommend using the notebooks of the Arkhangelsk Pulp and Paper Mill (18 sheets, cage) or Pyaterochka, although it is more expensive. It is advisable to choose a gel pen, even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smears or tears paper. The only "competitive" ballpoint pen in my memory is the Erich Krause. She writes clearly, beautifully and stably - either with a full stem, or with an almost empty one.

Additionally: the vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Vector basis, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.

3D case

It's almost the same here.

1) We draw coordinate axes. Standard: applicate axis – directed upwards, axis – directed to the right, axis – downwards to the left strictly at an angle of 45 degrees.

2) We sign the axes.

3) Set the scale along the axes. Scale along the axis - two times smaller than the scale along the other axes. Also note that in the right drawing, I used a non-standard "serif" along the axis (this possibility has already been mentioned above). From my point of view, it’s more accurate, faster and more aesthetically pleasing - you don’t need to look for the middle of the cell under a microscope and “sculpt” the unit right up to the origin.

When doing a 3D drawing again - give priority to scale
1 unit = 2 cells (drawing on the left).

What are all these rules for? Rules are there to be broken. What am I going to do now. The fact is that the subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect in terms of proper design. I could draw all the graphs by hand, but it’s really scary to draw them, as Excel is reluctant to draw them much more accurately.

Graphs and basic properties of elementary functions

Linear function is given by the equation. Linear function graph is direct. In order to construct a straight line, it is enough to know two points.

Example 1

Plot the function. Let's find two points. It is advantageous to choose zero as one of the points.

If , then

We take some other point, for example, 1.

If , then

When preparing tasks, the coordinates of points are usually summarized in a table:

And the values ​​themselves are calculated orally or on a draft, calculator.

Two points are found, let's draw:

When drawing up a drawing, we always sign the graphics.

It will not be superfluous to recall special cases of a linear function:

Notice how I placed the captions, signatures should not be ambiguous when studying the drawing. AT this case it was extremely undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.

1) A linear function of the form () is called direct proportionality. For example, . The direct proportionality graph always passes through the origin. Thus, the construction of a straight line is simplified - it is enough to find only one point.

2) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is built immediately, without finding any points. That is, the entry should be understood as follows: "y is always equal to -4, for any value of x."

3) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also built immediately. The entry should be understood as follows: "x is always, for any value of y, equal to 1."

Some will ask, well, why remember the 6th grade?! That's how it is, maybe so, only during the years of practice I met a good dozen students who were baffled by the task of constructing a graph like or .

Drawing a straight line is the most common action when making drawings.

The straight line is discussed in detail in the course of analytic geometry, and those who wish can refer to the article Equation of a straight line on a plane.

Quadratic function graph, cubic function graph, polynomial graph

Parabola. Schedule quadratic function () is a parabola. Consider famous case:

Let's recall some properties of the function.

So, the solution to our equation: - it is at this point that the vertex of the parabola is located. Why this is so can be learned from the theoretical article on the derivative and the lesson on the extrema of the function. In the meantime, we calculate the corresponding value of "y":

So the vertex is at the point

Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function – is not even, but, nevertheless, no one canceled the symmetry of the parabola.

In what order to find the remaining points, I think it will be clear from the final table:

This algorithm construction can be figuratively called a "shuttle" or the principle of "back and forth" with Anfisa Chekhova.

Let's make a drawing:

From the considered graphs, another useful feature comes to mind:

For a quadratic function () the following is true:

If , then the branches of the parabola are directed upwards.

If , then the branches of the parabola are directed downwards.

In-depth knowledge of the curve can be obtained in the lesson Hyperbola and parabola.

The cubic parabola is given by the function . Here is a drawing familiar from school:

Let's list basic properties functions

Function Graph

It represents one of the branches of the parabola. Let's make a drawing:

The main properties of the function:

In this case, the axis is vertical asymptote for the hyperbola graph at .

Will BAD mistake, if, when making a drawing, by negligence, we allow the graph to intersect with the asymptote .

Also one-sided limits, tell us that a hyperbole not limited from above and not limited from below.

Let's explore the function at infinity: , that is, if we start to move along the axis to the left (or right) to infinity, then the “games” will be a slender step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.

So the axis is horizontal asymptote for the graph of the function, if "x" tends to plus or minus infinity.

The function is odd, which means that the hyperbola is symmetrical with respect to the origin. This fact is obvious from the drawing, moreover, it can be easily verified analytically: .

The graph of a function of the form () represents two branches of a hyperbola.

If , then the hyperbola is located in the first and third coordinate quadrants(see picture above).

If , then the hyperbola is located in the second and fourth coordinate quadrants.

It is not difficult to analyze the specified regularity of the place of residence of the hyperbola from the point of view of geometric transformations of graphs.

Example 3

Construct the right branch of the hyperbola

We use the pointwise construction method, while it is advantageous to select the values ​​so that they divide completely:

Let's make a drawing:

It will not be difficult to construct the left branch of the hyperbola, here the oddness of the function will just help. Roughly speaking, in the pointwise construction table, mentally add a minus to each number, put the corresponding dots and draw the second branch.

Detailed geometric information about the considered line can be found in the article Hyperbola and parabola.

Graph of an exponential function

In this paragraph, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponent that occurs.

I remind you that this is irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points probably enough:

Let's leave the graph of the function alone for now, about it later.

The main properties of the function:

Fundamentally, the graphs of functions look the same, etc.

I must say that the second case is less common in practice, but it does occur, so I felt it necessary to include it in this article.

Graph of a logarithmic function

Consider a function with natural logarithm.
Let's do a line drawing:

If you forgot what a logarithm is, please refer to school textbooks.

The main properties of the function:

Domain:

Range of values: .

The function is not limited from above: , albeit slowly, but the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: . So the axis is vertical asymptote for the graph of the function with "x" tending to zero on the right.

Be sure to know and remember the typical value of the logarithm: .

Fundamentally, the graph of the logarithm at the base looks the same: , , ( decimal logarithm in base 10), etc. At the same time, the larger the base, the flatter the chart will be.

We will not consider the case, something I don’t remember when the last time I built a graph with such a basis. Yes, and the logarithm seems to be a very rare guest in problems of higher mathematics.

In conclusion of the paragraph, I will say one more fact: Exponential function and logarithmic function are two mutual inverse functions . If you look closely at the graph of the logarithm, you can see that this is the same exponent, just it is located a little differently.

Graphs of trigonometric functions

How does trigonometric torment begin at school? Correctly. From the sine

Let's plot the function

This line is called sinusoid.

I remind you that “pi” is an irrational number:, and in trigonometry it dazzles in the eyes.

The main properties of the function:

This function is an periodical with a period. What does it mean? Let's look at the cut. To the left and to the right of it, exactly the same piece of the graph repeats endlessly.

Domain: , that is, for any value of "x" there is a sine value.

Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but said equations don't have a solution.

The scope and range of the function. In elementary mathematics, functions are studied only on the set real numbers R.This means that the function argument can only take on those real values ​​for which the function is defined, i.e. it also only accepts real values. A bunch of X all valid valid values ​​of the argument x, for which the function y= f(x) is defined, called function scope. A bunch of Y all real values y that the function accepts is called function range. Now you can give more precise definition features: rule(law) of correspondence between sets X and Y, by which for each element from the setX can find one and only one element from the set Y, is called a function.

It follows from this definition that a function is considered given if:

The scope of the function is set X ;

The scope of the function is set Y ;

The rule (law) of correspondence is known, and such that for each

Only one function value can be found for an argument value.

This requirement of uniqueness of the function is mandatory.

monotonic function. If for any two values ​​of the argument x 1 and x 2 of the condition x 2 > x 1 follows f(x 2) > f(x 1), then the function f(x) is called increasing; if for any x 1 and x 2 of the condition x 2 > x 1 follows f(x 2) < f(x 1), then the function f(x) is called waning. A function that only increases or only decreases is called monotonous.

Limited and unlimited functions. The function is called limited if there is such positive number M what | f(x) | M for all values x . If no such number exists, then the function is unlimited.

EXAMPLES.

The function depicted in Fig. 3 is bounded, but not monotonic. The function in Figure 4 is just the opposite, monotonic, but unlimited. (Explain this please!)

Continuous and discontinuous functions. Function y = f (x) is called continuous at the pointx = a, if:

1) the function is defined for x = a, i.e. f (a) exist;

2) exists finite limit lim f (x) ;

x→a

(See "Limits of Functions")

3) f (a) = lim f (x) .

x→a

If at least one of these conditions is not met, then the function is called discontinuous at the point x = a.

If the function is continuous in all points of its domain of definition, then it is called continuous function.

Even and odd functions. If for any x f(- x) = f (x), then the function is called even; if it does: f(- x) = - f (x), then the function is called odd. Schedule even functionsymmetrical about the Y axis(Fig.5), a graph odd function Simmetric about the origin(Fig. 6).

Periodic function. Function f (x) - periodical if there is such non-zero number T what for any x from the scope of the function definition takes place: f (x + T) = f (x). Such least the number is called function period. All trigonometric functions are periodic.

EXAMPLE 1. Prove that sin x has a period of 2.

SOLUTION We know that sin ( x+ 2n) = sin x, where n= 0, ± 1, ± 2, …

Therefore, adding 2 n to the sine argument

Changes its value. Is there another number with this

Same property?

Let's pretend that P- such a number, i.e. equality:

Sin ( x+ P) = sin x,

Valid for any value x. But then it has

Location and x= / 2 , i.e.

sin(/2 + P) = sin / 2 = 1.

But according to the reduction formula sin ( / 2 + P) = cos P. Then

It follows from the last two equalities that cos P= 1, but we

We know that this is true only when P = 2n. Since the smallest

A non-zero number out of 2 n is 2, then this number

And there is a sin period x. It is proved similarly that 2 from n is , so this is the period sin 2 x.

Function nulls. The value of the argument for which the function is equal to 0 is called zero (root) functions. A function can have multiple zeros. For example, the function y = x (x + 1) (x-3) has three zeros: x= 0, x= -1, x= 3. Geometrically function null - is the abscissa of the point of intersection of the graph of the function with the axis X .

Figure 7 shows the graph of the function with zeros: x= a, x = b and x= c.

Asymptote. If the graph of a function approaches a certain straight line indefinitely as it moves away from the origin, then this straight line is called asymptote.

Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

The following are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

• When you submit an application on the site, we may collect various information, including your name, phone number, address Email etc.

How we use your personal information:

• Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send you important notices and communications.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.

Disclosure to third parties

We do not disclose information received from you to third parties.

Exceptions:

• If necessary - in accordance with the law, judicial order, in legal proceedings, and/or based on public requests or requests from government agencies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest reasons.
• In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.

Maintaining your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.

Russian gymnasium

ABSTRACT

Fulfilled

student 10"F" class Burmistrov Sergey

Supervisor

mathematic teacher

Yulina O.A.

Nizhny Novgorod

Function and its properties

Function- variable dependency at from a variable x , if each value X matches a single value at .

Variable x- independent variable or argument.

Variable y- dependent variable

Function value- meaning at corresponding set value X .

Function scope- all the values ​​that the independent variable takes.

Function range (set of values) - all the values ​​that the function takes.

The function is even- if for any X f(x)=f(-x)

The function is odd- if for any X from the scope of the function, the equality f(-x)=-f(x)

Increasing function- if for any x 1 and x 2, such that x 1 < x 2, the inequality f( x 1 ) x 2 )

Decreasing function- if for any x 1 and x 2, such that x 1 < x 2, the inequality f( x 1 )>f( x 2 )

Ways to set a function

¨ To define a function, you need to specify the way in which for each argument value you can find the corresponding function value. The most common is the way to define a function using the formula at =f(x), where f(x)- some expression with a variable X. In this case, we say that the function is given by a formula or that the function is given by analytically.

¨ In practice, it is often used tabular the way the function is defined. With this method, a table is provided indicating the values ​​of the function for the values ​​of the argument present in the table. Examples of a tabular function definition are a table of squares, a table of cubes.

Types of functions and their properties

1) Permanent function- function, given by the formula y= b , where b- some number. schedule permanent function y \u003d b is a straight line parallel to the x-axis and passing through the point (0; b) on the y-axis

2) Direct proportionality- function given by formula y= kx , where k¹0. Number k called coefficient of proportionality .

Function properties y=kx :

1. Domain of definition functions - set all real numbers

2. y=kx- odd function

3. For k>0, the function increases, and for k<0 убывает на всей числовой прямой

3)Linear function- the function that is given by the formula y=kx+b, where k and b - real numbers. If, in particular, k=0, then we get a constant function y=b; if b=0, then we get a direct proportionality y=kx .

Function Properties y=kx+b :

1. Domain of definition - the set of all real numbers

2. Function y=kx+b general view, i.e. neither even nor odd.

3. For k>0, the function increases, and for k<0 убывает на всей числовой прямой

The graph of the function is straight .

4)Inverse proportionality- function given by formula y=k /X, where k¹0 Number k called inverse proportionality factor.

Function Properties y=k / x:

1. Domain of definition - the set of all real numbers except zero

2. y=k / x - odd function

3. If k>0, then the function decreases on the interval (0;+¥) and on the interval (-¥;0). If k<0, то функция возрастает на промежутке (-¥;0) и на промежутке (0;+¥).

The graph of the function is hyperbola .

5)Function y=x2

Function Properties y=x2:

2. y=x2 - even function

3. The function decreases on the interval

The graph of the function is parabola .

6)Function y=x 3

Function Properties y=x3:

1. The domain of definition is the entire number line

2. y=x 3 - odd function

3. The function is increasing on the entire number line

The graph of the function is cubic parabola

7)Power function with natural exponent- function given by formula y=xn, where n- natural number. For n=1 we get the function y=x, its properties are considered in Section 2. For n=2;3 we get the functions y=x 2 ; y=x 3 . Their properties are discussed above.

Let n be an arbitrary even number greater than two: 4,6,8... In this case, the function y=xn has the same properties as the function y=x 2 . The graph of the function resembles a parabola y=x 2 , only the branches of the graph for |x|>1 go up the steeper, the larger n, and for |x|<1 тем “теснее прижимаются” к оси Х, чем больше n.

Let n be an arbitrary odd number greater than three: 5,7,9... In this case, the function y=xn has the same properties as the function y=x 3 . The function graph resembles a cubic parabola.

8)Power function with integer negative exponent - function given by formula y=x-n , where n- natural number. For n=1 we get y=1/x, the properties of this function are considered in Section 4.

Let n be an odd number greater than one: 3,5,7... In this case, the function y=x-n has basically the same properties as the function y=1/x.

Let n be an even number, for example n=2.

Function Properties y=x -2 :

1. The function is defined for all x¹0

2. y=x -2 - even function

3. The function decreases by (0;+¥) and increases by (-¥;0).

Any function with an even n greater than two has the same properties.

9)Function y= Ö X

Function Properties y= Ö X :

1. The domain of definition is a ray and increases on the interval )