The length of the segment on the coordinate axis is found by the formula:

The length of the segment coordinate plane searched by the formula:

To find the length of a segment in a three-dimensional coordinate system, the following formula is used:

The coordinates of the middle of the segment (for the coordinate axis only the first formula is used, for the coordinate plane - the first two formulas, for the three-dimensional coordinate system - all three formulas) are calculated by the formulas:

Function is a correspondence of the form y= f(x) between variables, due to which each considered value of some variable x(argument or independent variable) corresponds to a certain value of another variable, y(dependent variable, sometimes this value is simply called the value of the function). Note that the function assumes that one value of the argument X there can only be one value of the dependent variable at. However, the same value at can be obtained with various X.

Function scope are all values ​​of the independent variable (function argument, usually X) for which the function is defined, i.e. its meaning exists. The domain of definition is indicated D(y). By and large, you are already familiar with this concept. The scope of a function is otherwise called the domain of valid values, or ODZ, which you have been able to find for a long time.

Function range are all possible values ​​of the dependent variable of this function. Denoted E(at).

Function rises on the interval on which the larger value of the argument corresponds to the larger value of the function. Function Decreasing on the interval on which the larger value of the argument corresponds to the smaller value of the function.

Function intervals are the intervals of the independent variable at which the dependent variable retains its positive or negative sign.

Function zeros are those values ​​of the argument for which the value of the function is equal to zero. At these points, the graph of the function intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means simply solving the equation. Also, often the need to find intervals of constant sign means the need to simply solve the inequality.

Function y = f(x) are called even X

This means that for any opposite values ​​of the argument, the values ​​of the even function are equal. Schedule even function always symmetrical about the y-axis of the y.

Function y = f(x) are called odd, if it is defined on a symmetric set and for any X from the domain of definition the equality is fulfilled:

This means that for any opposite values ​​of the argument, the values ​​of the odd function are also opposite. The graph of an odd function is always symmetrical about the origin.

The sum of the roots of an even and odd features(points of intersection of the x-axis OX) is always zero, because for every positive root X account for negative root –X.

It is important to note that some function does not have to be even or odd. There are many functions that are neither even nor odd. Such functions are called functions general view , and none of the above equalities or properties hold for them.

Linear function is called a function that can be given by the formula:

The graph of a linear function is a straight line and in the general case looks like this (an example is given for the case when k> 0, in this case the function is increasing; for the occasion k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):

Graph of Quadratic Function (Parabola)

The graph of a parabola is given by a quadratic function:

A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x 1 ; 0) and ( x 2; 0). If there are no roots, then the quadratic function does not intersect the OX axis, if there is one root, then at this point ( x 0; 0) the quadratic function only touches the OX axis, but does not intersect it. A quadratic function always intersects the OY axis at a point with coordinates: (0; c). The graph of a quadratic function (parabola) may look like this (the figure shows examples that far from exhaust all possible types of parabolas):

Wherein:

• if the coefficient a> 0, in the function y = ax 2 + bx + c, then the branches of the parabola are directed upwards;
• if a < 0, то ветви параболы направлены вниз.

Parabola vertex coordinates can be calculated using the following formulas. X tops (p- in the figures above) of a parabola (or the point at which the square trinomial reaches its maximum or minimum value):

Y tops (q- in the figures above) of a parabola or the maximum if the branches of the parabola are directed downwards ( a < 0), либо минимальное, если ветви параболы направлены вверх (a> 0), value square trinomial:

Graphs of other functions

power function

Here are some examples of graphs of power functions:

Inversely proportional dependence call the function given by the formula:

Depending on the sign of the number k An inversely proportional graph can have two fundamental options:

Asymptote is the line to which the line of the graph of the function approaches infinitely close, but does not intersect. Asymptotes for Graphs inverse proportionality shown in the figure above are the coordinate axes to which the graph of the function approaches infinitely close, but does not intersect them.

exponential function with base A call the function given by the formula:

a schedule exponential function can have two fundamental options (we will also give examples, see below):

logarithmic function call the function given by the formula:

Depending on whether the number is greater or less than one a The graph of a logarithmic function can have two fundamental options:

Function Graph y = |x| as follows:

Graphs of periodic (trigonometric) functions

Function at = f(x) is called periodical, if there exists such a non-zero number T, What f(x + T) = f(x), for anyone X out of function scope f(x). If the function f(x) is periodic with period T, then the function:

Where: A, k, b are constant numbers, and k not equal to zero, also periodic with a period T 1 , which is determined by the formula:

Most examples of periodic functions are trigonometric functions. Here are the graphs of the main trigonometric functions. The following figure shows part of the graph of the function y= sin x(the whole graph continues indefinitely to the left and right), the graph of the function y= sin x called sinusoid:

Function Graph y= cos x called cosine wave. This graph is shown in the following figure. Since the graph of the sine, it continues indefinitely along the OX axis to the left and to the right:

Function Graph y=tg x called tangentoid. This graph is shown in the following figure. Like the graphs of other periodic functions, this graph repeats indefinitely along the OX axis to the left and to the right.

And finally, the graph of the function y=ctg x called cotangentoid. This graph is shown in the following figure. Like the graphs of other periodic and trigonometric functions, this graph repeats indefinitely along the OX axis to the left and to the right.

Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.

Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and the knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own name. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.

Found an error?

If you think you have found an error in training materials, then write, please, about it by mail. You can also report a bug in social network(). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.

Elementary functions and their graphs

Straight proportionality. Linear function.

Inverse proportion. Hyperbola.

quadratic function. Square parabola.

Power function. Exponential function.

logarithmic function. trigonometric functions.

Inverse trigonometric functions.

1.	proportional values. If variables y And x directly proportional, then the functional dependence between them is expressed by the equation: y = k x , Where k- constant value ( proportionality factor). Schedule straight proportionality- a straight line passing through the origin and forming with the axis X angle whose tangent is k:tan= k(Fig. 8). Therefore, the coefficient of proportionality is also called slope factor. Figure 8 shows three graphs for k = 1/3, k= 1 and k = 3 .
2.	Linear function. If variables y And x connected by the equation of the 1st degree: Ax + By = C , where at least one of the numbers A or B is not equal to zero, then the graph of this functional dependence is straight line. If C= 0, then it passes through the origin, otherwise it does not. Linear Function Graphs for Various Combinations A,B,C are shown in Fig.9.
3.	Reverse proportionality. If variables y And x back proportional, then the functional dependence between them is expressed by the equation: y = k / x , Where k- a constant value. Inverse Proportional Plot - hyperbola (Fig. 10). This curve has two branches. Hyperbolas are obtained when a circular cone is intersected by a plane (for conic sections, see the section "Cone" in the chapter "Stereometry"). As shown in Fig. 10, the product of the coordinates of the points of the hyperbola is a constant value, in our example equal to 1. In the general case, this value is equal to k, which follows from the hyperbola equation: xy = k. The main characteristics and properties of a hyperbola: Function scope: x 0, range: y 0 ; The function is monotonic (decreasing) at x< 0 and at x > 0, but not monotonic overall due to break point x= 0 (think why?); Unbounded function, discontinuous at a point x= 0, odd, non-periodic; - The function has no zeros.
4.	Quadratic function. This is the function: y = ax 2 + bx + c, Where a, b, c- permanent, a 0. In the simplest case, we have: b=c= 0 and y = ax 2. Graph of this function square parabola - curve passing through the origin (Fig. 11). Every parabola has an axis of symmetry OY, which is called parabola axis. Dot O the intersection of a parabola with its axis is called top of the parabola. Function Graph y = ax 2 + bx + c is also a square parabola of the same type as y = ax 2 , but its vertex lies not at the origin, but at the point with coordinates: The shape and location of a square parabola in the coordinate system depends entirely on two parameters: the coefficient a at x 2 and discriminant D:D = b 2 – 4ac. These properties follow from the analysis of the roots of the quadratic equation (see the corresponding section in the Algebra chapter). All possible different cases for a square parabola are shown in Fig.12.

Please draw a square parabola for the case a > 0, D > 0 .

Main characteristics and properties of a square parabola:

Function scope:  < x+ (i.e. x R ), and the area

values: … (Please answer this question yourself!);

The function as a whole is not monotonic, but to the right or left of the vertex

behaves like a monotone;

The function is unbounded, everywhere continuous, even for b = c = 0,

and non-periodic;

- at D< 0 не имеет нулей. (А что при D 0 ?) .

5.	Power function. This is the function: y=ax n, Where a, n- permanent. At n= 1 we get direct proportionality: y=ax; at n = 2 - square parabola; at n = 1 - inverse proportionality or hyperbole. Thus, these functions are special cases of a power function. We know that the zero power of any number other than zero is equal to 1, therefore, when n= 0 the power function becomes a constant: y= a, i.e. its graph is a straight line parallel to the axis X, excluding the origin of coordinates (please explain why?). All these cases (with a= 1) are shown in Fig. 13 ( n 0) and Fig.14 ( n < 0). Отрицательные значения x are not considered here, because then some functions: If n– entire, power functions make sense even when x < 0, но их графики имеют different kind depending on whether n an even number or an odd number. Figure 15 shows two such power functions: for n= 2 and n = 3. At n= 2 the function is even and its graph is symmetrical about the axis Y. At n= 3 the function is odd and its graph is symmetrical with respect to the origin. Function y = x 3 called cubic parabola. Figure 16 shows the function . This function is the inverse of the square parabola y = x 2 , its graph is obtained by rotating the graph of a square parabola around the bisector of the 1st coordinate angleThis is a way to obtain the graph of any inverse function from the graph of its original function. We can see from the graph that this is a two-valued function (this is also indicated by the  sign in front of the square root). Such functions are not studied in elementary mathematics, therefore, as a function, we usually consider one of its branches: upper or lower.
6.	Demonstration function. Function y = a x, Where a is a positive constant number, called exponential function. Argument x accepts any valid values; as function values ​​are considered only positive numbers, since otherwise we have a multivalued function. Yes, the function y = 81 x has at x= 1/4 four different meanings: y = 3, y = 3, y = 3 i And y = 3 i(Check, please!). But we consider as the value of the function only y= 3. Graphs of the exponential function for a= 2 and a= 1/2 are shown in Fig.17. They pass through the point (0, 1). At a= 1 we have a graph of a straight line parallel to the axis X, i.e. the function turns into a constant value equal to 1. When a> 1, the exponential function increases, and at 0< a < 1 – убывает. The main characteristics and properties of the exponential function:  < x+ (i.e. x R ); range: y> 0 ; The function is monotonic: it increases with a> 1 and decreases at 0< a < 1; - The function has no zeros.
7.	Logarithmic function. Function y= log a x, Where a is a constant positive number, not equal to 1 is called logarithmic. This function is the inverse of the exponential function; its graph (Fig. 18) can be obtained by rotating the graph of the exponential function around the bisector of the 1st coordinate angle. The main characteristics and properties of the logarithmic function: Function scope: x> 0, and the range of values:  < y+ (i.e. y R ); This is a monotonic function: it increases as a> 1 and decreases at 0< a < 1; The function is unbounded, everywhere continuous, non-periodic; The function has one zero: x = 1.
8.	trigonometric functions. When constructing trigonometric functions, we use radian measure of angles. Then the function y= sin x represented by a graph (Fig. 19). This curve is called sinusoid. Function Graph y= cos x shown in Fig.20; it is also a sine wave resulting from moving the graph y= sin x along the axis X to the left by 2 From these graphs, the characteristics and properties of these functions are obvious: Domain:  < x+  range: -1 y +1; These functions are periodic: their period is 2; Limited functions (| y| , everywhere continuous, not monotone, but having so-called intervals monotony, inside which they behave like monotonic functions (see graphs in Fig. 19 and Fig. 20); Functions have an infinite number of zeros (for more details, see the section "Trigonometric Equations"). Function Graphs y= tan x And y= cot x shown respectively in Fig.21 and Fig.22 It can be seen from the graphs that these functions are: periodic (their period , unbounded, generally not monotonic, but have intervals of monotonicity (what?), discontinuous (what break points do these functions have?). Region definitions and range of these functions:
9.	Inverse trigonometric functions. Definitions of inverses trigonometric functions and their main properties are given in section of the same name in the chapter "Trigonometry". Therefore, here we restrict ourselves only short comments regarding their graphs received by rotating the graphs of trigonometric functions around the bisector of the 1st coordinate angle.

Functions y= Arcsin x(fig.23) and y= Arccos x(fig.24) many-valued, unlimited; their domain of definition and range of values, respectively: 1 x+1 and  < y+ . Since these functions are multivalued,

Coordinate system - these are two mutually perpendicular coordinate lines intersecting at the point that is the origin for each of them.

Coordinate axes are the lines that form the coordinate system.

abscissa(x-axis) is the horizontal axis.

Y-axis(y-axis) is the vertical axis.

Function

Function is a mapping of the elements of the set X to the set Y . In this case, each element x of the set X corresponds to one single value y of the set Y .

Straight

Linear function is a function of the form y = a x + b where a and b are any numbers.

The graph of a linear function is a straight line.

Consider how the graph will look depending on the coefficients a and b:

If a > 0 , the line will pass through the I and III coordinate quarters.

If a< 0 , прямая будет проходить через II и IV координатные четверти.

b is the point of intersection of the line with the y-axis.

If a = 0 , the function becomes y = b .

Separately, we select the graph of the equation x \u003d a.

Important: this equation is not a function, since the definition of the function is violated (the function associates each element x of the set X with a single value y of the set Y). This equation associates one element x with an infinite set of elements y . However, the graph of this equation can be plotted. Let's just not call it the proud word "Function".

Parabola

The graph of the function y = a x 2 + b x + c is parabola .

In order to unambiguously determine how the parabola graph is located on the plane, you need to know what the coefficients a, b, c affect:

1. The coefficient a indicates where the branches of the parabola are directed.

• If a > 0 , the branches of the parabola are directed upwards.
• If a< 0 , ветки параболы направлены вниз.

1. The coefficient c indicates at what point the parabola intersects the y axis.
2. The coefficient b helps to find x in - the coordinate of the top of the parabola.

x in \u003d - b 2 a

1. The discriminant allows you to determine how many points of intersection a parabola has with an axis.

• If D > 0 - two points of intersection.
• If D = 0 - one point of intersection.
• If D< 0 — нет точек пересечения.

The graph of the function y = k x is hyperbola .

A characteristic feature of a hyperbola is that it has asymptotes.

Asymptotes of a hyperbola - straight lines, to which it tends, going to infinity.

The x-axis is the horizontal asymptote of the hyperbola

The y-axis is the vertical asymptote of the hyperbola.

On the graph, the asymptotes are marked with a green dotted line.

If the coefficient k > 0, then the branches of the hyperola pass through the I and III quarters.

If k<     0, ветви гиперболы проходят через II и IV четверти.

The smaller the absolute value of the coefficient k (the coefficient k without taking into account the sign), the closer the branches of the hyperbola to the x and y axes.

Square root

The function y     =     x has the following graph:

Increasing/decreasing functions

Function y   =   f(x) increases over the interval if a larger value of the argument (a larger x value) corresponds to a larger function value (a larger y value) .

That is, the more (to the right) x, the more (higher) y. Graph rises (look from left to right)

Function y   =   f(x) decreases over the interval if a larger argument value (a larger x value) corresponds to a smaller function value (a larger y value) .

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 issue - how to correctly and FAST build a graph. In the course of studying higher mathematics without knowing the graphs of the main 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. We will also talk about some properties of the main 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 the 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

The 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. In this case, it was highly 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. Graph of a quadratic function () is a parabola. Consider the 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 construction algorithm can be figuratively called a "shuttle" or the "back and forth" principle 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:

We list the main properties of the function

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 .

It will be a BIG mistake if, when drawing up a drawing, by negligence, you 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 an irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points is 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 Functionare 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? Right. 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 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 these equations do not have a solution.

The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values ​​of the argument, and the ordinates are equal to the corresponding values ​​of the function.

The following table shows the average monthly temperatures in the capital of our country, the city of Minsk.

P

t,V

Here the argument is the ordinal number of the month, and the value of the function is the air temperature in degrees Celsius. For example, from this table we learn that in April the average monthly temperature is 5.3 °C.

Functional dependence can be given by a graph.

Figure 1 shows a graph of the movement of a body thrown at an angle of 6СГ to the horizon with an initial velocity of 20 m/s.

Using the function graph, you can find the corresponding value of the function by the value of the argument. According to the graph in Figure 1, we determine that, for example, after 2 s from the start of movement, the body was at a height of 15 m, and after 3 s at a height of 7.8 m (Fig. 2).

It is also possible to solve the inverse problem, namely, by the given value a of the function, find those values ​​of the argument for which the function takes this value a. For example, according to the graph in Figure 1, we find that at a height of 10 m the body was in 0.7 s and 2.8 s from the start of movement (Fig. 3),

There are devices that draw graphs of dependencies between quantities. These are barographs - devices for fixing the dependence of atmospheric pressure on time, thermographs - devices for fixing the dependence of temperature on time, cardiographs - devices for graphic recording of the activity of the heart, etc. Figure 102 schematically shows a thermograph. Its drum rotates evenly. The paper wound on the drum is touched by a recorder, which, depending on the temperature, rises and falls and draws a certain line on the paper.

From the representation of a function by a formula, you can move on to its representation in a table and graph.