This teaching material is for reference only and relates to a wide range of topics. The article provides an overview of graphs of basic elementary functions and considers the most important issue - how to build a graph correctly and QUICKLY. In the course of studying higher mathematics without knowledge of the graphs of basic elementary functions, it will be difficult, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, and remember some of the meanings of the functions. We will also talk about some properties of the main functions.

I do not claim completeness and scientific thoroughness of the materials; the emphasis will be placed, first of all, on practice - those things with which one encounters literally at every step, in any topic of higher mathematics. Charts for dummies? One could say so.

Due to numerous requests from readers clickable table of contents:

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

Seriously, six, even I was surprised. This summary 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 let's start right away:

How to construct coordinate axes correctly?

In practice, tests are almost always completed by students in separate notebooks, lined in a square. 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 high-quality and accurate design of drawings.

Any drawing of a function graph begins with coordinate axes.

Drawings can be two-dimensional or three-dimensional.

Let's first consider the two-dimensional case Cartesian rectangular coordinate system:

1) Draw coordinate axes. The axis is called x-axis , and the axis is 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 large letters “X” and “Y”. Don't forget to label the axes.

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

There is NO NEED to “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 “mark” 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 define the coordinate grid.

It is better to estimate the estimated dimensions of the drawing BEFORE constructing the drawing. So, for example, if the task requires drawing a triangle with vertices , , , then it is completely clear that the popular scale of 1 unit = 2 cells will not work. Why? Let's look at the point - here you will 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 30 notebook cells contain 15 centimeters? For fun, measure 15 centimeters in your notebook with a ruler. In the USSR, this may have been true... It is interesting to note that if you measure these same centimeters horizontally and vertically, the results (in the cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. This may seem 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 automobile industry, falling planes or exploding power plants.

Speaking of quality, or a brief recommendation on stationery. Today, most of the notebooks on sale are, to say the least, complete crap. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! They save money on paper. To complete tests, I recommend using notebooks from the Arkhangelsk Pulp and Paper Mill (18 sheets, square) 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 smudges or tears the paper. The only “competitive” ballpoint pen I can remember is the Erich Krause. She writes clearly, beautifully and consistently – whether with a full core 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. Basis of vectors, 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) Draw coordinate axes. Standard: axis applicate – directed upwards, axis – directed to the right, axis – directed downwards to the left strictly at an angle of 45 degrees.

2) Label the axes.

3) Set the scale along the axes. The scale along the axis is two times smaller than the scale along the other axes. Also note that in the right drawing I used a non-standard "notch" along the axis (this possibility has already been mentioned above). From my point of view, this is more accurate, faster and more aesthetically pleasing - there is no need to look for the middle of the cell under a microscope and “sculpt” a unit close to the origin of coordinates.

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

What are all these rules for? Rules are made to be broken. That's what I'll do now. The fact is that subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect from the point of view of correct design. I could draw all the graphs by hand, but it’s actually scary to draw them as Excel is reluctant to draw them much more accurately.

Graphs and basic properties of elementary functions

A linear function is given by the equation. The graph of linear functions is direct. In order to construct a straight line, it is enough to know two points.

Example 1

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

If , then

Let's take another point, for example, 1.

If , then

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

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

Two points have been found, let's make a drawing:

When preparing a drawing, we always sign the graphics.

It would be useful to recall special cases of a linear function:

Notice how I placed the signatures, signatures should not allow discrepancies when studying the drawing. In 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, . A direct proportionality graph always passes through the origin. Thus, constructing a straight line is simplified - it is enough to find just one point.

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

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

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

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

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

Graph of a quadratic, cubic function, graph of a polynomial

Parabola. Graph of a quadratic function () represents 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 found in the theoretical article on the derivative and the lesson on extrema of the function. In the meantime, let’s calculate the corresponding “Y” value:

Thus, 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 figuratively be called a “shuttle” or the “back and forth” principle with Anfisa Chekhova.

Let's make the drawing:

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

For a quadratic function () the following is true:

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

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

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

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

Let us list the main properties of the function

Graph of a function

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

Main properties of the function:

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

It would be a GROSS mistake if, when drawing up a drawing, you carelessly allow the graph to intersect with an asymptote.

Also one-sided limits tell us that the hyperbola not limited from above And not limited from below.

Let’s examine the function at infinity: , that is, if we start moving along the axis to the left (or right) to infinity, then the “games” will be in an orderly 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 a function, if “x” tends to plus or minus infinity.

The function is odd, and, therefore, the hyperbola is symmetrical about the origin. This fact is obvious from the drawing, in addition, it is 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 quarters(see picture above).

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

The indicated pattern of hyperbola residence is easy to analyze from the point of view of geometric transformations of graphs.

Example 3

Construct the right branch of the hyperbola

We use the point-wise construction method, and it is advantageous to select the values ​​so that they are divisible by a whole:

Let's make the drawing:

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

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

Graph of an Exponential Function

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

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

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

Main properties of the function:

Function graphs, etc., look fundamentally the same.

I must say that the second case occurs less frequently in practice, but it does occur, so I considered it necessary to include it in this article.

Graph of a logarithmic function

Consider a function with a natural logarithm.
Let's make a point-by-point drawing:

If you have forgotten what a logarithm is, please refer to your school textbooks.

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 a function as “x” tends to zero from the right.

It is imperative to know and remember the typical value of the logarithm: .

In principle, the graph of the logarithm to the base looks the same: , , (decimal logarithm to the base 10), etc. Moreover, the larger the base, the flatter the graph will be.

We won’t consider the case; I don’t remember the last time I built a graph with such a basis. And the logarithm seems to be a very rare guest in problems of higher mathematics.

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

Graphs of trigonometric functions

Where does trigonometric torment begin at school? Right. From sine

Let's plot the function

This line is called sinusoid.

Let me remind you that “pi” is an irrational number: , and in trigonometry it makes your eyes dazzle.

Main properties of the function:

This function is periodic with period . What does it mean? Let's look at the segment. To the left and right of it, exactly the same piece of the graph is repeated 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.

Presentation and lesson on the topic:
"Hyperbole, definition, property of a function"

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Hyperbole, definition

Guys, today we will study a new function and build its graph.
Consider the function: $y=\frac(k)(x)$, $k≠0$.
Coefficient $k$ – can take any real value except zero. For simplicity, let's start analyzing the function from the case when $k=1$.
Let's plot the function: $y=\frac(1)(x)$.
As always, let's start by building a table. True, this time we will have to divide our table into two parts. Let's consider the case when $x>0$.
We need to mark six points with coordinates $(x;y)$, which are given in the table and connect them with a line.
Now let's see what we get for negative x. Let's do the same thing, mark the points and connect them with a line. We have built two pieces of the graph, let's combine them.

Graph of the function $y=\frac(1)(x)$.
The graph of such a function is called a “Hyperbola”.

Properties of a hyperbola

Agree, the graph looks pretty nice, and it is symmetrical about the origin. If we draw any straight line passing through the origin of coordinates from the first to the third quarter, then it will intersect our graph at two points that will be equally distant from the origin of coordinates.
A hyperbola consists of two parts, symmetrical about the origin. These parts are called branches of the hyperbola.
The branches of a hyperbola in one direction (left and right) tend more and more towards the x-axis, but never cross it. In the other direction (up and down) they tend to the ordinate axis, but will also never cross it (since it is impossible to divide by zero). In such cases, the corresponding lines are called asymptotes. The graph of a hyperbola has two asymptotes: the x-axis and the y-axis.

A hyperbola has not only a center of symmetry, but also an axis of symmetry. Guys, draw the line $y=x$ and see how our graph is divided. You can notice that if the part that is located above the straight line $y=x$ is superimposed on the part that is located below, then they will coincide, this means symmetry with respect to the straight line.

We have plotted the function $y=\frac(1)(x)$, but what will happen in the general case is $y=\frac(k)(x)$, $k>0$.
The graphs will be practically no different. The result will be a hyperbola with the same branches, only the more $k$, the further the branches will be removed from the origin, and the less $k$, the closer to the origin.

For example, the graph of the function $y=\frac(10)(x)$ looks like this. The graph became “wider” and moved away from the origin.
But what about negative $k$? The graph of the function $y=-f(x)$ is symmetrical to the graph of $y=f(x)$ relative to the x-axis; you need to turn it upside down.
Let's take advantage of this property and plot the function $y=-\frac(1)(x)$.

Let us summarize the knowledge gained.
The graph of the function $y=\frac(k)(x)$, $k≠0$ is a hyperbola located in the first and third (second and fourth) coordinate quarters, for $k>0$ ($k

Properties of the function $y=\frac(k)(x)$, $k>0$

1. Domain of definition: all numbers except $x=0$.
2. $y>0$ for $x>0$, and $y 3. The function decreases on the intervals $(-∞;0)$ and $(0;+∞)$.



7. Range of values: $(-∞;0)U(0;+∞)$.

Properties of the function $y=\frac(k)(x)$, $k
1. Domain of definition: all numbers except $x=0$.
2. $y>0$ for $x 0$.
3. The function increases on the intervals $(-∞;0)$ and $(0;+∞)$.
4. The function is not limited either above or below.
5. There is no maximum or minimum value.
6. The function is continuous on the intervals $(-∞;0)U(0;+∞)$ and has a discontinuity at the point $x=0$.
7. Range of values: $(-∞;0)U(0;+∞)$.

The function Coefficient k can take any value except k = 0. Let us first consider the case when k = 1; so we'll talk about the function first.

To build a graph of the function, we will do the same as in the previous paragraph: we will give the independent variable x several specific values ​​and calculate (using the formula) the corresponding values ​​of the dependent variable variable u. True, this time it is more convenient to carry out calculations and constructions gradually, first giving the argument only positive values, and then only negative ones.

First stage. If x = 1, then y = 1 (recall that we use the formula);

Second phase.

In short, we have compiled the following table:

Now let’s combine the two stages into one, that is, we’ll make one from two figures 24 and 26 (Fig. 27). That's what it is graph of a function it is called a hyperbole.
Let's try to describe the geometric properties of a hyperbola using the drawing.

Firstly, we notice that this line looks as beautiful as a parabola because it has symmetry. Any line passing through the origin of coordinates O and located in the first and third coordinate angles intersects the hyperbola at two points that lie on this line on opposite sides of the point O, but at equal distances from it (Fig. 28). This is inherent, in particular, to points (1; 1) and (- 1; - 1),

Etc. This means - O is the center of symmetry of the hyperbola. They also say that a hyperbola is symmetrical about the origin coordinates.

Secondly, we see that the hyperbola consists of two parts that are symmetrical with respect to the origin; they are usually called branches of a hyperbola.

Thirdly, we notice that each branch of the hyperbola in one direction comes closer and closer to the abscissa axis, and in the other direction to the ordinate axis. In such cases, the corresponding straight lines are called asymptotes.

This means that the graph of the function, i.e. hyperbola has two asymptotes: the x-axis and the y-axis.

If you carefully analyze the plotted graph, you can discover one more geometric property, not as obvious as the three previous ones (mathematicians usually say this: “a more subtle property”). A hyperbola has not only a center of symmetry, but also an axes of symmetry.

In fact, let's construct a straight line y = x (Fig. 29). Now look: dots located on opposite sides of the conducted straight, but at equal distances from it. They are symmetrical relative to this straight line. The same can be said about points where, of course, this means that the straight line y = x is the axis of symmetry of the hyperbola (as well as y = -x)

Example 1. Find the smallest and largest values ​​of the function a) on the segment ; b) on the segment [- 8, - 1].
Solution, a) Let's construct a graph of the function and select that part of it that corresponds to the values ​​of the variable x from the segment (Fig. 30). For the selected part of the graph we find:

b) Construct a graph of the function and select that part of it that corresponds to the values ​​of the variable x from segment[- 8, - 1] (Fig. 31). For the selected part of the graph we find:

So, we looked at the function for the case when k= 1. Now let k be a positive number different from 1, for example k = 2.

Let's look at the function and make a table of the values ​​of this function:

Let's construct points (1; 2), (2; 1), (-1; -2), (-2; -1),

on the coordinate plane (Fig. 32). They outline a certain line consisting of two branches; Let's carry it out (Fig. 33). Like the graph of a function, this line is called a hyperbola.

Let us now consider the case when k< 0; пусть, например, k = - 1. Построим график функции (здесь k = - 1).

In the previous paragraph, we noted that the graph of the function y = -f(x) is symmetrical to the graph of the function y = f(x) about the x-axis. In particular, this means that the graph of the function y = - f(x) is symmetrical to the graph of the function y = f(x) with respect to the x-axis. In particular, this means that schedule, is symmetrical to the graph relative to the x-axis (Fig. 34) Thus, we obtain a hyperbola, the branches of which are located in the second and fourth coordinate angles.

In general, the graph of the function is a hyperbola, the branches of which are located in the first and third coordinate angles if k > 0 (Fig. 33), and in the second and fourth coordinate angles if k< О (рис. 34). Точка (0; 0) - центр симметрии гиперболы, оси координат - асимптоты гиперболы.

It is usually said that two quantities x and y are inversely proportional if they are related by the relation xy = k (where k is a number other than 0), or, what is the same, . For this reason, the function is sometimes called inverse proportionality (by analogy with the function y - kx, which, as you probably know,
remember, it is called direct proportionality); number k - inverse coefficient proportionality.

Properties of the function for k > 0

Describing the properties of this function, we will rely on its geometric model - a hyperbola (see, Fig. 33).

2. y > 0 for x>0;y<0 при х<0.

3. The function decreases on the intervals (-°°, 0) and (0, +°°).

5. Neither the smallest nor the largest values ​​​​of a function

Properties of the function at k< 0
Describing the properties of this function, we will rely on its geometric model- hyperbole (see Fig. 34).

1. The domain of a function consists of all numbers except x = 0.

2. y > 0 at x< 0; у < 0 при х > 0.

3. The function increases on the intervals (-oo, 0) and (0, +oo).

4. The function is not limited either from below or from above.

5. The function has neither the smallest nor the largest values.

6. The function is continuous on the intervals (-oo, 0) and (0, +oo) and undergoes a discontinuity at x = 0.

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The function is written in general form as y = or f(x) =

y and x are inversely proportional quantities, i.e. when one increases, the other decreases (check by plugging numbers into the function)

Unlike the previous function, in which x 2 always creates positive values, here we cannot say that - = because they will be completely opposite numbers. Such functions are called odd.

For example, let's build a graph y =

Naturally, x cannot be equal to zero (x ≠ 0)

Branches hyperbolas lie in the 1st and 3rd parts of the coordinates.

They can endlessly approach the abscissa and ordinate axes and never reach them, even if “x” becomes equal to a billion. The hyperbola will be infinitely close, but still will not intersect with the axes (such a mathematical sadness).

Let's build a graph for y = -

​​​​​​​​​​​​​And now the branches of the hyperbola are in the second and 4th quarters of the coordinate plane.

As a result, complete symmetry can be observed between all branches.