Given methodological material is for reference only and refers to to a wide circle topics The article provides an overview of graphs of basic elementary functions and discusses the most important question – 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!

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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) Label the axes in capital 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 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 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 notebooks are on sale, bad words not to mention complete rubbish. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! They save money on paper. For registration 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 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 the 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 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. Schedule 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 learned from 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 to move along the axis to the left (or right) to infinity, then the “games” will be 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 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 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 of definition:

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 will not consider the case, I don’t remember when last time I built a graph on this 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- the two are mutual 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 of definition: , that is, for any value of “x” there is a sine value.

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

In mathematics lessons at school, you have already become acquainted with the simplest properties and graph of a function y = x 2. Let's expand our knowledge on quadratic function.

Task 1.

Graph the function y = x 2. Scale: 1 = 2 cm. Mark a point on the Oy axis F(0; 1/4). Using a compass or strip of paper, measure the distance from the point F to some point M parabolas. Then pin the strip at point M and rotate it around that point until it is vertical. The end of the strip will fall slightly below the x-axis (Fig. 1). Mark on the strip how far it extends beyond the x-axis. Now take another point on the parabola and repeat the measurement again. How far has the edge of the strip fallen below the x-axis?

Result: no matter what point on the parabola y = x 2 you take, the distance from this point to the point F(0; 1/4) will be greater than the distance from the same point to the abscissa axis by always the same number - by 1/4.

We can say it differently: the distance from any point of the parabola to the point (0; 1/4) is equal to the distance from the same point of the parabola to the straight line y = -1/4. This wonderful point F(0; 1/4) is called focus parabolas y = x 2, and straight line y = -1/4 – headmistress this parabola. Every parabola has a directrix and a focus.

Interesting properties of a parabola:

1. Any point of the parabola is equidistant from some point, called the focus of the parabola, and some straight line, called its directrix.

2. If you rotate a parabola around the axis of symmetry (for example, the parabola y = x 2 around the Oy axis), you will get a very interesting surface called a paraboloid of revolution.

The surface of the liquid in a rotating vessel has the shape of a paraboloid of revolution. You can see this surface if you stir vigorously with a spoon in an incomplete glass of tea, and then remove the spoon.

3. If you throw a stone into the void at a certain angle to the horizon, it will fly in a parabola (Fig. 2).

4. If you intersect the surface of a cone with a plane parallel to any one of its generatrices, then the cross section will result in a parabola (Fig. 3).

5. Amusement parks sometimes have a fun ride called Paraboloid of Wonders. It seems to everyone standing inside the rotating paraboloid that he is standing on the floor, while the rest of the people are somehow miraculously holding on to the walls.

6. In reflecting telescopes, parabolic mirrors are also used: the light of a distant star, coming in a parallel beam, falling on the telescope mirror, is collected into focus.

7. Spotlights usually have a mirror in the shape of a paraboloid. If you place a light source at the focus of a paraboloid, then the rays, reflected from the parabolic mirror, form a parallel beam.

Graphing a Quadratic Function

In mathematics lessons, you studied how to obtain graphs of functions of the form from the graph of the function y = x 2:

1) y = ax 2– stretching the graph y = x 2 along the Oy axis in |a| times (with |a|< 0 – это сжатие в 1/|a| раз, rice. 4).

2) y = x 2 + n– shift of the graph by n units along the Oy axis, and if n > 0, then the shift is upward, and if n< 0, то вниз, (или же можно переносить ось абсцисс).

3) y = (x + m) 2– shift of the graph by m units along the Ox axis: if m< 0, то вправо, а если m >0, then left, (Fig. 5).

4) y = -x 2– symmetrical display relative to the Ox axis of the graph y = x 2 .

Let's take a closer look at plotting the function y = a(x – m) 2 + n.

A quadratic function of the form y = ax 2 + bx + c can always be reduced to the form

y = a(x – m) 2 + n, where m = -b/(2a), n = -(b 2 – 4ac)/(4a).

Let's prove it.

Really,

y = ax 2 + bx + c = a(x 2 + (b/a) x + c/a) =

A(x 2 + 2x · (b/a) + b 2 /(4a 2) – b 2 /(4a 2) + c/a) =

A((x + b/2a) 2 – (b 2 – 4ac)/(4a 2)) = a(x + b/2a) 2 – (b 2 – 4ac)/(4a).

Let us introduce new notations.

Let m = -b/(2a), A n = -(b 2 – 4ac)/(4a),

then we get y = a(x – m) 2 + n or y – n = a(x – m) 2.

Let's make some more substitutions: let y – n = Y, x – m = X (*).

Then we obtain the function Y = aX 2, the graph of which is a parabola.

The vertex of the parabola is at the origin. X = 0; Y = 0.

Substituting the coordinates of the vertex into (*), we obtain the coordinates of the vertex of the graph y = a(x – m) 2 + n: x = m, y = n.

Thus, in order to plot a quadratic function represented as

y = a(x – m) 2 + n

through transformations, you can proceed as follows:

a) plot the function y = x 2 ;

b) by parallel translation along the Ox axis by m units and along the Oy axis by n units - transfer the vertex of the parabola from the origin to the point with coordinates (m; n) (Fig. 6).

Recording transformations:

y = x 2 → y = (x – m) 2 → y = a(x – m) 2 → y = a(x – m) 2 + n.

Example.

Using transformations, construct a graph of the function y = 2(x – 3) 2 in the Cartesian coordinate system – 2.

Solution.

Chain of transformations:

y = x 2 (1) → y = (x – 3) 2 (2) → y = 2(x – 3) 2 (3) → y = 2(x – 3) 2 – 2 (4) .

The plotting is shown in rice. 7.

You can practice graphing quadratic functions on your own. For example, build a graph of the function y = 2(x + 3) 2 + 2 in one coordinate system using transformations. If you have any questions or want to get advice from a teacher, then you have the opportunity to conduct free 25 minute lesson with online tutor after registration . To further work with the teacher, you can choose the tariff plan that suits you.

Still have questions? Don't know how to graph a quadratic function?
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- — [] quadratic function Function of the form y= ax2 + bx + c (a ? 0). Graph K.f. - a parabola, the vertex of which has coordinates [ b/ 2a, (b2 4ac) / 4a], with a>0 branches of the parabola ... ...

QUADRATIC FUNCTION, a mathematical FUNCTION whose value depends on the square of the independent variable, x, and is given, respectively, by a quadratic POLYNOMIAL, for example: f(x) = 4x2 + 17 or f(x) = x2 + 3x + 2. see also SQUARE EQUATION... Scientific and technical encyclopedic dictionary

Quadratic function- Quadratic function - a function of the form y= ax2 + bx + c (a ≠ 0). Graph K.f. - a parabola, the vertex of which has coordinates [ b/ 2a, (b2 4ac) / 4a], for a> 0 the branches of the parabola are directed upward, for a< 0 –вниз… …

- (quadratic) Function having the following form: y=ax2+bx+c, where a≠0 and highest degree x is a square. Quadratic equation y=ax2 +bx+c=0 can also be solved using the following formula: x= –b+ √ (b2–4ac) /2a. These roots are real... Economic dictionary

An affine quadratic function on an affine space S is any function Q: S→K that has the form Q(x)=q(x)+l(x)+c in vectorized form, where q is a quadratic function, l is a linear function, c is a constant. Contents 1 Shifting the reference point 2 ... ... Wikipedia

An affine quadratic function on an affine space is any function that has the form in vectorized form, where is a symmetric matrix, a linear function, a constant. Contents... Wikipedia

A function on a vector space defined by a homogeneous polynomial of the second degree in the coordinates of the vector. Contents 1 Definition 2 Related definitions... Wikipedia

- is a function that, in the theory of statistical decisions, characterizes losses due to incorrect decision-making based on observed data. If the problem of estimating a signal parameter against a background of noise is being solved, then the loss function is a measure of the discrepancy... ... Wikipedia

objective function- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] objective function In extremal problems, a function whose minimum or maximum is required to be found. This… … Technical Translator's Guide

Objective function- in extremal problems, a function whose minimum or maximum must be found. This is a key concept in optimal programming. Having found the extremum of C.f. and, therefore, having determined the values ​​of the controlled variables that go to it... ... Economic-mathematical dictionary

Books

• Set of tables. Mathematics. Graphs of functions (10 tables), . Educational album of 10 sheets. Linear function. Graphical and analytical assignment of functions. Quadratic function. Transforming the graph of a quadratic function. Function y=sinx. Function y=cosx.…
• The most important function of school mathematics is quadratic - in problems and solutions, Petrov N.N.. The quadratic function is the main function school course mathematics. This is not surprising. On the one hand, the simplicity of this function, and on the other, the deep meaning. Many tasks of school...

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