The 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!

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. For registration tests I recommend using notebooks from the Arkhangelsk Pulp and Paper Mill (18 sheets, grid) 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: Seeing a rectangular coordinate system with the eyes analytical geometry 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 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 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 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.

I remind you that this is irrational number: , this will be required when constructing a graph, which, in fact, I will build without ceremony. Three points, perhaps that's 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:

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– 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 the above equations don't have a solution.

Definition: A numerical function is a correspondence that associates each number x from a given set singular y.

Designation:

where x is the independent variable (argument), y is the dependent variable (function). The set of values ​​of x is called the domain of the function (denoted D(f)). The set of values ​​of y is called the range of values ​​of the function (denoted E(f)). The graph of a function is the set of points in the plane with coordinates (x, f(x))

Methods for specifying a function.

1. analytical method (using a mathematical formula);
2. tabular method (using a table);
3. descriptive method (using verbal description);
4. graphical method (using a graph).

Basic properties of the function.

1. Even and odd

A function is called even if
– the domain of definition of the function is symmetrical about zero
f(-x) = f(x)

Schedule even function symmetrical about the axis 0y

A function is called odd if
– the domain of definition of the function is symmetrical about zero
– for any x from the domain of definition f(-x) = –f(x)

The graph of an odd function is symmetrical about the origin.

2. Frequency

A function f(x) is called periodic with period if for any x from the domain of definition f(x) = f(x+T) = f(x-T) .

The graph of a periodic function consists of unlimitedly repeating identical fragments.

3. Monotony (increasing, decreasing)

The function f(x) is increasing on the set P if for any x 1 and x 2 from this set such that x 1

The function f(x) decreases on the set P if for any x 1 and x 2 from this set, such that x 1 f(x 2) .

4. Extremes

The point X max is called the maximum point of the function f(x) if for all x from some neighborhood of X max the inequality f(x) f(X max) is satisfied.

The value Y max =f(X max) is called the maximum of this function.

X max – maximum point
At max - maximum

A point X min is called a minimum point of the function f(x) if for all x from some neighborhood of X min, the inequality f(x) f(X min) is satisfied.

The value Y min =f(X min) is called the minimum of this function.

X min – minimum point
Y min – minimum

X min , X max – extremum points
Y min , Y max – extrema.

5. Zeros of the function

The zero of a function y = f(x) is the value of the argument x at which the function becomes zero: f(x) = 0.

X 1, X 2, X 3 – zeros of the function y = f(x).

Tasks and tests on the topic "Basic properties of a function"

• Function Properties - Numerical functions 9th grade

  Lessons: 2 Assignments: 11 Tests: 1

• Properties of logarithms - Exponential and logarithmic functions grade 11

  Lessons: 2 Assignments: 14 Tests: 1

• Square root function, its properties and graph - Function square root. Properties of square root grade 8

  Lessons: 1 Assignments: 9 Tests: 1

• Functions - Important Topics for repeating the Unified State Examination in mathematics

  Tasks: 24

• Power functions, their properties and graphs - Degrees and roots. Power functions Grade 11

  Lessons: 4 Assignments: 14 Tests: 1

After studying this topic, you should be able to find the domain of definition various functions, determine intervals of monotonicity of a function using graphs, examine functions for evenness and oddness. Let's consider solving similar problems using the following examples.

Examples.

1. Find the domain of definition of the function.

Solution: the domain of definition of the function is found from the condition

therefore, the function f(x) is even.

Answer: even

D(f) = [-1; 1] – symmetrical about zero.

hence the function is neither even nor odd.

Answer: neither even nor uneven.

1) Function domain and function range.

The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Function zero is the value of the argument at which the value of the function is equal to zero.

3) Intervals of constant sign of a function.

Intervals of constant sign of a function are sets of argument values ​​on which the function values ​​are only positive or only negative.

4) Monotonicity of the function.

An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) function.

An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). The graph of an even function is symmetrical about the ordinate.

An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). The graph of an odd function is symmetrical about the origin.

6) Limited and unlimited functions.

A function is called bounded if there is such a positive number M such that |f(x)| ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited.

7) Periodicity of the function.

A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

19. Basic elementary functions, their properties and graphs. Application of functions in economics.

Basic elementary functions. Their properties and graphs

1. Linear function.

Linear function is called a function of the form , where x is a variable, a and b are real numbers.

Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.

Properties of a Linear Function

1. Domain of definition - the set of all real numbers: D(y)=R

2. The set of values ​​is the set of all real numbers: E(y)=R

3. The function takes a zero value when or.

4. The function increases (decreases) over the entire domain of definition.

5. A linear function is continuous over the entire domain of definition, differentiable and .

2. Quadratic function.

A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic

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