Basic elementary functions: their properties and graphs. Online graphing How to determine the type of function

A linear function is a function of the form y=kx+b, where x is the independent variable, k and b are any numbers.
The graph of a linear function is a straight line.

1. To plot a function graph, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the function equation, and use them to calculate the corresponding y values.

For example, to plot the function y= x+2, it is convenient to take x=0 and x=3, then the ordinates of these points will be equal to y=2 and y=3. We get points A(0;2) and B(3;3). Let's connect them and get a graph of the function y= x+2:

2. In the formula y=kx+b, the number k is called the proportionality coefficient:
if k>0, then the function y=kx+b increases
if k
Coefficient b shows the displacement of the function graph along the OY axis:
if b>0, then the graph of the function y=kx+b is obtained from the graph of the function y=kx by shifting b units upward along the OY axis
if b
The figure below shows the graphs of the functions y=2x+3; y= ½ x+3; y=x+3

Note that in all these functions the coefficient k Above zero, and the functions are increasing. Moreover, the greater the value of k, the greater the angle of inclination of the straight line to the positive direction of the OX axis.

In all functions b=3 - and we see that all graphs intersect the OY axis at point (0;3)

Now consider the graphs of the functions y=-2x+3; y=- ½ x+3; y=-x+3

This time in all functions the coefficient k less than zero and functions are decreasing. Coefficient b=3, and the graphs, as in the previous case, intersect the OY axis at point (0;3)

Consider the graphs of the functions y=2x+3; y=2x; y=2x-3

Now in all function equations the coefficients k are equal to 2. And we got three parallel lines.

But the coefficients b are different, and these graphs intersect the OY axis at different points:
The graph of the function y=2x+3 (b=3) intersects the OY axis at point (0;3)
The graph of the function y=2x (b=0) intersects the OY axis at the point (0;0) - the origin.
The graph of the function y=2x-3 (b=-3) intersects the OY axis at point (0;-3)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function y=kx+b looks like.
If k 0

If k>0 and b>0, then the graph of the function y=kx+b looks like:

If k>0 and b, then the graph of the function y=kx+b looks like:

If k, then the graph of the function y=kx+b looks like:

If k=0, then the function y=kx+b turns into the function y=b and its graph looks like:

The ordinates of all points on the graph of the function y=b are equal to b If b=0, then the graph of the function y=kx (direct proportionality) passes through the origin:

3. Let us separately note the graph of the equation x=a. The graph of this equation is a straight line parallel to the OY axis, all points of which have an abscissa x=a.

For example, the graph of the equation x=3 looks like this:
Attention! The equation x=a is not a function, so one value of the argument corresponds to different values ​​of the function, which does not correspond to the definition of a function.

4. Condition for parallelism of two lines:

The graph of the function y=k 1 x+b 1 is parallel to the graph of the function y=k 2 x+b 2 if k 1 =k 2

5. The condition for two straight lines to be perpendicular:

The graph of the function y=k 1 x+b 1 is perpendicular to the graph of the function y=k 2 x+b 2 if k 1 *k 2 =-1 or k 1 =-1/k 2

6. Points of intersection of the graph of the function y=kx+b with the coordinate axes.

With OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero in the equation of the function instead of x. We get y=b. That is, the point of intersection with the OY axis has coordinates (0; b).

With OX axis: The ordinate of any point belonging to the OX axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero in the equation of the function instead of y. We get 0=kx+b. Hence x=-b/k. That is, the point of intersection with the OX axis has coordinates (-b/k;0):

First, try to find the domain of the function:

Did you manage? Let's compare the answers:

Is everything right? Well done!

Now let's try to find the range of values ​​of the function:

Found? Let's compare:

Got it? Well done!

Let's work with graphs again, only now it's a little more complicated - find both the domain of definition of the function and the range of values ​​of the function.

How to find both the domain and range of a function (advanced)

Here's what happened:

I think you've figured out the graphs. Now let’s try to find the domain of definition of a function in accordance with the formulas (if you don’t know how to do this, read the section about):

Did you manage? Let's check answers:

1. , since the radical expression must be greater than or equal to zero.
2. , since you cannot divide by zero and the radical expression cannot be negative.
3. , since, respectively, for all.
4. , since you cannot divide by zero.

However, we still have one more unanswered point...

I will repeat the definition once again and emphasize it:

Did you notice? The word "single" is a very, very important element of our definition. I'll try to explain it to you with my fingers.

Let's say we have a function defined by a straight line. . At, we substitute this value into our “rule” and get that. One value corresponds to one value. We can even make a table of the different values ​​and graph this function to see for ourselves.

"Look! - you say, ““ occurs twice!” So maybe a parabola is not a function? No, it is!

The fact that “ ” appears twice is not a reason to accuse the parabola of ambiguity!

The fact is that, when calculating for, we received one game. And when calculating with, we received one game. So that's right, a parabola is a function. Look at the graph:

Got it? If not, here is a life example that is very far from mathematics!

Let's say we have a group of applicants who met while submitting documents, each of whom told in a conversation where he lives:

Agree, it is quite possible for several guys to live in one city, but it is impossible for one person to live in several cities at the same time. This is like a logical representation of our “parabola” - Several different X's correspond to the same game.

Now let's come up with an example where the dependency is not a function. Let’s say these same guys told us what specialties they applied for:

Here we have a completely different situation: one person can easily submit documents for one or several directions. That is one element sets are put into correspondence several elements multitudes. Respectively, this is not a function.

Let's test your knowledge in practice.

Determine from the pictures what is a function and what is not:

Got it? And here it is answers:

• The function is - B, E.
• The function is not - A, B, D, D.

You ask why? Yes, here's why:

In all pictures except IN) And E) There are several for one!

I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the range of permissible values ​​of an argument and the range of definition of a function. Let's move on to the next section - how to set a function?

Methods for specifying a function

What do you think the words mean? "set function"? That's right, this means explaining to everyone what function we are talking about in this case. Moreover, explain it in such a way that everyone understands you correctly and the function graphs drawn by people based on your explanation are the same.

How can I do that? How to set a function? The simplest method, which has already been used more than once in this article, is using the formula. We write a formula, and by substituting a value into it, we calculate the value. And as you remember, a formula is a law, a rule by which it becomes clear to us and to another person how an X turns into a Y.

Usually, this is exactly what they do - in tasks we see ready-made functions specified by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” baffles. Let's understand everything in order, and let's start with the analytical method.

Analytical method of specifying a function

The analytical method is to specify a function using a formula. This is the most universal, comprehensive and unambiguous method. If you have a formula, then you know absolutely everything about a function - you can make a table of values ​​​​from it, you can build a graph, determine where the function increases and where it decreases, in general, study it in full.

Let's consider the function. What's the difference?

"What does it mean?" - you ask. I'll explain now.

Let me remind you that in the notation the expression in brackets is called an argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (the expression in brackets) is, we will write it instead in the expression.

In our example it will look like this:

Let's consider another task related to the analytical method of specifying a function, which you will have on the exam.

Find the value of the expression at.

I'm sure that at first you were scared when you saw such an expression, but there is absolutely nothing scary about it!

Everything is the same as in the previous example: whatever the argument (the expression in brackets) is, we will write it instead in the expression. For example, for a function.

What needs to be done in our example? Instead you need to write, and instead -:

shorten the resulting expression:

That's all!

Independent work

Now try to find the meaning of the following expressions yourself:

1. , If
2. , If

Did you manage? Let's compare our answers: We are used to the fact that the function has the form

Even in our examples, we define the function in exactly this way, but analytically it is possible to specify the function in an implicit form, for example.

Try building this function yourself.

Did you manage?

This is how I built it.

What equation did we finally derive?

Right! Linear, which means that the graph will be a straight line. Let's make a table to determine which points belong to our line:

This is exactly what we were talking about... One corresponds to several.

Let's try to draw what happened:

Is what we got a function?

That's right, no! Why? Try to answer this question with the help of a drawing. What did you get?

“Because one value corresponds to several values!”

What conclusion can we draw from this?

That's right, a function cannot always be expressed explicitly, and what is “disguised” as a function is not always a function!

Tabular method of specifying a function

As the name suggests, this method is a simple sign. Yes Yes. Like the one you and I have already made. For example:

Here you immediately noticed a pattern - the Y is three times larger than the X. And now the task to “think very carefully”: do you think that a function given in the form of a table is equivalent to a function?

Let's not talk for a long time, but let's draw!

So. We draw the function specified by the wallpaper in the following ways:

Do you see the difference? It's not all about the marked points! Take a closer look:

Have you seen it now? When we define a function in a tabular way, we display on the graph only those points that we have in the table and the line (as in our case) passes only through them. When we define a function analytically, we can take any points, and our function is not limited to them. This is the peculiarity. Remember!

Graphical method of constructing a function

The graphical method of constructing a function is no less convenient. We draw our function, and another interested person can find what y is equal to at a certain x and so on. Graphical and analytical methods are among the most common.

However, here you need to remember what we talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Do you remember? Just in case, I’ll copy here the definition of what a function is:

As a rule, people usually name exactly the three ways of specifying a function that we have discussed - analytical (using a formula), tabular and graphical, completely forgetting that a function can be described verbally. Like this? Yes, very simple!

Verbal description of the function

How to describe a function verbally? Let's take our recent example - . This function can be described as “every real value of x corresponds to its triple value.” That's all. Nothing complicated. You, of course, will object - “there are such complex functions that it is simply impossible to specify verbally!” Yes, there are such, but there are functions that are easier to describe verbally than to define with a formula. For example: “each natural value of x corresponds to the difference between the digits of which it consists, while the minuend is taken to be the largest digit contained in the notation of the number.” Now let's look at how our verbal description of the function is implemented in practice:

The largest digit in a given number is, respectively, the minuend, then:

Main types of functions

Now let's move on to the most interesting part - let's look at the main types of functions with which you have worked/are working and will work in the course of school and college mathematics, that is, let's get to know them, so to speak, and give them a brief description. Read more about each function in the corresponding section.

Linear function

A function of the form where, are real numbers.

The graph of this function is a straight line, so constructing a linear function comes down to finding the coordinates of two points.

The position of the straight line on the coordinate plane depends on the angular coefficient.

The scope of a function (aka the scope of valid argument values) is .

Range of values ​​- .

Quadratic function

Function of the form, where

The graph of the function is a parabola; when the branches of the parabola are directed downwards, when the branches are directed upwards.

Many properties of a quadratic function depend on the value of the discriminant. The discriminant is calculated using the formula

The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:

Domain

The range of values ​​depends on the extremum of the given function (vertex point of the parabola) and the coefficient (direction of the branches of the parabola)

Inverse proportionality

The function given by the formula, where

The number is called the coefficient of inverse proportionality. Depending on the value, the branches of the hyperbola are in different squares:

Domain - .

Range of values ​​- .

SUMMARY AND BASIC FORMULAS

1. A function is a rule according to which each element of a set is associated with a single element of the set.

• - this is a formula denoting a function, that is, the dependence of one variable on another;
• - variable value, or argument;
• - dependent quantity - changes when the argument changes, that is, according to any specific formula reflecting the dependence of one quantity on another.

2. Valid argument values, or the domain of a function, is what is associated with the possibilities in which the function makes sense.

3. Function range- this is what values ​​it takes, given acceptable values.

4. There are 4 ways to set a function:

• analytical (using formulas);
• tabular;
• graphic
• verbal description.

5. Main types of functions:

• : , where, are real numbers;
• : , Where;
• : , Where.

Build function

We offer to your attention a service for constructing graphs of functions online, all rights to which belong to the company Desmos. Use the left column to enter functions. You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the window with the graph, you can hide both the left column and the virtual keyboard.

Benefits of online charting

• Visual display of entered functions
• Building very complex graphs
• Construction of graphs specified implicitly (for example, ellipse x^2/9+y^2/16=1)
• The ability to save charts and receive a link to them, which becomes available to everyone on the Internet
• Control of scale, line color
• Possibility of plotting graphs by points, using constants
• Plotting several function graphs simultaneously
• Plotting in polar coordinates (use r and θ(\theta))

With us it’s easy to build charts of varying complexity online. Construction is done instantly. The service is in demand for finding intersection points of functions, for depicting graphs for further moving them into a Word document as illustrations when solving problems, and for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this website page is Google Chrome. Correct operation is not guaranteed when using other browsers.

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 to 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 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 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 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 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 bounded from above: although slowly, 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.