Elementary functions and their graphs

Straight proportionality. Linear function.

Inverse proportionality. Hyperbola.

Quadratic function. Square parabola.

Power function. Exponential function.

Logarithmic function. Trigonometric functions.

Inverse trigonometric functions.

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

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

Main characteristics and properties of a square parabola:

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

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

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

behaves as monotonous;

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

and non-periodic;

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

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

Functions y= Arcin x(Fig.23) and y= Arccos x(Fig.24) multi-valued, unlimited; their domain of definition and range of values, respectively: 1 x+1 and  < y+ . Since these functions are multi-valued, do not

considered in elementary mathematics, their main values ​​are considered as inverse trigonometric functions: y= arcsin x And y= arccos x; their graphs are highlighted in Fig. 23 and Fig. 24 with thick lines.

Functions y= arcsin x And y= arccos x have the following characteristics and properties:

Both functions have the same domain of definition: 1 x +1 ;

their ranges: /2 y/2 for y= arcsin x and 0 y For y= arccos x;

(y= arcsin x– increasing function; y= arccos x – decreasing);

Each function has one zero ( x= 0 for function y= arcsin x And

x= 1 for function y= arccos x).

Functions y= Arctan x(Fig.25) and y= Arccot x (Fig.26) - multi-valued, unlimited functions; their domain of definition:  x+ . Their main meanings y= arctan x And y= arccot x are considered as inverse trigonometric functions; their graphs are highlighted in Fig. 25 and Fig. 26 with bold branches.

Functions y= arctan x And y= arccot x have the following characteristics and properties:

Both functions have the same domain of definition:  x + ;

their ranges: /2 <y < /2 для y= arctan x and 0< y < для y= arccos x;

Functions are limited, non-periodic, continuous and monotonic

(y= arctan x– increasing function; y= arccot x – decreasing);

Function only y= arctan x has a single zero ( x = 0);

function y = arccot x has no zeros.

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 plotted 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 downwards.

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 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 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 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 “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.

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 greater than zero and the functions are increasing. Moreover, than more value 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 different meanings functions, 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):

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 page of the site is Google Chrome. Correct operation is not guaranteed when using other browsers.

Basic elementary functions are: constant function (constant), root n-th degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

Permanent function.

A constant function is given on the set of all real numbers by the formula , where C– some real number. A constant function associates each actual value of the independent variable x same value of the dependent variable y- meaning WITH. A constant function is also called a constant.

The graph of a constant function is a straight line parallel to the x-axis and passing through the point with coordinates (0,C). For example, let's show graphs of constant functions y=5,y=-2 and , which in the figure below correspond to the black, red and blue lines, respectively.

Properties of a constant function.

Domain: the entire set of real numbers.

The constant function is even.

Range of values: set consisting of a singular number WITH.

A constant function is non-increasing and non-decreasing (that’s why it’s constant).

It makes no sense to talk about convexity and concavity of a constant.

There are no asymptotes.

The function passes through the point (0,C) coordinate plane.

Root of the nth degree.

Let's consider the basic elementary function, which is given by the formula, where n– a natural number greater than one.

The nth root, n is an even number.

Let's start with the root function n-th power for even values ​​of the root exponent n.

As an example, here is a picture with images of function graphs and , they correspond to black, red and blue lines.

The graphs of even-degree root functions have a similar appearance for other values ​​of the exponent.

Properties of the root functionn -th power for evenn .

The nth root, n is an odd number.

Root function n-th power with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs and , they correspond to black, red and blue curves.

For other odd values ​​of the root exponent, the function graphs will have a similar appearance.

Properties of the root functionn -th power for oddn .