Schoolchildren are faced with the task of constructing a graph of a function at the very beginning of studying algebra and continue to build them year after year. Starting from the graph of a linear function, for which you need to know only two points, to a parabola, which already requires 6 points, a hyperbola and a sine wave. Every year the functions become more and more complex and it is no longer possible to construct their graphs using a template; it is necessary to carry out more complex studies using derivatives and limits.

Let's figure out how to find the graph of a function? To do this, let's start with the simplest functions, the graphs of which are plotted point by point, and then consider a plan for constructing more complex functions.

Graphing a Linear Function

To construct the simplest graphs, use a table of function values. The graph of a linear function is a straight line. Let's try to find the points on the graph of the function y=4x+5.

1. To do this, let’s take two arbitrary values ​​of the variable x, substitute them one by one into the function, find the value of the variable y and enter everything into the table.
2. Take the value x=0 and substitute it into the function instead of x - 0. We get: y=4*0+5, that is, y=5, write this value in the table under 0. Similarly, take x=0, we get y=4*1+5 , y=9.
3. Now, to build a graph of the function, you need to plot these points on the coordinate plane. Then you need to draw a straight line.

Graphing a Quadratic Function

A quadratic function is a function of the form y=ax 2 +bx +c, where x is a variable, a,b,c are numbers (a is not equal to 0). For example: y=x 2, y=x 2 +5, y=(x-3) 2, y=2x 2 +3x+5.

To construct the simplest quadratic function y=x 2, 5-7 points are usually taken. Let's take the values ​​for the variable x: -2, -1, 0, 1, 2 and find the values ​​of y in the same way as when constructing the first graph.

The graph of a quadratic function is called a parabola. After constructing graphs of functions, students have new tasks related to the graph.

Example 1: find the abscissa of the graph point of the function y=x 2 if the ordinate is 9. To solve the problem, you need to substitute its value 9 into the function instead of y. We get 9=x 2 and solve this equation. x=3 and x=-3. This can also be seen on the graph of the function.

Studying a function and plotting its graph

To plot graphs of more complex functions, it is necessary to perform several steps aimed at studying it. To do this you need:

1. Find the domain of definition of the function. The domain of definition is all the values ​​that the variable x can take. Those points at which the denominator becomes 0 or the radical expression becomes negative should be excluded from the definition domain.
2. Set whether the function is even or odd. Recall that an even function is one that meets the condition f(-x)=f(x). Its graph is symmetrical with respect to Oy. A function will be odd if it meets the condition f(-x)=-f(x). In this case, the graph is symmetrical about the origin.
3. Find the points of intersection with the coordinate axes. In order to find the abscissa of the point of intersection with the Ox axis, it is necessary to solve the equation f(x) = 0 (the ordinate is equal to 0). To find the ordinate of the point of intersection with the Oy axis, it is necessary to substitute 0 in the function instead of the variable x (the abscissa is 0).
4. Find the asymptotes of the function. An asyptote is a straight line that the graph approaches indefinitely, but never crosses. Let's figure out how to find the asymptotes of the graph of a function.
   • Vertical asymptote of the straight line x=a
   • Horizontal asymptote - straight line y=a
   • Oblique asymptote - straight line of the form y=kx+b
5. Find the extremum points of the function, the intervals of increase and decrease of the function. Let's find the extremum points of the function. To do this, you need to find the first derivative and equate it to 0. It is at these points that the function can change from increasing to decreasing. Let us determine the sign of the derivative on each interval. If the derivative is positive, then the graph of the function increases; if it is negative, it decreases.
6. Find the inflection points of the function graph, the upward and downward convexity intervals.

Finding inflection points is now easier than ever. You just need to find the second derivative, then equate it to zero. Next we find the sign of the second derivative on each interval. If it is positive, then the graph of the function is convex downward, if it is negative, it is convex upward.

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 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 point-by-point construction table, 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.

Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication tables. They are like the foundation, everything is based on them, everything is built from them and everything comes down to them.

In this article we will list all the main elementary functions, provide their graphs and give without conclusion or proof properties of basic elementary functions according to the scheme:

• behavior of a function at the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of discontinuity points of a function);
• even and odd;
• intervals of convexity (convexity upward) and concavity (convexity downward), inflection points (if necessary, see the article convexity of a function, direction of convexity, inflection points, conditions of convexity and inflection);
• oblique and horizontal asymptotes;
• singular points of functions;
• special properties of some functions (for example, the smallest positive period of trigonometric functions).

If you are interested in or, then you can go to these sections of the theory.

Basic elementary functions are: constant function (constant), nth root, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

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Permanent function.

A constant function is defined on the set of all real numbers by the formula , where C is some real number. A constant function associates each real value of the independent variable x with the same value of the dependent variable y - the value C. 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: a set consisting of the singular number C.
• 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) of the coordinate plane.

Root of the nth degree.

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

Root of the nth degree, n is an even number.

Let's start with the nth root function 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 nth root function for even n.

The nth root, n is an odd number.

The nth root function 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 nth root function for odd n.

Power function.

The power function is given by a formula of the form .

Let's consider the form of graphs of a power function and the properties of a power function depending on the value of the exponent.

Let's start with a power function with an integer exponent a. In this case, the type of graphs of power functions and the properties of the functions depend on the evenness or oddness of the exponent, as well as on its sign. Therefore, we will first consider power functions for odd positive values ​​of the exponent a, then for even positive exponents, then for odd negative exponents, and finally, for even negative a.

The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, for a from zero to one, secondly, for a greater than one, thirdly, for a from minus one to zero, fourthly, for a less than minus one.

At the end of this section, for completeness, we will describe a power function with zero exponent.

Power function with odd positive exponent.

Let's consider a power function with an odd positive exponent, that is, with a = 1,3,5,....

The figure below shows graphs of power functions - black line, - blue line, - red line, - green line. For a=1 we have linear function y=x.

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

Let's consider a power function with an even positive exponent, that is, for a = 2,4,6,....

As an example, we give graphs of power functions – black line, – blue line, – red line. For a=2 we have a quadratic function, the graph of which is quadratic parabola.

Properties of a power function with an even positive exponent.

Power function with odd negative exponent.

Look at the graphs of the power function for odd negative values ​​of the exponent, that is, for a = -1, -3, -5,....

The figure shows graphs of power functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.

Properties of a power function with an odd negative exponent.

Power function with even negative exponent.

Let's move on to the power function for a=-2,-4,-6,….

The figure shows graphs of power functions – black line, – blue line, – red line.

Properties of a power function with an even negative exponent.

A power function with a rational or irrational exponent whose value is greater than zero and less than one.

Pay attention! If a is a positive fraction with an odd denominator, then some authors consider the domain of definition of the power function to be the interval. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. We will adhere to precisely this view, that is, we will consider the set to be the domains of definition of power functions with fractional positive exponents. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.

Let us consider a power function with a rational or irrational exponent a, and .

Let us present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).

A power function with a non-integer rational or irrational exponent greater than one.

Let us consider a power function with a non-integer rational or irrational exponent a, and .

Let us present graphs of power functions given by the formulas (black, red, blue and green lines respectively).

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

Properties of the power function at .

A power function with a real exponent that is greater than minus one and less than zero.

Pay attention! If a is a negative fraction with an odd denominator, then some authors consider the domain of definition of a power function to be the interval . It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. We will adhere to precisely this view, that is, we will consider the domains of definition of power functions with fractional fractional negative exponents to be a set, respectively. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.

Let's move on to the power function, kgod.

To have a good idea of ​​the form of graphs of power functions for , we give examples of graphs of functions (black, red, blue and green curves, respectively).

Properties of a power function with exponent a, .

A power function with a non-integer real exponent that is less than minus one.

Let us give examples of graphs of power functions for , they are depicted by black, red, blue and green lines, respectively.

Properties of a power function with a non-integer negative exponent less than minus one.

When a = 0 and we have a function - this is a straight line from which the point (0;1) is excluded (it was agreed not to attach any significance to the expression 0 0).

Exponential function.

One of the main elementary functions is the exponential function.

The graph of the exponential function, where and takes different forms depending on the value of the base a. Let's figure this out.

First, consider the case when the base of the exponential function takes a value from zero to one, that is, .

As an example, we present graphs of the exponential function for a = 1/2 – blue line, a = 5/6 – red line. The graphs of the exponential function have a similar appearance for other values ​​of the base from the interval.

Properties of an exponential function with a base less than one.

Let us move on to the case when the base of the exponential function is greater than one, that is, .

As an illustration, we present graphs of exponential functions - blue line and - red line. For other values ​​of the base greater than one, the graphs of the exponential function will have a similar appearance.

Properties of an exponential function with a base greater than one.

Logarithmic function.

The next basic elementary function is the logarithmic function, where , . The logarithmic function is defined only for positive values ​​of the argument, that is, for .

The graph of a logarithmic function takes different forms depending on the value of the base a.

Power function. This is the function: y = axn, 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 non-zero number is 1, therefore, at n= 0 the power function turns into a constant value:y = a, i.e. her schedule is straight line parallel to the axisX, excluding the origin (please explain Why ? ). All these cases (with a= 1 ) shown in Fig. 13 (n 0) and Fig. 14 ( n < 0). Отрицательные значения xare not considered here, so like then some functions:

If n– integer, power functions make sense even whenx< 0, но их графики имеют различный вид в зависимости от того, является ли neven or odd number. Figure 15 shows two such power functions: For n= 2 and n = 3.


At n= 2 function is even andits graph is symmetrical relative to the axis Y. At n= 3 the function is odd and its graph is symmetrical relative to the origin coordinates Functiony = x 3 called cubic parabola.

Figure 16 shows the function. This function is inverse to a 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 method of obtaining 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.

The coordinate of absolutely any point on the plane is determined by its two quantities: along the abscissa axis and the ordinate axis. The collection of many such points represents the graph of the function. From it you can see how the Y value changes depending on the change in the X value. You can also determine in which section (interval) the function increases and in which it decreases.

Instructions

• What can you say about a function if its graph is a straight line? See if this line passes through the coordinate origin point (that is, the one where the X and Y values ​​​​are equal to 0). If it passes, then such a function is described by the equation y = kx. It is easy to understand that the greater the value of k, the closer to the ordinate axis this straight line will be located. And the Y axis itself actually corresponds to an infinitely large value of k.
• Look at the direction of the function. If it goes “from bottom left to top right,” that is, through the 3rd and 1st coordinate quarters, it is increasing, but if it goes “from top left to bottom right” (through the 2nd and 4th quarters), then it decreasing.
• When a line does not pass through the origin, it is described by the equation y = kx + b. The straight line intersects the y-axis at the point where y = b, and the value of y can be either positive or negative.
• A function is called a parabola if it is described by the equation y = x^n, and its form depends on the value of n. If n is any even number (the simplest case is a quadratic function y = x^2), the graph of the function is a curve passing through the origin point, as well as through points with coordinates (1;1), (-1;1), since a unit to any degree will remain a unit. All y values ​​corresponding to any nonzero X values ​​can only be positive. The function is symmetrical about the Y axis, and its graph is located in the 1st and 2nd coordinate quarters. It is easy to understand that the larger the value of n, the closer the graph will be to the Y axis.
• If n is an odd number, the graph of this function is a cubic parabola. The curve is located in the 1st and 3rd coordinate quarters, is symmetrical about the Y axis and passes through the origin of coordinates, as well as through the points (-1;-1), (1;1). When the quadratic function is the equation y = ax^2 + bx + c, the shape of the parabola is the same as in the simplest case (y = x^2), but its vertex is not at the origin.
• A function is called a hyperbola if it is described by the equation y = k/x. You can easily see that as the value of x tends to 0, the value of y increases to infinity. The graph of a function is a curve consisting of two branches and located in different coordinate quarters.