We choose a rectangular coordinate system on the plane and plot the values ​​of the argument on the abscissa axis X, and on the y-axis - the values ​​of the function y = f(x).

Function Graph y = f(x) the set of all points is called, for which the abscissas belong to the domain of the function, and the ordinates are equal to the corresponding values ​​of the function.

In other words, the graph of the function y \u003d f (x) is the set of all points in the plane, the coordinates X, at which satisfy the relation y = f(x).

On fig. 45 and 46 are graphs of functions y = 2x + 1 and y \u003d x 2 - 2x.

Strictly speaking, one should distinguish between the graph of a function (exact mathematical definition which was given above) and the drawn curve, which always gives only a more or less accurate sketch of the graph (and even then, as a rule, not the entire graph, but only its part located in the final part of the plane). In what follows, however, we will usually refer to "chart" rather than "chart sketch".

Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the scope of the function y = f(x), then to find the number f(a)(i.e. the function values ​​at the point x = a) should do so. Need through a dot with an abscissa x = a draw a straight line parallel to the y-axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will be, by virtue of the definition of the graph, equal to f(a)(Fig. 47).

For example, for the function f(x) = x 2 - 2x using the graph (Fig. 46) we find f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0, etc.

A function graph visually illustrates the behavior and properties of a function. For example, from a consideration of Fig. 46 it is clear that the function y \u003d x 2 - 2x takes positive values ​​when X< 0 and at x > 2, negative - at 0< x < 2; smallest value function y \u003d x 2 - 2x accepts at x = 1.

To plot a function f(x) you need to find all points of the plane, coordinates X,at which satisfy the equation y = f(x). In most cases, this is impossible, since there are infinitely many such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the multi-point plotting method. It consists in the fact that the argument X attach finite number values ​​- say, x 1 , x 2 , x 3 ,..., x k and make up a table that includes the selected values ​​of the function.

The table looks like this:

Having compiled such a table, we can outline several points on the graph of the function y = f(x). Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y = f(x).

However, it should be noted that the multi-point plotting method is very unreliable. In fact, the behavior of the graph between the marked points and its behavior outside the segment between the extreme points taken remains unknown.

Example 1. To plot a function y = f(x) someone compiled a table of argument and function values:

The corresponding five points are shown in Fig. 48.

Based on the location of these points, he concluded that the graph of the function is a straight line (shown in Fig. 48 by a dotted line). Can this conclusion be considered reliable? Unless there are additional considerations to support this conclusion, it can hardly be considered reliable. reliable.

To substantiate our assertion, consider the function

.

Calculations show that the values ​​of this function at points -2, -1, 0, 1, 2 are just described by the table above. However, the graph of this function is not at all a straight line (it is shown in Fig. 49). Another example is the function y = x + l + sinx; its meanings are also described in the table above.

These examples show that in its "pure" form, the multi-point plotting method is unreliable. Therefore, to plot a given function, as a rule, proceed as follows. First, the properties of this function are studied, with the help of which it is possible to construct a sketch of the graph. Then, by calculating the values ​​of the function at several points (the choice of which depends on the set properties of the function), the corresponding points of the graph are found. And, finally, a curve is drawn through the constructed points using the properties of this function.

We will consider some (the most simple and frequently used) properties of functions used to find a sketch of a graph later, but now we will analyze some commonly used methods for plotting graphs.

Graph of the function y = |f(x)|.

It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Recall how this is done. A-priory absolute value numbers can be written

This means that the graph of the function y=|f(x)| can be obtained from the graph, functions y = f(x) as follows: all points of the graph of the function y = f(x), whose ordinates are non-negative, should be left unchanged; further, instead of the points of the graph of the function y = f(x), having negative coordinates, one should construct the corresponding points of the graph of the function y = -f(x)(i.e. part of the function graph
y = f(x), which lies below the axis X, should be reflected symmetrically about the axis X).

Example 2 Plot a function y = |x|.

We take the graph of the function y = x(Fig. 50, a) and part of this graph when X< 0 (lying under the axis X) is symmetrically reflected about the axis X. As a result, we get the graph of the function y = |x|(Fig. 50, b).

Example 3. Plot a function y = |x 2 - 2x|.

First we plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upwards, the top of the parabola has coordinates (1; -1), its graph intersects the abscissa axis at points 0 and 2. On the interval (0; 2) the function takes negative values, therefore this part of the graph reflect symmetrically about the x-axis. Figure 51 shows a graph of the function y \u003d |x 2 -2x |, based on the graph of the function y = x 2 - 2x

Graph of the function y = f(x) + g(x)

Consider the problem of plotting the function y = f(x) + g(x). if graphs of functions are given y = f(x) and y = g(x).

Note that the domain of the function y = |f(x) + g(x)| is the set of all those values ​​of x for which both functions y = f(x) and y = g(x) are defined, i.e. this domain of definition is the intersection of the domains of definition, the functions f(x) and g(x).

Let the points (x 0, y 1) and (x 0, y 2) respectively belong to the function graphs y = f(x) and y = g(x), i.e. y 1 \u003d f (x 0), y 2 \u003d g (x 0). Then the point (x0;. y1 + y2) belongs to the graph of the function y = f(x) + g(x)(for f(x 0) + g(x 0) = y 1+y2),. and any point of the graph of the function y = f(x) + g(x) can be obtained in this way. Therefore, the graph of the function y = f(x) + g(x) can be obtained from function graphs y = f(x). and y = g(x) by replacing each point ( x n, y 1) function graphics y = f(x) dot (x n, y 1 + y 2), where y 2 = g(x n), i.e., by shifting each point ( x n, y 1) function graph y = f(x) along the axis at by the amount y 1 \u003d g (x n). In this case, only such points are considered. X n for which both functions are defined y = f(x) and y = g(x).

This method of plotting a function graph y = f(x) + g(x) is called the addition of graphs of functions y = f(x) and y = g(x)

Example 4. In the figure, by the method of adding graphs, a graph of the function is constructed
y = x + sinx.

When plotting a function y = x + sinx we assumed that f(x) = x, a g(x) = sinx. To build a function graph, we select points with abscissas -1.5π, -, -0.5, 0, 0.5,, 1.5, 2. Values f(x) = x, g(x) = sinx, y = x + sinx we will calculate at the selected points and place the results in the table.

Lesson on the topic: "Graph and properties of the function $y=x^3$. Examples of plotting"

Additional materials
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Teaching aids and simulators in the online store "Integral" for grade 7
Electronic textbook for grade 7 "Algebra in 10 minutes"
Educational complex 1C "Algebra, grades 7-9"

Properties of the function $y=x^3$

Let's describe the properties of this function:

1. x is the independent variable, y is the dependent variable.

2. Domain of definition: it is obvious that for any value of the argument (x) it is possible to calculate the value of the function (y). Accordingly, the domain of definition of this function is the entire number line.

3. Range of values: y can be anything. Accordingly, the range is also the entire number line.

4. If x= 0, then y= 0.

Graph of the function $y=x^3$

1. Let's make a table of values:

2. For positive values ​​of x, the graph of the function $y=x^3$ is very similar to a parabola, the branches of which are more "pressed" to the OY axis.

3. Because for negative values x function $y=x^3$ has opposite values, then the graph of the function is symmetrical with respect to the origin.

Now let's mark the points on the coordinate plane and build a graph (see Fig. 1).

This curve is called a cubic parabola.

Examples

I. Completely finished on a small ship fresh water. It is necessary to bring enough water from the city. Water is ordered in advance and paid for a full cube, even if you fill it a little less. How many cubes should be ordered so as not to overpay for an extra cube and completely fill the tank? It is known that the tank has the same length, width and height, which are equal to 1.5 m. Let's solve this problem without performing calculations.

Decision:

1. Let's plot the function $y=x^3$.
2. Find point A, coordinate x, which is equal to 1.5. We see that the function coordinate is between the values ​​3 and 4 (see Fig. 2). So you need to order 4 cubes.

Unfortunately, not all students and schoolchildren know and love algebra, but everyone has to prepare homework, solve tests and take exams. It is especially difficult for many to find tasks for plotting function graphs: if somewhere you don’t understand something, don’t finish it, miss it, mistakes are inevitable. But who wants to get bad grades?

Would you like to join the cohort of tailers and losers? To do this, you have 2 ways: sit down for textbooks and fill in the gaps in knowledge, or use a virtual assistant - a service for automatically plotting function graphs according to specified conditions. With or without decision. Today we will introduce you to a few of them.

The best thing about Desmos.com is a highly customizable interface, interactivity, the ability to spread the results into tables and store your work in the resource database for free without time limits. And the disadvantage is that the service is not fully translated into Russian.

Grafikus.ru

Grafikus.ru is another worthy of attention Russian-language calculator for plotting graphs. Moreover, he builds them not only in two-dimensional, but also in three-dimensional space.

Here is an incomplete list of tasks that this service successfully copes with:

• Drawing 2D graphs of simple functions: lines, parabolas, hyperbolas, trigonometric, logarithmic, etc.
• Drawing 2D graphs parametric functions: circles, spirals, Lissajous figures and others.
• Drawing 2D graphs in polar coordinates.
• Construction of 3D surfaces of simple functions.
• Construction of 3D surfaces of parametric functions.

The finished result opens in a separate window. The user has options to download, print and copy a link to it. For the latter, you will have to log in to the service through the buttons of social networks.

Coordinate plane Grafikus.ru supports changing the boundaries of the axes, their labels, the grid spacing, as well as the width and height of the plane itself and the font size.

The biggest strength of Grafikus.ru is the ability to create 3D graphs. Otherwise, it works no worse and no better than analogue resources.

Onlinecharts.ru

The Onlinecharts.ru online assistant does not build charts, but charts of almost all existing types. Including:

• Linear.
• Columnar.
• Circular.
• with regions.
• Radial.
• XY charts.
• Bubble.
• Point.
• Polar Bulls.
• Pyramids.
• Speedometers.
• Column-linear.

The resource is very easy to use. The appearance of the chart (background color, grid, lines, pointers, corner shape, fonts, transparency, special effects, etc.) is completely user-defined. Data for building can be entered either manually or imported from a table in a CSV file stored on a computer. The finished result is available for download on a PC as an image, PDF, CSV or SVG file, as well as for saving online on ImageShack.Us photo hosting or in personal account Onlinecharts.ru. The first option can be used by everyone, the second - only registered ones.

Graphing online is a very useful way to graphically display something that cannot be expressed in words.

Information is the future of email marketing, and the right visuals are a powerful tool to engage your target audience.

This is where infographics come to the rescue, allowing you to present various kinds of information in a simple and expressive form.

However, the construction of infographic images requires a certain analytical thinking and a wealth of imagination.

We hasten to please you - there are enough resources on the Internet that provide online charting.

Yotx.ru

A wonderful Russian-language service that provides plotting online by points (by values) and graphs of functions (normal and parametric).

This site has an intuitive interface and is easy to use. It does not require registration, which significantly saves the user's time.

Allows you to quickly save ready-made graphics on your computer, and also generates code for posting on a blog or website.

Yotx.ru has a tutorial and chart examples that were created by users.

Perhaps, for people who study mathematics or physics in depth, this service will not be enough (for example, it is impossible to plot in polar coordinates, since the service does not have a logarithmic scale), but to perform the simplest laboratory work quite enough.

The advantage of the service is that it does not force, like many other programs, to search for the result obtained over the entire two-dimensional plane.

The size of the graph and the intervals along the coordinate axes are automatically generated so that the graph is easy to view.

At the same time on the same plane it is possible to build several graphs.

Additionally, on the site you can use the matrix calculator, with which it is easy to perform various actions and transformations.

ChartGo

English-language service for the development of multifunctional and multi-colored histograms, line charts, pie charts.

A detailed manual and demo videos are presented to users for training.

ChartGo will be useful for those who need it regularly. Among similar resources, “Create a graph online quickly” is distinguished by its simplicity.

Charting online is carried out according to the table.

At the beginning of the work, you must select one of the types of charts.

The application provides users with a number of simple charting customization options various functions in 2D and 3D coordinates.

You can select one of the chart types and switch between 2D and 3D.

Size settings provide maximum control between vertical and horizontal orientation.

Users can customize their charts with a unique title, as well as name the X and Y elements.

To plot online xyz graphs in the "Example" section, many layouts are available that you can change to your liking.

Note! In ChartGo, many charts can be built in one rectangular system. Each graph is made up of points and lines. Functions of a real variable (analytical) are set by the user in a parametric form.

Additional functionality has also been developed, which includes monitoring and displaying coordinates on a plane or in a three-dimensional system, importing and exporting numerical data in certain formats.

The program has a highly customizable interface.

After creating a diagram, the user can use the function to print the result and save the graph as a static picture.

OnlineCharts.ru

You can find another great application for a spectacular presentation of information on the OnlineCharts.ru website, where you can plot a function graph online for free.

The service is able to work with many types of charts, including line, bubble, pie, column and radial.

The system has a very simple and intuitive interface. All available functions are separated by tabs in the form of a horizontal menu.

To get started, you need to select the type of chart you want to build.

After that, you can configure some additional options appearance, depending on the selected chart type.

In the "Add data" tab, the user is prompted to set the number of rows and, if necessary, the number of groups.

You can also define a color.

Note! The “Signatures and fonts” tab offers to set the properties of the signatures (should they be displayed at all, if so, what color and font size). It also provides the ability to select the font type and size for the main text of the chart.

Everything is extremely simple.

Aiportal.ru

The simplest and least functional of all the online services presented here. Create 3D graph online on this site will not succeed.

It is designed for plotting complex functions in the coordinate system at a certain range of values.

For the convenience of users, the service provides reference data on the syntax various mathematical operations, as well as the list of supported functions and constant values.

All data necessary for drawing up the schedule is entered into the "Functions" window. At the same time, the user can build several graphs on the same plane.

Therefore, it is allowed to add several functions in a row, but after each function, you must insert a semicolon. The construction area is also set.

It is possible to build graphs online according to the table or without it. Color legend supported.

Despite the poor functionality, it is still an online service, so you do not have to search, download and install any software for a long time.

To build a graph, you just need to have it from any available device: PC, laptop, tablet or smartphone.

Plotting a function online

TOP 4 the best service for plotting graphs online