Unfortunately, not all students and schoolchildren know and love algebra, but everyone has to prepare homework, solve tests and take exams. Many people find it especially difficult to construct graphs of functions: if somewhere you don’t understand something, don’t finish learning it, or miss it, mistakes are inevitable. But who wants to get bad grades?

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

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

Grafikus.ru

Grafikus.ru is another Russian-language graphing calculator worthy of attention. 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: straight lines, parabolas, hyperbolas, trigonometric, logarithmic, etc.
• Drawing 2D graphs of 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 the options of downloading, printing and copying a link to it. For the latter, you will have to log in to the service through the social network buttons.

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

The most strong point Grafikus.ru - the ability to create 3D graphs. Otherwise, it works no worse and no better than analogous resources.

Onlinecharts.ru

The online assistant Onlinecharts.ru builds not graphs, but diagrams of almost all existing types. Including:

• Linear.
• Columnar.
• Circular.
• With areas.
• Radial.
• XY-graphs.
• Bubble.
• Spot.
• Polar bubbles.
• Pyramids.
• Speedometers.
• Columnar-linear.

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

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

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Properties of the function $y=x^3$

Let's describe the properties of this function:

1. x is an independent variable, y is a dependent variable.

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

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

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

Graph of the function $y=x^3$

1. Let's create a table of values:

2. For positive values x 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 about 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. The small ship completely ran out of fresh water. It is necessary to bring a sufficient amount of 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 I order 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 us solve this problem without performing calculations.

Solution:

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

Constructing graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, everything is not so bad. It is enough to remember a few algorithms for solving such problems, and you can easily build a graph of even the most seemingly complex function. Let's figure out what kind of algorithms these are.

1. Plotting a graph of the function y = |f(x)|

Note that the set of function values ​​y = |f(x)| : y ≥ 0. Thus, the graphs of such functions are always located entirely in the upper half-plane.

Plotting a graph of the function y = |f(x)| consists of the following simple four steps.

1) Carefully and carefully construct a graph of the function y = f(x).

2) Leave unchanged all points on the graph that are above or on the 0x axis.

3) Display the part of the graph that lies below the 0x axis symmetrically relative to the 0x axis.

Example 1. Draw a graph of the function y = |x 2 – 4x + 3|

1) We build a graph of the function y = x 2 – 4x + 3. Obviously, the graph of this function is a parabola. Let's find the coordinates of all points of intersection of the parabola with the coordinate axes and the coordinates of the vertex of the parabola.

x 2 – 4x + 3 = 0.

x 1 = 3, x 2 = 1.

Therefore, the parabola intersects the 0x axis at points (3, 0) and (1, 0).

y = 0 2 – 4 0 + 3 = 3.

Therefore, the parabola intersects the 0y axis at the point (0, 3).

Parabola vertex coordinates:

x in = -(-4/2) = 2, y in = 2 2 – 4 2 + 3 = -1.

Therefore, point (2, -1) is the vertex of this parabola.

Draw a parabola using the data obtained (Fig. 1)

2) The part of the graph lying below the 0x axis is displayed symmetrically relative to the 0x axis.

3) We get a graph of the original function ( rice. 2, shown in dotted line).

2. Plotting the function y = f(|x|)

Note that functions of the form y = f(|x|) are even:

y(-x) = f(|-x|) = f(|x|) = y(x). This means that the graphs of such functions are symmetrical about the 0y axis.

Plotting a graph of the function y = f(|x|) consists of the following simple chain of actions.

1) Graph the function y = f(x).

2) Leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the part of the graph specified in point (2) symmetrically to the 0y axis.

4) As the final graph, select the union of the curves obtained in points (2) and (3).

Example 2. Draw a graph of the function y = x 2 – 4 · |x| + 3

Since x 2 = |x| 2, then the original function can be rewritten in the following form: y = |x| 2 – 4 |x| + 3. Now we can apply the algorithm proposed above.

1) We carefully and carefully build a graph of the function y = x 2 – 4 x + 3 (see also rice. 1).

2) We leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the right side of the graph symmetrically to the 0y axis.

(Fig. 3).

Example 3. Draw a graph of the function y = log 2 |x|

We apply the scheme given above.

1) Build a graph of the function y = log 2 x (Fig. 4).

3. Plotting the function y = |f(|x|)|

Note that functions of the form y = |f(|x|)| are also even. Indeed, y(-x) = y = |f(|-x|)| = y = |f(|x|)| = y(x), and therefore, their graphs are symmetrical about the 0y axis. The set of values ​​of such functions: y ≥ 0. This means that the graphs of such functions are located entirely in the upper half-plane.

To plot the function y = |f(|x|)|, you need to:

1) Carefully construct a graph of the function y = f(|x|).

2) Leave unchanged the part of the graph that is above or on the 0x axis.

3) Display the part of the graph located below the 0x axis symmetrically relative to the 0x axis.

4) As the final graph, select the union of the curves obtained in points (2) and (3).

Example 4. Draw a graph of the function y = |-x 2 + 2|x| – 1|.

1) Note that x 2 = |x| 2. This means that instead of the original function y = -x 2 + 2|x| - 1

you can use the function y = -|x| 2 + 2|x| – 1, since their graphs coincide.

We build a graph y = -|x| 2 + 2|x| – 1. For this we use algorithm 2.

a) Graph the function y = -x 2 + 2x – 1 (Fig. 6).

b) We leave that part of the graph that is located in the right half-plane.

c) We display the resulting part of the graph symmetrically to the 0y axis.

d) The resulting graph is shown in the dotted line in the figure (Fig. 7).

2) There are no points above the 0x axis; we leave the points on the 0x axis unchanged.

3) The part of the graph located below the 0x axis is displayed symmetrically relative to 0x.

4) The resulting graph is shown in the figure with a dotted line (Fig. 8).

Example 5. Graph the function y = |(2|x| – 4) / (|x| + 3)|

1) First you need to plot the function y = (2|x| – 4) / (|x| + 3). To do this, we return to Algorithm 2.

a) Carefully plot the function y = (2x – 4) / (x + 3) (Fig. 9).

Note that this function is fractional linear and its graph is a hyperbola. To plot a curve, you first need to find the asymptotes of the graph. Horizontal – y = 2/1 (the ratio of the coefficients of x in the numerator and denominator of the fraction), vertical – x = -3.

2) We will leave that part of the graph that is above the 0x axis or on it unchanged.

3) The part of the graph located below the 0x axis will be displayed symmetrically relative to 0x.

4) The final graph is shown in the figure (Fig. 11).

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A function graph is a visual representation of the behavior of a function on a coordinate plane. Graphs help you understand various aspects of a function that cannot be determined from the function itself. You can build graphs of many functions, and each of them will be given a specific formula. The graph of any function is built using a specific algorithm (if you have forgotten the exact process of graphing a specific function).

Steps

Graphing a Linear Function

Determine whether the function is linear. The linear function is given by a formula of the form F (x) = k x + b (\displaystyle F(x)=kx+b) or y = k x + b (\displaystyle y=kx+b)(for example, ), and its graph is a straight line. Thus, the formula includes one variable and one constant (constant) without any exponents, root signs, or the like. Given a function of a similar type, it is quite simple to plot a graph of such a function. Here are other examples of linear functions:

Use a constant to mark a point on the Y axis. The constant (b) is the “y” coordinate of the point where the graph intersects the Y axis. That is, it is a point whose “x” coordinate is equal to 0. Thus, if x = 0 is substituted into the formula, then y = b (constant). In our example y = 2 x + 5 (\displaystyle y=2x+5) the constant is equal to 5, that is, the point of intersection with the Y axis has coordinates (0.5). Plot this point on the coordinate plane.

Find the slope of the line. It is equal to the multiplier of the variable. In our example y = 2 x + 5 (\displaystyle y=2x+5) with the variable “x” there is a factor of 2; thus, the slope coefficient is equal to 2. The slope coefficient determines the angle of inclination of the straight line to the X axis, that is, the greater the slope coefficient, the faster the function increases or decreases.

Write the slope as a fraction. The angular coefficient is equal to the tangent of the angle of inclination, that is, the ratio of the vertical distance (between two points on a straight line) to the horizontal distance (between the same points). In our example, the slope is 2, so we can state that the vertical distance is 2 and the horizontal distance is 1. Write this as a fraction: 2 1 (\displaystyle (\frac (2)(1))).

• If the slope is negative, the function is decreasing.

1. From the point where the straight line intersects the Y axis, plot a second point using vertical and horizontal distances.

   A linear function can be graphed using two points. In our example, the intersection point with the Y axis has coordinates (0.5); From this point, move 2 spaces up and then 1 space to the right. Mark a point; it will have coordinates (1,7). Now you can draw a straight line. Using a ruler, draw a straight line through two points.

   To avoid mistakes, find the third point, but in most cases the graph can be plotted using two points. Thus, you have plotted a linear function.

   1. Plotting points on the coordinate plane Define a function.

      The function is denoted as f(x). All possible values ​​of the variable "y" are called the domain of the function, and all possible values ​​of the variable "x" are called the domain of the function. For example, consider the function y = x+2, namely f(x) = x+2. Draw two intersecting perpendicular lines.

      The horizontal line is the X axis. The vertical line is the Y axis. Label the coordinate axes.

      Divide each axis into equal segments and number them. The intersection point of the axes is 0. For the X axis: positive numbers are plotted to the right (from 0), and negative numbers to the left. For the Y axis: positive numbers are plotted on top (from 0), and negative numbers on the bottom. Find the values ​​of "y" from the values ​​of "x".

      • -1: -1 + 2 = 1
      • 0: 0 +2 = 2
      • 1: 1 + 2 = 3

   2. In our example, f(x) = x+2. Substitute specific x values ​​into this formula to calculate the corresponding y values. If given a complex function, simplify it by isolating the “y” on one side of the equation. Plot the points on the coordinate plane.

      For each pair of coordinates, do the following: find the corresponding value on the X axis and draw a vertical line (dotted); find the corresponding value on the Y axis and draw a horizontal line (dashed line). Mark the intersection point of the two dotted lines; thus, you have plotted a point on the graph. Erase the dotted lines.

   Do this after plotting all the points on the graph on the coordinate plane. Note: the graph of the function f(x) = x is a straight line passing through the coordinate center [point with coordinates (0,0)]; the graph f(x) = x + 2 is a line parallel to the line f(x) = x, but shifted upward by two units and therefore passing through the point with coordinates (0,2) (because the constant is 2).

   Graphing a Complex Function The zeros of a function are the values ​​of the x variable where y = 0, that is, these are the points where the graph intersects the X-axis. Keep in mind that not all functions have zeros, but they are the first step in the process of graphing any function. To find the zeros of a function, equate it to zero. For example:

   Find and mark the horizontal asymptotes. An asymptote is a line that the graph of a function approaches but never intersects (that is, in this region the function is not defined, for example, when dividing by 0). Mark the asymptote with a dotted line. If the variable "x" is in the denominator of a fraction (for example, y = 1 4 − x 2 (\displaystyle y=(\frac (1)(4-x^(2))))), set the denominator to zero and find “x”. In the obtained values ​​of the variable “x” the function is not defined (in our example, draw dotted lines through x = 2 and x = -2), because you cannot divide by 0. But asymptotes exist not only in cases where the function contains a fractional expression. Therefore, it is recommended to use common sense:

Online graphing is a very useful way to graphically display what you cannot convey in words.

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

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

However, constructing infographic images requires a certain amount of 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 creates online graphs by points (by values) and graphs of functions (regular and parametric).

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

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

Yotx.ru has a tutorial and examples of charts 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 construct a graph in polar coordinates, since the service does not have a logarithmic scale), but it is quite sufficient for performing the simplest laboratory work.

The advantage of the service is that it does not force you, like many other programs, to search for the result across 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 convenient for viewing.

It is possible to construct several graphs simultaneously on one plane.

Additionally, on the site you can use a matrix calculator, with which you can easily perform various actions and transformations.

ChartGo

English-language service for developing multifunctional and multi-colored histograms, line graphs, and pie charts.

For training, users are provided with a detailed manual and demos.

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

Online graphs are constructed using a table.

To get started, you need to select one of the types of diagrams.

The application provides users with a number of simple options for customizing the plotting of 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 and also assign titles to X and Y elements.

To create online xyz charts, there are many layouts available in the “Example” section that you can change to your liking.

Note! In ChartGo, many charts can be plotted in one rectangular system. Moreover, each graph is made using points and lines. Functions of a real variable (analytical) are specified by the user in 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 chart, the user can use the function of printing the result and saving the graph as a static drawing.

OnlineCharts.ru

Another excellent application for effectively presenting information can be found on the website OnlineCharts.ru, where you can build a graph of a function online for free.

The service is capable of working 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 this, you can configure some additional settings appearance, depending on the selected chart type.

In the “Add Data” tab, the user is prompted to specify the number of rows and, if necessary, the number of groups.

You can also determine the color.

Note! The “Captions and Fonts” tab offers to set the properties of signatures (whether they need to be displayed at all, if so, what color and font size). You also have the option 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. It is not possible to create a 3D chart online on this site.

It is intended for plotting graphs of complex functions in a coordinate system over a certain range of values.

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

All data necessary for drawing up a schedule is entered into the “Functions” window. The user can construct several graphs simultaneously on one plane.

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

It is possible to build graphs online using a table or without it. Color legend supported.

Despite the poor functionality, it is still an online service, so you don’t have to spend a long time searching, downloading and installing any software.

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

Graphing a function online

TOP 4 best online charting services