Graph of the function y 4x2. Functions

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

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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), the value of the function (y) can be calculated. Accordingly, the domain 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 ​​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. Since for negative values ​​of x the function $ y = x ^ 3 $ has opposite values, the graph of the function is symmetric 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 of

I. The small ship has completely run out of 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 with a little less. How many cubes do you need to order so as not to overpay for an extra cubic meter 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 any calculations.

Solution:

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

Online charting is very useful way to graphically display what cannot be conveyed in words.

Information is the future of email marketing, and well-presented visuals are a powerful tool for reaching target audiences.

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

However, building 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 builds graphs online by points (by values) and graphs of functions (regular and parametric).

This site has an intuitive interface and 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 examples of graphs 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 a graph in polar coordinates, since the service does not have a logarithmic scale), but it is quite enough for performing the simplest laboratory work.

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.

It is possible to construct several graphs at the same time on one plane.

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

ChartGo

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

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

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

Online charting is carried out according to the table.

At the beginning of the work, you need to choose one of the types of diagrams.

The application provides users with a number of easy options for customizing the plotting of various functions in 2D and 3D coordinates.

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

Size settings provide maximum control between vertical and horizontal orientation.

Users can customize their charts with a unique name, and assign names to X and Y elements.

There are many layouts available for plotting online xyz charts in the "Example" section, which you can change at your discretion.

Note! In ChartGo, many charts can be plotted in one rectangular system. Moreover, each graph is drawn using dots and lines. Functions of a real variable (analytical) are specified by the user in a parametric form.

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

The program has a flexible interface.

After creating a diagram, the user can use the function of printing the result and saving the graph as a static picture.

OnlineCharts.ru

Another great application for effective presentation of information can be found on the OnlineCharts.ru website, where you can plot a function graph online for free.

The service is capable of working with many types of charts, including line, bubble, pie, bar 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 that you want to build.

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

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

You can also define the color.

Note! The tab "Signatures and fonts" offers to set the properties of the signatures (whether they should be displayed at all, if so, what color and font size). It also provides the ability to select the type of font and its 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 will not be possible to create a 3D graph online on this site.

It is intended for plotting 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 on the list of supported functions and constant values.

All data required for scheduling is entered into the "Functions" window. The user can draw several graphs at the same time on one plane.

Therefore, it is allowed to enter several functions in a row, but after each function it is necessary to insert a semicolon. The area of ​​construction is also set.

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

Despite the limited functionality, it is still an online service, so you don't 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 best online charting services

The study of the properties of functions and their graphs takes a significant place both in school mathematics and in subsequent courses. And not only in the courses of mathematical and functional analysis, and even not only in other sections higher mathematics but also in most narrowly professional subjects. For example, in economics - utility functions, costs, demand, supply and consumption functions ..., in radio engineering - control functions and response functions, in statistics - distribution functions ... To facilitate further study of special functions, you need to learn how to freely operate with graphs elementary functions... To do this, after studying the following table, I recommend following the link "Function graph transformations".

In the school mathematics course, the following are studied
elementary functions.

Function name	Function formula	Function graph	Chart name	A comment
Linear	y = kx		Straight	The simplest particular case of linear dependence is direct proportionality y = kx, where k≠ 0 - proportionality coefficient. The figure shows an example for k= 1, i.e. in fact, the given graph illustrates the functional dependence, which sets the equality of the value of the function to the value of the argument.
Linear	y = kx + b		Straight	General case of linear dependence: coefficients k and b- any real numbers. Here k = 0.5, b = -1.
Quadratic	y = x 2		Parabola	The simplest case of a quadratic dependence is a symmetric parabola with apex at the origin.
Quadratic	y = ax 2 + bx + c		Parabola	General case of quadratic dependence: coefficient a- an arbitrary real number not equal to zero ( a belongs to R, a ≠ 0), b, c- any real numbers.
Power	y = x 3		Cubic parabola	The simplest case is for an odd integer degree. Cases with coefficients are studied in the section "Movement of function graphs".
Power	y = x 1/2		Function graph y = √x	The simplest case for a fractional power ( x 1/2 = √x). Cases with coefficients are studied in the section "Movement of function graphs".
Power	y = k / x		Hyperbola	The simplest case for a negative integer power ( 1 / x = x-1) - inversely proportional relationship. Here k = 1.
Indicative	y = e x		Exhibitor	The exponential dependence is called the exponential function for the base e- an irrational number approximately equal to 2.7182818284590 ...
Indicative	y = a x		Exponential function graph	a> 0 and a a... Here's an example for y = 2 x (a = 2 > 1).
Indicative	y = a x		Exponential function graph	Exponential function defined for a> 0 and a≠ 1. The graphs of the function essentially depend on the value of the parameter a... Here's an example for y = 0.5 x (a = 1/2 < 1).
Logarithmic	y= ln x			Graph of the logarithmic function for the base e(natural logarithm) is sometimes called logarithm.
Logarithmic	y= log a x		Logarithmic function graph	Logarithms are defined for a> 0 and a≠ 1. The graphs of the function essentially depend on the value of the parameter a... Here's an example for y= log 2 x (a = 2 > 1).
Logarithmic	y = log a x		Logarithmic function graph	Logarithms are defined for a> 0 and a≠ 1. The graphs of the function essentially depend on the value of the parameter a... Here's an example for y= log 0.5 x (a = 1/2 < 1).
Sinus	y= sin x		Sinusoid	Trigonometric function sinus. Cases with coefficients are studied in the section "Movement of function graphs".
Cosine	y= cos x		Cosine	Trigonometric cosine function. Cases with coefficients are studied in the section "Movement of function graphs".
Tangent	y= tg x		Tangentoid	Trigonometric tangent function. Cases with coefficients are studied in the section "Movement of function graphs".
Cotangent	y= ctg x		Cotangensoid	Trigonometric cotangent function. Cases with coefficients are studied in the section "Movement of function graphs".

Inverse trigonometric functions.

Function name	Function formula	Function graph	Chart name

In the golden age of information technology, few people will buy graph paper and spend hours drawing a function or arbitrary dataset, and why bother doing such a dreary job when you can plot a function online. In addition, it is almost impossible and difficult to calculate millions of values ​​of an expression for correct display, and despite all the efforts, you will get a broken line, not a curve. Therefore, the computer in this case is an irreplaceable assistant.

What is a graph of functions

A function is a rule according to which each element of one set is associated with some element of another set, for example, the expression y = 2x + 1 establishes a connection between the sets of all values ​​of x and all values ​​of y, therefore, this is a function. Accordingly, the graph of a function will be called a set of points whose coordinates satisfy a given expression.

In the figure, we see the graph of the function y = x... This is a straight line and each point has its own coordinates on the axis X and on the axis Y... Based on the definition, if we substitute the coordinate X some point into the given equation, then we get the coordinate of this point on the axis Y.

Services for plotting functions online

Let's take a look at some of the most popular and best-performing services that allow you to quickly draw a graph of a function.

Opens the list of the most common service that allows you to build a graph of a function by an equation online. Umath contains only the necessary tools, such as scaling, moving along the coordinate plane and viewing the coordinate of the point at which the mouse is pointing.

Instructions:

1. Enter your equation in the box after the "=" sign.
2. Click the button "Build a graph".

As you can see, everything is extremely simple and accessible, the syntax for writing complex mathematical functions: with a module, trigonometric, exponential - is shown right below the graph. Also, if necessary, you can set the equation by the parametric method or build graphs in polar system coordinates.

Yotx has all the functions of the previous service, but at the same time it contains such interesting innovations as creating an interval for displaying a function, the ability to build a graph using tabular data, and also display a table with entire solutions.

Instructions:

1. Select the desired method for setting the schedule.
2. Enter your equation.
3. Set the interval.
4. Click the button "Build".

For those who are too lazy to figure out how to write down certain functions, this position presents a service with the ability to select the one you need from the list with one click of the mouse.

Instructions:

1. Find the function you need in the list.
2. Left-click on it
3. If necessary, enter the coefficients in the field "Function:".
4. Click the button "Build".

In terms of visualization, it is possible to change the color of the chart, as well as hide it or delete it altogether.

Desmos is by far the most sophisticated equation building service online. By moving the cursor while holding the left mouse button on the graph, you can see in detail all the solutions to the equation with an accuracy of 0.001. The built-in keyboard allows you to quickly write exponents and fractions. The most important plus is the ability to write the equation in any state, without leading to the form: y = f (x).

Instructions:

1. In the left column, right-click on a free line.
2. In the lower left corner, click on the keyboard icon.
3. On the panel that appears, type the required equation (to write the names of the functions, go to the section "A B C").
4. The graph is plotted in real time.

The visualization is just perfect, adaptive, you can see that the designers worked on the application. On the plus side, there is a huge abundance of opportunities, for the development of which you can see examples in the menu in the upper left corner.

There are a great many sites for plotting functions, but everyone is free to choose for themselves based on the required functionality and personal preferences. The list of the best has been formed to satisfy the requirements of any mathematician, young and old. I wish you success in comprehending the "queen of sciences"!