Graph x 0. How to graph a function. Lesson on the topic: "Graph and properties of the function $y=x3$. Examples of plotting"

First, try to find the scope of the function:

Did you manage? Let's compare the answers:

All right? Well done!

Now let's try to find the range of the function:

Found? Compare:

Did it agree? Well done!

Let's work with the graphs again, only now it's a little more difficult - to find both the domain of the function and the range of the function.

How to Find Both the Domain and Range of a Function (Advanced)

Here's what happened:

With graphics, I think you figured it out. Now let's try to find the domain of the function in accordance with the formulas (if you don't know how to do this, read the section about):

Did you manage? Checking answers:

1. , since the root expression must be greater than or equal to zero.
2. , since it is impossible to divide by zero and the radical expression cannot be negative.
3. , since, respectively, for all.
4. because you can't divide by zero.

However, we still have one more moment that has not been sorted out ...

Let me reiterate the definition and focus on it:

Noticed? The word "only" is a very, very important element of our definition. I will try to explain to you on the fingers.

Let's say we have a function given by a straight line. . When, we substitute this value into our "rule" and get that. One value corresponds to one value. We can even make a table of various values ​​and plot a given function to verify this.

"Look! - you say, - "" meets twice!" So maybe the parabola is not a function? No, it is!

The fact that "" occurs twice is far from a reason to accuse the parabola of ambiguity!

The fact is that, when calculating for, we got one game. And when calculating with, we got one game. So that's right, the parabola is a function. Look at the chart:

Got it? If not, here's a real-life example for you, far from mathematics!

Let's say we have a group of applicants who met when submitting documents, each of whom told in a conversation where he lives:

Agree, it is quite realistic that several guys live in the same city, but it is impossible for one person to live in several cities at the same time. This is, as it were, a logical representation of our "parabola" - Several different x's correspond to the same y.

Now let's come up with an example where the dependency is not a function. Let's say these same guys told what specialties they applied for:

Here we have a completely different situation: one person can easily apply for one or several directions. That is one element sets are put in correspondence multiple elements sets. Respectively, it's not a function.

Let's test your knowledge in practice.

Determine from the pictures what is a function and what is not:

Got it? And here is answers:

• The function is - B,E.
• Not a function - A, B, D, D.

You ask why? Yes, here's why:

In all figures except AT) and E) there are several for one!

I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the scope of the argument and the scope of the function. Let's move on to the next section - how to define a function?

Ways to set a function

What do you think the words mean "set function"? That's right, it means explaining to everyone what function we are talking about in this case. Moreover, explain in such a way that everyone understands you correctly and the graphs of functions drawn by people according to your explanation were the same.

How can I do that? How to set a function? The easiest way, which has already been used more than once in this article - using a formula. We write a formula, and by substituting a value into it, we calculate the value. And as you remember, a formula is a law, a rule according to which it becomes clear to us and to another person how an X turns into a Y.

Usually, this is exactly what they do - in tasks we see ready-made functions defined by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” confuses. Let's take a look at everything in order, and start with the analytical method.

Analytical way of defining a function

The analytical method is the task of a function using a formula. This is the most universal and comprehensive and unambiguous way. If you have a formula, then you know absolutely everything about the function - you can make a table of values ​​​​on it, you can build a graph, determine where the function increases and where it decreases, in general, explore it in full.

Let's consider a function. What does it matter?

"What does it mean?" - you ask. I'll explain now.

Let me remind you that in the notation, the expression in brackets is called the argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (expression in brackets), we will write it instead in the expression.

In our example, it will look like this:

Consider another task related to the analytical method of specifying a function that you will have on the exam.

Find the value of the expression, at.

I'm sure that at first, you were scared when you saw such an expression, but there is absolutely nothing scary in it!

Everything is the same as in the previous example: whatever the argument (expression in brackets), we will write it instead in the expression. For example, for a function.

What should be done in our example? Instead, you need to write, and instead of -:

shorten the resulting expression:

That's all!

Independent work

Now try to find the meaning of the following expressions yourself:

1. , if
2. , if

Did you manage? Let's compare our answers: We are used to the fact that the function has the form

Even in our examples, we define the function in this way, but analytically it is possible to define the function implicitly, for example.

Try building this function yourself.

Did you manage?

Here's how I built it.

What equation did we end up with?

Correctly! Linear, which means that the graph will be a straight line. Let's make a table to determine which points belong to our line:

That's just what we were talking about ... One corresponds to several.

Let's try to draw what happened:

Is what we got a function?

That's right, no! Why? Try to answer this question with a picture. What did you get?

“Because one value corresponds to several values!”

What conclusion can we draw from this?

That's right, a function can't always be expressed explicitly, and what's "disguised" as a function isn't always a function!

Tabular way of defining a function

As the name suggests, this method is a simple plate. Yes Yes. Like the one we already made. For example:

Here you immediately noticed a pattern - Y is three times larger than X. And now the “think very well” task: do you think that a function given in the form of a table is equivalent to a function?

Let's not talk for a long time, but let's draw!

So. We draw a function given in both ways:

Do you see the difference? It's not about the marked points! Take a closer look:

Have you seen it now? When we set the function in a tabular way, we reflect on the graph only those points that we have in the table and the line (as in our case) passes only through them. When we define a function in an analytical way, we can take any points, and our function is not limited to them. Here is such a feature. Remember!

Graphical way to build a function

The graphical way of constructing a function is no less convenient. We draw our function, and another interested person can find what y is equal to at a certain x, and so on. Graphical and analytical methods are among the most common.

However, here you need to remember what we talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Remembered? Just in case, I'll copy here the definition of what a function is:

As a rule, people usually name exactly those three ways of specifying a function that we have analyzed - analytical (using a formula), tabular and graphic, completely forgetting that a function can be described verbally. Like this? Yes, very easy!

Verbal description of the function

How to describe the function verbally? Let's take our recent example - . This function can be described as "each real value of x corresponds to its triple value." That's all. Nothing complicated. Of course, you will object - “there are such complex functions that it is simply impossible to set verbally!” Yes, there are some, but there are functions that are easier to describe verbally than to set with a formula. For example: "each natural value of x corresponds to the difference between the digits of which it consists, while the largest digit contained in the number entry is taken as the minuend." Now consider how our verbal description of the function is implemented in practice:

The largest digit in a given number -, respectively, - is reduced, then:

Main types of functions

Now let's move on to the most interesting - we will consider the main types of functions with which you worked / work and will work in the course of school and institute mathematics, that is, we will get to know them, so to speak, and give them a brief description. Read more about each function in the corresponding section.

Linear function

A function of the form, where, are real numbers.

The graph of this function is a straight line, so the construction of a linear function is reduced to finding the coordinates of two points.

The position of the straight line on the coordinate plane depends on the slope.

Function scope (aka argument range) - .

The range of values ​​is .

quadratic function

Function of the form, where

The graph of the function is a parabola, when the branches of the parabola are directed downwards, when - upwards.

Many properties of a quadratic function depend on the value of the discriminant. The discriminant is calculated by the formula

The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:

Domain

The range of values ​​depends on the extremum of the given function (the vertex of the parabola) and the coefficient (the direction of the branches of the parabola)

Inverse proportionality

The function given by the formula, where

The number is called the inverse proportionality factor. Depending on what value, the branches of the hyperbola are in different squares:

Domain - .

The range of values ​​is .

SUMMARY AND BASIC FORMULA

1. A function is a rule according to which each element of a set is assigned a unique element of the set.

• - this is a formula denoting a function, that is, the dependence of one variable on another;
• - variable, or argument;
• - dependent value - changes when the argument changes, that is, according to some specific formula that reflects the dependence of one value on another.

2. Valid argument values, or the scope of a function, is what is related to the possible under which the function makes sense.

3. Range of function values- this is what values ​​it takes, with valid values.

4. There are 4 ways to set the function:

• analytical (using formulas);
• tabular;
• graphic
• verbal description.

5. Main types of functions:

• : , where, are real numbers;
• : , where;
• : , where.

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) 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. Since the function $y=x^3$ has opposite values ​​for negative values ​​of x, 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. The small ship ran 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 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.

Solution:

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.

Plotting a function dependency graph is a characteristic mathematical problem. Everyone who is familiar with mathematics at least at the school level has built such dependencies on paper. The graph shows how the function changes depending on the value of the argument. Modern electronic applications allow this procedure to be carried out with a few mouse clicks. Microsoft Excel will help you in building an accurate graph for any mathematical function. Let's take a look at the steps on how to graph a function in excel using its formula

Plotting a Linear Function in Excel

Graphing in Excel 2016 has been greatly improved and made even easier than in previous versions. Let's analyze an example of plotting a linear function graph y=kx+b on a small interval [-4;4].

Preparation of the calculation table

We enter the names of the constants k and b in our function into the table. This is necessary to quickly change the schedule without altering the calculation formulas.

• In cells A5 and A6, we enter the notation for the argument and the function itself, respectively. The formula entry will be used as the title of the chart.
• Enter in cells B5 and C5 two values ​​of the function argument with a given step (in our example, the step is equal to one).
• Select these cells.
• Move the mouse pointer over the lower right corner of the selection. When a cross appears (see the figure above), hold down the left mouse button and drag to the right to column J.

The cells will automatically be filled with numbers whose values ​​differ by the given step.

Attention! The formula entry begins with an equal sign (=). Cell addresses are written on the English layout. Notice the absolute addresses with the dollar sign.

To finish entering the formula, press the Enter key or the check mark to the left of the formula bar at the top above the table.

We copy this formula for all values ​​of the argument. We stretch the frame to the right from the cell with the formula to the column with the final values ​​of the function argument.

Plotting a Function

Select a rectangular range of cells A5:J6.

Go to tab Insert in the toolbox. In chapter Diagram choose Spot with smooth curves(see figure below). Let's get a diagram.

After construction, the coordinate grid has unit segments of different lengths. Change it by dragging the side markers to get square cells.

Now you can enter new values ​​for the constants k and b to change the graph. And we see that when you try to change the coefficient, the graph remains unchanged, but the values ​​​​on the axis change. Fixing. Click on the diagram to activate it. Further on the ribbon of tools in the tab Working with charts tab Constructor choose Add chart element - Axes - Additional axis options..

A settings sidebar will appear on the right side of the window. Axis Format.

• Click the Axis Options drop-down list.
• Select Vertical Axis (values).
• Click the green chart icon.
• Set the interval of the axis values ​​and the unit of measurement (circled in red). We set the units of measurement Maximum and minimum (Preferably symmetrical) and the same for the vertical and horizontal axes. Thus, we make a single segment smaller and, accordingly, we observe a larger range of the graph on the diagram. And the main unit of measurement is the value 1.
• Repeat the same for the horizontal axis.

Now, if we change the values ​​of K and b , we get a new graph with a fixed grid of coordinates.

Plotting Other Functions

Now that we have a basic table and chart, we can plot other functions by making small adjustments to our table.

Quadratic function y=ax 2 +bx+c

Do the following:

• =$B3*B5*B5+$D3*B5+$F3

We get the result

Cubic parabola y=ax 3

To build, follow these steps:

• Change the title on the first line
• In the third line we indicate the coefficients and their values
• In cell A6 we write the designation of the function
• In cell B6, enter the formula =$B3*B5*B5*B5
• Copy it to the entire range of argument values ​​to the right

We get the result

Hyperbola y=k/x

To build a hyperbola, fill in the table manually (see the figure below). Where before there was a zero value of the argument, we leave an empty cell.

• Change the title on the first line.
• In the third line, we indicate the coefficients and their values.
• In cell A6 we write the designation of the function.
• In cell B6, enter the formula =$B3/B5
• We copy it to the entire range of values ​​of the argument to the right.
• Removing a formula from a cell I6.

To correctly display the graph, you need to change the range of initial data for the chart, since in this example it is larger than in the previous ones.

• Click Chart
• On the tab Working with charts go to Constructor and in the section Data click Select data.
• The data entry wizard window will open.
• Select a rectangular range of cells with the mouse A5:P6
• Click OK in the wizard window.

We get the result

Construction of trigonometric functions sin(x) and cos(x)

Consider an example of plotting a trigonometric function y=a*sin(b*x).
First fill in the table as in the picture below

The first line contains the name of the trigonometric function.
The third line contains the coefficients and their values. Pay attention to the cells in which the values ​​​​of the coefficients are entered.
The fifth line of the table contains the values ​​of the angles in radians. These values ​​will be used for chart labels.
The sixth line contains the numerical values ​​of the angles in radians. They can be written manually or using formulas of the appropriate form =-2*PI(); =-3/2*PI(); =-PI(); =-PI()/2; …
The seventh line contains the calculation formulas of the trigonometric function.

In our example =$B$3*SIN($D$3*B6). Addresses B3 and D3 are absolute. Their values ​​are the coefficients a and b, which are set to one by default.
After filling in the table, we proceed to plotting the graph.

Select a range of cells A6:J7. Select a tab in the ribbon Insert In chapter Diagrams specify the type dotted and view Spot with smooth curves and markers.

As a result, we get a diagram.

Now let's set up the correct display of the grid, so that the graph points lie at the intersection of the grid lines. Follow the steps Working with charts -Designer - Add chart element - Grid and enable three line display modes as shown in the figure.

Now go to point Additional grid line options. You will have a sidebar Construction area format. Let's make the settings here.

Click in the diagram on the main vertical Y-axis (should be highlighted with a box). In the sidebar, set the axis format as shown in the figure.

Click on the main horizontal axis X (should be highlighted) and also make settings according to the figure.

Now let's make data labels over the points. Execute again Working with charts -Designer - Add chart element - Data labels - Top. You will be substituted with the numbers 1 and 0, but we will replace them with values ​​from the range B5:J5.
Click on any value 1 or 0 (picture step 1) and in the signature parameters check the Values ​​from cells box (picture step 2). You will immediately be prompted to provide a range with new values ​​(Figure step 3). Specify B5:J5.

That's all. If done correctly, then the schedule will be wonderful. Here's one.

To get the graph of a function cos(x), replace in the calculation formula and in the title sin(x) on the cos(x).

In a similar way, you can build graphs of other functions. The main thing is to write down the computational formulas correctly and build a table of function values. I hope you found this information useful.

PS: Interesting facts about the logos of famous companies

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Building charts 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 plots online graphs 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 build a graph in polar coordinates, since the service does not have a logarithmic scale), but it is quite enough to perform 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.

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

An English-language service for the development of multifunctional and multi-colored histograms, line graphs, 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 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, 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 build a graph of a function 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 appearance options, 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. It will not be possible to create a three-dimensional graph online on this site.

It is designed to plot complex functions in a coordinate system at 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 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 best online charting services