Build a graph x 2 4. We build a graph of functions online. What is a function graph

A function graph is a visual representation of the behavior of some function on coordinate plane. Plots help to 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 by a specific formula. The graph of any function is built according to a certain algorithm (if you forgot the exact process of plotting a graph of a particular function).

Steps

Plotting a Linear Function

Determine if the function is linear. A 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, and the like. Given a function of a similar form, plotting such a function is quite simple. 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 intersection point of the graph with the Y-axis. That is, it is a point whose “x” coordinate is 0. Thus, if x = 0 is substituted into the formula, then y = b (constant). In our example y = 2x + 5 (\displaystyle y=2x+5) the constant is 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 = 2x + 5 (\displaystyle y=2x+5) with the variable "x" is a factor of 2; thus, the slope is 2. The slope determines the angle of inclination of the straight line to the X-axis, that is, the larger the slope, the faster the function increases or decreases.

Write the slope as a fraction. The slope 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 say 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.

From the point where the line intersects with the Y axis, draw a second point using the vertical and horizontal distances. Schedule linear function can be built from two points. In our example, the point of intersection 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.

Use a ruler to draw a straight line through two points. To avoid mistakes, find the third point, but in most cases the graph can be built using two points. Thus, you have plotted a linear function.

Drawing points on the coordinate plane
1. Define a function. The function is denoted as f(x). All possible values of the variable "y" are called the range 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. Break each axis into equal segments and number them. The intersection point of the axes is 0. For the X axis: positive numbers are plotted on the right (from 0), and negative numbers on the left. For the Y-axis: positive numbers are plotted on top (from 0), and negative numbers on the bottom.
  
  Find the "y" values from the "x" values. In our example f(x) = x+2. Substitute certain "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.
  - -1: -1 + 2 = 1
  - 0: 0 +2 = 2
  - 1: 1 + 2 = 3
2. Draw 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 line); find the corresponding value on the y-axis and draw a horizontal line (dotted line). Mark the point of intersection of the two dotted lines; thus, you have plotted a graph point.
  
  Erase the dotted lines. Do this after plotting all the graph points on the coordinate plane. Note: the graph of the function f(x) = x is a straight line passing through the center of coordinates [point with coordinates (0,0)]; the graph f(x) = x + 2 is a line parallel to the line f(x) = x, but shifted up by two units and therefore passing through the point with coordinates (0,2) (because the constant is 2).
Plotting a complex function

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 plot 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 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 in one rectangular system multiple graphs can be generated. 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 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 the best service for plotting graphs online

Graphing functions is one of the features of Excel. In this article, we will look at the process of plotting graphs of some mathematical functions: linear, quadratic and inverse proportionality.

A function is a set of points (x, y) that satisfies the expression y=f(x). Therefore, we need to fill in an array of such points, and Excel will build a function graph based on them.

1) Consider an example of plotting a graph of a linear function: y=5x-2

The graph of a linear function is a straight line that can be drawn from two points. Let's create a sign

In our case y=5x-2. To the cell with the first value y let's enter the formula: =5*D4-2. In another cell, the formula can be entered in the same way (by changing D4 on the D5) or use the autocomplete token.

As a result, we will get a table:

Now you can start creating a chart.

Choose: INSERT -> SPINT -> SPOT WITH SMOOTH CURVES AND MARKERS (I recommend using this particular type of chart)

An empty chart area will appear. Press the SELECT DATA button

Let's select the data: the range of cells of the abscissa axis (x) and the ordinate axis (y). As the name of the series, we can enter the function itself in quotes "y=5x-2" or something else. Here's what happened:

We press OK. Before us is a graph of a linear function.

2) Consider the process of constructing a graph of a quadratic function - a parabola y \u003d 2x 2 -2

A parabola cannot be built from two points, unlike a straight line.

Let's set the spacing on the axis x on which our parabola will be built. I'll choose [-5; 5].

I'll take a step. The smaller the step, the more accurate the plot will be. I will choose 0,2 .

Populate a column with values X, using the autocomplete token to the value x=5.

Value column at calculated by the formula: =2*B4^2-2. Using the autocomplete marker, we calculate the values at for others X.

Choose: INSERT -> POINT -> POINT WITH SMOOTH CURVES AND MARKERS and act in the same way as plotting a linear function graph.

To avoid dots on the chart, change the chart type to SPOT WITH SMOOTH CURVES.

Any other graphs of continuous functions are constructed in a similar way.

3) If the function is piecewise, then it is necessary to combine each “piece” of the graph in one area of the diagrams.

Let's look at this using the function as an example. y=1/x.

The function is defined on the intervals (- ins; 0) and (0; + ins)

Let's create a graph of the function on the intervals: [-4; 0) and (0; 4].

Let's prepare two tables, where x changes in steps 0,2 :

Find function values from each argument X similar to the examples above.

You must add two rows to the diagram - for the first and second plates, respectively.

We get the graph of the function y=1/x

In addition, I give a video - which shows the procedure described above.

In the next article I will tell you how to create 3-dimensional graphs in Excel.

Thank you for your attention!

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:

, since the root expression must be greater than or equal to zero.
, since it is impossible to divide by zero and the radical expression cannot be negative.
, since, respectively, for all.
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 very, very important element 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 life example far from math!

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:

, if
, 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 define a function tabular way, we reflect on the chart 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 "every 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 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 - .