"Natural logarithm" - 0.1. natural logarithms. 4. "Logarithmic darts". 0.04. 7.121.
"Power function grade 9" - U. Cubic parabola. Y = x3. Grade 9 teacher Ladoshkina I.A. Y = x2. Hyperbola. 0. Y \u003d xn, y \u003d x-n where n is the given natural number. X. The exponent is an even natural number (2n).
"Quadratic Function" - 1 Definition quadratic function 2 Function properties 3 Function graphs 4 Quadratic inequalities 5 Conclusion. Properties: Inequalities: Prepared by Andrey Gerlitz, a student of grade 8A. Plan: Graph: -Intervals of monotonicity at a > 0 at a< 0. Квадратичная функция. Квадратичные функции используются уже много лет.
"Quadratic function and its graph" - Decision. y \u003d 4x A (0.5: 1) 1 \u003d 1 A-belongs. When a=1, the formula y=ax takes the form.
"Class 8 quadratic function" - 1) Construct the top of the parabola. Plotting a quadratic function. x. -7. Plot the function. Algebra Grade 8 Teacher 496 school Bovina TV -1. Construction plan. 2) Construct the axis of symmetry x=-1. y.
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 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
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 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 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. Create 3D graph online on this site will not succeed.
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 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!
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 the link to it. For the latter, you will have to log in to the service through the buttons of social networks.
The Grafikus.ru coordinate plane 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. Appearance charts (background color, grid, lines, pointers, corner shape, fonts, transparency, special effects, etc.) are 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.
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 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. I.e 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 in this case in question. 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 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 so many complex functions which it is simply impossible to verbally ask!” 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 functions are 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.
Direct position on coordinate plane depends on the slope factor.
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 values argument, 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.
