How to build a graph x 2. Functions and graphs

We choose a rectangular coordinate system on the plane and plot the values ​​of the argument on the abscissa axis X, and on the y-axis - the values ​​of the function y = f(x).

Function Graph y = f(x) the set of all points is called, for which the abscissas belong to the domain of the function, and the ordinates are equal to the corresponding values ​​of the function.

In other words, the graph of the function y \u003d f (x) is the set of all points in the plane, the coordinates X, at which satisfy the relation y = f(x).

On fig. 45 and 46 are graphs of functions y = 2x + 1 and y \u003d x 2 - 2x.

Strictly speaking, one should distinguish between the graph of a function (the exact mathematical definition of which was given above) and the drawn curve, which always gives only a more or less accurate sketch of the graph (and even then, as a rule, not of the entire graph, but only of its part located in the final parts of the plane). In what follows, however, we will usually refer to "chart" rather than "chart sketch".

Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the scope of the function y = f(x), then to find the number f(a)(i.e. the function values ​​at the point x = a) should do so. Need through a dot with an abscissa x = a draw a straight line parallel to the y-axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will be, by virtue of the definition of the graph, equal to f(a)(Fig. 47).

For example, for the function f(x) = x 2 - 2x using the graph (Fig. 46) we find f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0, etc.

A function graph visually illustrates the behavior and properties of a function. For example, from a consideration of Fig. 46 it is clear that the function y \u003d x 2 - 2x takes positive values ​​when X< 0 and at x > 2, negative - at 0< x < 2; smallest value function y \u003d x 2 - 2x accepts at x = 1.

To plot a function f(x) you need to find all points of the plane, coordinates X,at which satisfy the equation y = f(x). In most cases, this is impossible, since there are infinitely many such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the multi-point plotting method. It consists in the fact that the argument X give a finite number of values ​​- say, x 1 , x 2 , x 3 ,..., x k and make a table that includes the selected values ​​of the function.

The table looks like this:

Having compiled such a table, we can outline several points on the graph of the function y = f(x). Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y = f(x).

However, it should be noted that the multi-point plotting method is very unreliable. In fact, the behavior of the graph between the marked points and its behavior outside the segment between the extreme points taken remains unknown.

Example 1. To plot a function y = f(x) someone compiled a table of argument and function values:

The corresponding five points are shown in Fig. 48.

Based on the location of these points, he concluded that the graph of the function is a straight line (shown in Fig. 48 by a dotted line). Can this conclusion be considered reliable? Unless there are additional considerations to support this conclusion, it can hardly be considered reliable. reliable.

To substantiate our assertion, consider the function

.

Calculations show that the values ​​of this function at points -2, -1, 0, 1, 2 are just described by the above table. However, the graph of this function is not at all a straight line (it is shown in Fig. 49). Another example is the function y = x + l + sinx; its meanings are also described in the table above.

These examples show that in its "pure" form, the multi-point plotting method is unreliable. Therefore, to plot a given function, as a rule, proceed as follows. First, the properties of this function are studied, with the help of which it is possible to construct a sketch of the graph. Then, by calculating the values ​​of the function at several points (the choice of which depends on the set properties of the function), the corresponding points of the graph are found. And, finally, a curve is drawn through the constructed points using the properties of this function.

We will consider some (the most simple and frequently used) properties of functions used to find a sketch of a graph later, but now we will analyze some commonly used methods for plotting graphs.

Graph of the function y = |f(x)|.

It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Recall how this is done. By definition of the absolute value of a number, one can write

This means that the graph of the function y=|f(x)| can be obtained from the graph, functions y = f(x) as follows: all points of the graph of the function y = f(x), whose ordinates are non-negative, should be left unchanged; further, instead of the points of the graph of the function y = f(x), having negative coordinates, one should construct the corresponding points of the graph of the function y = -f(x)(i.e. part of the function graph
y = f(x), which lies below the axis X, should be reflected symmetrically about the axis X).

Example 2 Plot a function y = |x|.

We take the graph of the function y = x(Fig. 50, a) and part of this graph when X< 0 (lying under the axis X) is symmetrically reflected about the axis X. As a result, we get the graph of the function y = |x|(Fig. 50, b).

Example 3. Plot a function y = |x 2 - 2x|.

First we plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upwards, the top of the parabola has coordinates (1; -1), its graph intersects the abscissa axis at points 0 and 2. On the interval (0; 2) the function takes negative values, therefore this part of the graph reflect symmetrically about the x-axis. Figure 51 shows a graph of the function y \u003d |x 2 -2x |, based on the graph of the function y = x 2 - 2x

Graph of the function y = f(x) + g(x)

Consider the problem of plotting the function y = f(x) + g(x). if graphs of functions are given y = f(x) and y = g(x).

Note that the domain of the function y = |f(x) + g(х)| is the set of all those values ​​of x for which both functions y = f(x) and y = g(x) are defined, i.e. this domain of definition is the intersection of the domains of definition, the functions f(x) and g(x).

Let the points (x 0, y 1) and (x 0, y 2) respectively belong to the function graphs y = f(x) and y = g(x), i.e. y 1 \u003d f (x 0), y 2 \u003d g (x 0). Then the point (x0;. y1 + y2) belongs to the graph of the function y = f(x) + g(x)(for f(x 0) + g(x 0) = y 1+y2),. and any point of the graph of the function y = f(x) + g(x) can be obtained in this way. Therefore, the graph of the function y = f(x) + g(x) can be obtained from function graphs y = f(x). and y = g(x) by replacing each point ( x n, y 1) function graphics y = f(x) dot (x n, y 1 + y 2), where y 2 = g(x n), i.e., by shifting each point ( x n, y 1) function graph y = f(x) along the axis at by the amount y 1 \u003d g (x n). In this case, only such points are considered. X n for which both functions are defined y = f(x) and y = g(x).

This method of plotting a function graph y = f(x) + g(x) is called the addition of graphs of functions y = f(x) and y = g(x)

Example 4. In the figure, by the method of adding graphs, a graph of the function is constructed
y = x + sinx.

When plotting a function y = x + sinx we assumed that f(x) = x, a g(x) = sinx. To build a function graph, we select points with abscissas -1.5π, -, -0.5, 0, 0.5,, 1.5, 2. Values f(x) = x, g(x) = sinx, y = x + sinx we will calculate at the selected points and place the results in the table.

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 something is not understood, not completed, or missed somewhere, 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 noteworthy Russian-language charting calculator. 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 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 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 most forte Grafikus.ru - the ability to build 3D graphs. Otherwise, it works no worse and no better than analogue resources.

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

Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions. All materials are checked by an antivirus program.

Teaching aids and simulators in the online store "Integral" for grade 7
Electronic textbook for grade 7 "Algebra in 10 minutes"
Educational complex 1C "Algebra, grades 7-9"

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. Completely finished on a small ship 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.

Decision:

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.

It is not difficult to find calculators on the Internet for plotting a function graph, which are offered to your attention in this review.

http://www.yotx.ru/

This service can build:

• regular graphs (like y = f(x)),
• given parametrically,
• dot charts,
• graphs of functions in the polar coordinate system.

This is an online service one step:

• Enter the function to be built

In addition to plotting a function graph, you will receive the result of the study of the function.

Plotting functions:

http://matematikam.ru/calculate-online/grafik.php

You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the chart window, you can hide both the left column and the virtual keyboard.

Benefits of online charting:

• Visual display of introduced functions
• Building very complex graphs
• Plotting implicitly defined graphs (e.g. ellipse x^2/9+y^2/16=1)
• The ability to save charts and get a link to them, which becomes available to everyone on the Internet
• Scale control, line color
• The ability to plot graphs by points, the use of constants
• Construction of several graphs of functions at the same time
• Plotting in polar coordinates (use r and θ(\theta))

The service is in demand for finding intersection points of functions, for displaying graphs for their further movement to word document as illustrations in solving problems, for analyzing the behavioral features of function graphs. The best browser for working with charts on this page of the site is Google Chrome. When using other browsers, correct operation is not guaranteed.

http://graph.reshish.ru/

You can build an interactive function graph online. Thanks to this, the graph can be scaled, as well as moved along the coordinate plane, which will allow you not only to get a general idea about the construction of this graph, but also to study in more detail the behavior of the function graph on sections.

To build a graph, select the function you need (on the left) and click on it, or enter it yourself in the input field, and click 'Build'. The variable 'x' is used as an argument.

To set a function nth root from 'x', use the notation x^(1/n) - pay attention to the brackets: without them, following the mathematical logic, you will get (x^1)/n.

You can omit the multiplication sign in expressions with a number: 5x, 10sin(x), 3(x-1); between brackets:(x-7)(4+x); and also between the variable and brackets: x(x-3). Expressions like xsin(x) or xx will throw an error.

Consider the priority of operations and if you are not sure which will be executed first, put extra parentheses. For example: -x^2 and (-x)^2 are not the same.

Keep in mind that the graph may not draw if it tends to infinity in 'y' fast enough, due to the computer's inability to infinitely approach the asymptote in 'x'. This does not mean that the graph breaks off and does not continue to infinity.

AT trigonometric functions The default is the radian measure of the angle.

http://easyto.me/services/graphic/

In order to build multiple graphs in the same coordinate system, check the box "Build in the same coordinate system" and plot the graphs of the functions one by one.

The service allows you to build graphs of functions in which there are options.

For this:

1. Enter a function with parameters and click "Plot"
2. In the window that appears, select which of the variables to build a graph with respect to. Usually this is x.
3. Change the parameter values ​​in the History menu. The schedule will change before your eyes.

http://allcalc.ru/node/650

The service allows you to build function graphs in a rectangular coordinate system for a given range of values. In one coordinate plane, you can build several graphs of functions at once.
To build a graph of a function, you need to set the area for plotting the graph (for the variable x and the function y) and enter the value of the dependence of the function on the argument. It is possible to build several graphs at the same time, for this it is necessary to separate the functions with a semicolon. Graphs will be built on the same coordinate plane and will differ in color for clarity.

http://function-graph.ru/

To plot a function online, you just need to enter your function in special field and click somewhere outside of it. After that, the graph of the introduced function will be drawn automatically.

If you need to plot multiple functions at the same time, then click on the blue "Add more" button. After that, another field will open, in which you will need to enter the second function. Her schedule will also be built automatically.

You can adjust the color of the graph lines by clicking on the box located to the right of the function input field. The rest of the settings are right above the graph area. With their help, you can set the background color, the presence and color of the grid, the presence and color of the axes, as well as the presence and color of the numbering of chart segments. If necessary, you can scale the graph of the function using the mouse wheel or special icons in the lower right corner of the drawing area.

After plotting the graph and making the necessary changes to the settings, you can download chart using the big green "Download" button at the very bottom. You will be prompted to save the graph of the function as a PNG image.

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