Sinusitis

Online graphing calculator with solution. Graph of a function

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

Function graph y = f(x) is the set of all points whose abscissas belong to the domain of definition of the function, and the ordinates are equal to the corresponding values of the function.

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

In Fig. 45 and 46 show graphs of functions y = 2x + 1 And y = x 2 - 2x.

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

Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the domain of definition of the function y = f(x), then to find the number f(a)(i.e. the function values at the point x = a) you should do this. It is necessary through the abscissa point x = a draw a straight line parallel to the ordinate axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will, by virtue of the definition of the graph, be 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 clearly illustrates the behavior and properties of a function. For example, from consideration of Fig. 46 it is clear that the function y = x 2 - 2x accepts positive values at X< 0 and at x > 2, negative - at 0< x < 2; наименьшее значение функция y = x 2 - 2x accepts at x = 1.

To graph a function f(x) you need to find all the points of the plane, coordinates X,at which satisfy the equation y = f(x). In most cases this is impossible to do, since there are an infinite number of such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the method of plotting a graph using several points. 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 create a table that includes the selected function values.

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 approximate view function graphics y = f(x).

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

Example 1. To graph 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 with 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 statement, consider the function

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

These examples show that in its “pure” form the method of plotting a graph using several points is unreliable. Therefore, to plot a graph of a given function, one usually proceeds as follows. First, the properties of this function are studied, with the help of which you can build a sketch of the graph. Then, by calculating the values of the function at several points (the choice of which depends on the established 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 look at some (the simplest and most frequently used) properties of functions used to find a graph sketch later, but now we will look at some commonly used methods for constructing graphs.

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

It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Let us remind you how this is done. By defining the absolute value of a number, we can write

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

Example 2. Graph the function y = |x|.

Let's take the graph of the function y = x(Fig. 50, a) and part of this graph at X< 0 (lying under the axis X) symmetrically reflected relative to the axis X. As a result, we get the graph of the function y = |x|(Fig. 50, b).

Example 3. Graph the function y = |x 2 - 2x|.

First, let's plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upward, the vertex of the parabola has coordinates (1; -1), its graph intersects the x-axis at points 0 and 2. On the interval (0; 2) the function takes negative values, therefore, we will symmetrically display this part of the graph relative to the x-axis. Figure 51 shows the graph of the function y = |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 constructing a graph of a function y = f(x) + g(x). if function graphs are given y = f(x) And y = g(x).

Note that the domain of definition of the function y = |f(x) + g(x)| 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, functions f(x) and g(x).

Let the points (x 0 , y 1) And (x 0, y 2) respectively belong to the graphs of functions y = f(x) And y = g(x), i.e. y 1 = f(x 0), y 2 = 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 on the graph of the function y = f(x) + g(x) can be obtained 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) 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 = 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 y = f(x) + g(x) is called addition of graphs of functions y = f(x) And y = g(x)

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

When plotting a function y = x + sinx we thought that f(x) = x, A g(x) = sinx. To plot the 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 Let's calculate at the selected points and place the results in the table.

Into the golden age information technology few people will buy graph paper and spend hours drawing a function or an arbitrary set of data, and why bother with such tedious work when you can plot a function graph online. In addition, counting millions of expression values for correct display is almost unrealistic and difficult, and despite all efforts, the result will be a broken line, not a curve. Because the computer is in this case – indispensable assistant.

What is a function graph

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, it is a function. Accordingly, the graph of a function will be the set of points whose coordinates satisfy the given expression.

In the figure we see the graph of the function y = x. This is a straight line and each of its points 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 this equation, then we get the coordinate of this point on the axis Y.

Online services for plotting function graphs

Let's look at several popular and best services that allow you to quickly draw a graph of a function.

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

Instructions:

Enter your equation in the field after the "=" sign.
Click the button "Build a graph".

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

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

Instructions:

Select necessary method schedule assignments.
Enter your equation.
Set the interval.
Click the button "Build".

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

Instructions:

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

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

Desmos is by far the most sophisticated service for constructing equations online. By moving the cursor with the left mouse button held down along the graph, you can view in detail all the solutions to the equation with an accuracy of 0.001. The built-in keyboard allows you to quickly write powers and fractions. The most important advantage is the ability to write the equation in any state without reducing it to the form: y = f(x).

Instructions:

In the left column, right-click on an empty line.
In the lower left corner, click on the keyboard icon.
In the panel that appears, enter the required equation (to write the names of functions, go to the “A B C” section).
The schedule is built in real time.

The visualization is simply perfect, adaptive, it’s clear that designers worked on the application. On the plus side, we can note the huge abundance of possibilities, for mastering which you can see examples in the menu in the upper left corner.

There are a great many sites for constructing function graphs, but everyone is free to choose for themselves based on the required functionality and personal preferences. The list of the best was compiled in such a way as to satisfy the requirements of any mathematician, young and old. Good luck to you in comprehending the “queen of sciences”!

Unfortunately, not all students and schoolchildren know and love algebra, but everyone has to prepare homework, solve tests and take exams. Many people find it especially difficult to construct graphs of functions: if somewhere you don’t understand something, don’t finish learning it, or miss it, mistakes are inevitable. But who wants to get bad grades?

Would you like to join the cohort of tail-seekers and losers? To do this, you have 2 ways: sit down with textbooks and fill in knowledge gaps, or use a virtual assistant - a service for automatically plotting function graphs according to given conditions. With or without a solution. Today we will introduce you to several of them.

The best thing about Desmos.com is its highly customizable interface, interactivity, the ability to organize results into tables and store your work in the resource database for free without time limits. The drawback is that the service is not fully translated into Russian.

Grafikus.ru

Grafikus.ru is another noteworthy Russian-language calculator for creating 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 simple functions: straight 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 the options of downloading, printing and copying a link to it. For the latter, you will have to log in to the service through the social network buttons.

The Grafikus.ru coordinate plane supports changing the boundaries of axes, their labels, grid spacing, as well as the width and height of the plane itself and font size.

The most strong point Grafikus.ru - the ability to create 3D graphics. Otherwise, it works no worse and no better than analogue resources.