Function graph. Sketch a graph of a function (using a fractional-quadratic function as an example) Benefits of graphing online

Build function

We bring to your attention a service for drawing function charts online, all rights to which belong to the company Desmos... Use the left column to enter functions. You can enter it manually or using the virtual keyboard at the bottom of the window. To enlarge the window with the graph, you can hide both the left column and the virtual keyboard.

Benefits of charting online

  • Visual display of input functions
  • Building very complex graphs
  • Plotting implicitly (e.g. ellipse x ^ 2/9 + y ^ 2/16 \u003d 1)
  • The ability to save charts and receive a link to them, which becomes available to everyone on the Internet
  • Scale control, line color
  • Possibility of plotting graphs by points, using constants
  • Simultaneous construction of several graphs of functions
  • Plotting in polar coordinates (use r and θ (\\ theta))

It is easy with us to build charts of varying complexity online. Construction is done instantly. The service is in demand for finding intersection points of functions, for displaying graphs for their further movement in a Word document as illustrations when solving problems, for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this page of the site is Google Chrome. Operation is not guaranteed with other browsers.

Let us choose a rectangular coordinate system on the plane and plot the values \u200b\u200bof the argument on the abscissa axis x, and on the ordinate - the values \u200b\u200bof the function y \u003d f (x).

Function graph y \u003d f (x) is called the set of all points whose abscissas belong to the domain of the function, and the ordinates are equal to the corresponding values \u200b\u200bof the function.

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



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

Strictly speaking, one should distinguish between the graph of the 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 the entire graph, but only its part located in the final part of the plane). In what follows, however, we will usually say "graph" rather than "sketch graph".

Using the graph, you can find the value of a function at a point. Namely, if the point x \u003d a belongs to the domain of the function y \u003d f (x), then to find the number f (a) (i.e., the values \u200b\u200bof the function at the point x \u003d a) you should do this. It is necessary through a point with an abscissa x \u003d a draw a straight line parallel to the ordinate; this line will intersect the graph of the function y \u003d f (x) at one point; the ordinate of this point will, by virtue of the definition of the graph, equal f (a) (fig. 47).



For example, for the function f (x) \u003d x 2 - 2x using the graph (Fig. 46) we find f (-1) \u003d 3, f (0) \u003d 0, f (1) \u003d -l, f (2) \u003d 0, etc.

The function graph clearly illustrates the behavior and properties of the function. For example, from considering Fig. 46 it is clear that the function y \u003d x 2 - 2x takes positive values \u200b\u200bat x< 0 and at x\u003e 2, negative - at 0< x < 2; наименьшее значение функция y \u003d x 2 - 2x takes at x \u003d 1.

To plot the function f (x)you need to find all points of the plane, coordinates x, at which satisfy the equation y \u003d f (x)... In most cases, this cannot be done, since there are infinitely many such points. Therefore, the graph of the function is shown approximately - with more or less accuracy. The simplest is the multi-point plotting method. It consists in the fact that the argument x give a finite number of values \u200b\u200b- say, x 1, x 2, x 3, ..., x k and make up a table containing the selected values \u200b\u200bof the function.

The table looks like this:



Having compiled such a table, we can outline several points of the function graph y \u003d f (x)... Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y \u003d 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 designated points and its behavior outside the segment between the extreme of the points taken remains unknown.

Example 1... To plot the function y \u003d f (x) someone made 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? If there are no 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 \u200b\u200bof this function at points -2, -1, 0, 1, 2 are just described by the above table. However, the graph of this function is not a straight line at all (it is shown in Fig. 49). Another example is the function y \u003d x + l + sinπx; its values \u200b\u200bare also described in the table above.

These examples show that the pure multi-point charting method is not reliable. Therefore, to build a graph of a given function, as a rule, proceed as follows. First, we study the properties of this function, with which you can build a sketch of the graph. Then, calculating the values \u200b\u200bof 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.

Some (the simplest and most frequently used) properties of functions used to find a sketch of a graph will be discussed later, but now we will analyze some of the commonly used methods of plotting.


Graph of the function y \u003d | f (x) |.

Often you have to plot a function y \u003d | f (x)|, where f (x) -given function. Let us recall how this is done. By determining the absolute value of a number, you can write

This means that the graph of the function y \u003d | f (x) | can be obtained from graph, function y \u003d f (x) as follows: all points of the graph of the function y \u003d f (x)where ordinates are non-negative should be left unchanged; further, instead of the points of the graph of the function y \u003d f (x)with negative coordinates, you should build the corresponding points of the graph of the function y \u003d -f (x) (i.e. part of the graph of the function
y \u003d f (x)which lies below the axis x, should be symmetrically reflected about the axis x).



Example 2. Plot function y \u003d | x |.

Take the graph of the function y \u003d x(Fig. 50, a) and part of this graph at x< 0 (lying under the axis x) symmetrically reflect about the axis x... As a result, we get the graph of the function y \u003d | x | (Fig. 50, b).

Example 3... Plot function y \u003d | x 2 - 2x |.


First, let's plot the function y \u003d 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 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 abscissa axis. Figure 51 shows the graph of the function y \u003d | x 2 -2x |based on the function graph y \u003d x 2 - 2x

Graph of function y \u003d f (x) + g (x)

Consider the problem of plotting the function y \u003d f (x) + g (x). if function graphs are given y \u003d f (x) and y \u003d g (x).

Note that the domain of the function y \u003d | f (x) + g (x) | is the set of all those values \u200b\u200bof x for which both functions y \u003d f (x) and y \u003d g (x) are defined, that is, this domain of definition is the intersection of domains, 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 \u003d f (x) and y \u003d 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 \u003d f (x) + g (x) (for f (x 0) + g (x 0) \u003d y 1 + y2) ,. and any point on the graph of the function y \u003d f (x) + g (x) can be obtained this way. Therefore, the graph of the function y \u003d f (x) + g (x) can be obtained from function graphs y \u003d f (x)... and y \u003d g (x) replacing each point ( x n, y 1) function graphics y \u003d f (x) point (x n, y 1 + y 2), Where y 2 \u003d g (x n), i.e., by shifting each point ( x n, y 1) function graph y \u003d 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 \u003d f (x) and y \u003d g (x).

This method of plotting a function y \u003d f (x) + g (x) is called the addition of the graphs of the functions y \u003d f (x)and y \u003d g (x)

Example 4... In the figure, by adding graphs, a graph of the function is built
y \u003d x + sinx.

When plotting the function y \u003d x + sinx we believed that f (x) \u003d x,and g (x) \u003d sinx.To plot the function graph, select points with abscissas -1.5π, -, -0.5, 0, 0.5 ,, 1.5, 2. Values f (x) \u003d x, g (x) \u003d sinx, y \u003d x + sinxcalculate at the selected points and place the results in the table.


In this lesson, we will consider the technique for constructing a sketch of a function graph, we will give explanatory examples.

Topic: Repetition

Lesson: Sketching a graph of a function (using the example of a fractional-quadratic function)

Our goal is to sketch the graph of the fractional-quadratic function. For example, let's take a function already familiar to us:

A fractional function is given, in the numerator and denominator of which there are quadratic functions.

The sketching technique is as follows:

1. Let's select intervals of sign constancy and define on each sign of the function (Figure 1)

We examined in detail and found out that a function that is continuous in the ODZ can change sign only when the argument passes through the roots and break points of the ODZ.

The given function у is continuous in its ODZ, we indicate the ODV:

Find the roots:

Let's select intervals of constancy. We found the roots of the function and the breakpoints of the domain of definition - the roots of the denominator. It is important to note that the function preserves the sign within each interval.

Figure: 1. Intervals of constant sign function

To determine the sign of a function at each interval, you can take any point belonging to the interval, substitute it into the function, and determine its sign. For instance:

The function has a plus sign on the interval

The function has a minus sign on the interval.

This is the advantage of the interval method: we determine the sign at a single sample point and conclude that the function will have the same sign over the entire selected interval.

However, it is possible to set the signs automatically without calculating the values \u200b\u200bof the function; for this, determine the sign at the extreme interval, and then alternate the signs.

1. Let's build a graph in the neighborhood of each root. Recall that the roots of this function and:

Figure: 2. Graph in the vicinity of the roots

Since at a point the sign of the function changes from plus to minus, the curve is first located above the axis, then passes through zero and then is located below the x axis. The point is the opposite.

2. Let's construct a graph in the vicinity of each gap in the ODZ. Recall that the roots of the denominator of this function and:

Figure: 3. The graph of the function in the vicinity of the discontinuity points

When either the denominator of a fraction is practically zero, it means that when the value of the argument tends to these numbers, the value of the fraction tends to infinity. In this case, when the argument approaches the triple on the left, the function is positive and tends to plus infinity, on the right, the function is negative and goes out of minus infinity. About four, on the contrary, on the left the function tends to minus infinity, and on the right it leaves plus infinity.

According to the constructed sketch, we can guess at some intervals the behavior of the function.

Figure: 4. Sketch function graph

Consider the following important task - to build a sketch of the graph of a function in the vicinity of infinitely distant points, i.e. when the argument approaches plus or minus infinity. In this case, the constant terms can be neglected. We have:

Sometimes you can find such a record of this fact:

Figure: 5. Sketch of the graph of the function in the vicinity of infinitely distant points

We have obtained an approximate character of the behavior of the function throughout its entire domain of definition, then we need to refine the constructions using the derivative.

Example 1 - Sketch a graph of a function:

We have three points, when passing the argument through which the function can change sign.

Determine the signs of the function at each interval. We have a plus on the extreme right interval, then the signs alternate, since all roots have the first degree.

We build a sketch of the graph in the vicinity of the roots and break points of the ODZ. We have: since at the point the sign of the function changes from plus to minus, the curve is first located above the axis, then passes through zero and then is located below the x axis. When either the denominator of a fraction is practically zero, it means that when the value of the argument tends to these numbers, the value of the fraction tends to infinity. In this case, when the argument approaches minus two on the left, the function is negative and tends to minus infinity, on the right, the function is positive and goes out of plus infinity. About two is similar.

Find the derivative of the function:

It is obvious that the derivative is always less than zero, therefore, the function decreases in all sections. So, in the section from minus infinity to minus two, the function decreases from zero to minus infinity; in the section from minus two to zero, the function decreases from plus infinity to zero; in the area from zero to two, the function decreases from zero to minus infinity; in the range from two to plus infinity, the function decreases from plus infinity to zero.

Let's illustrate:

Figure: 6. Sketch of the function graph for example 1

Example 2 - Sketch a graph of a function:

We build a sketch of the function graph without using the derivative.

First, we examine the given function:

We have a single point, when passing the argument through which the function can change sign.

Note that the given function is odd.

Determine the signs of the function at each interval. We have a plus on the extreme right interval, then the sign changes, since the root has the first degree.

We build a sketch of the graph in the vicinity of the root. We have: since at the point the sign of the function changes from minus to plus, the curve is first located under the axis, then passes through zero and then is located above the x axis.

Now we build a sketch of the graph of the function in the vicinity of infinitely distant points, i.e. when the argument approaches plus or minus infinity. In this case, the constant terms can be neglected. We have:

After completing the above steps, we already imagine a function graph, but we need to refine it using a derivative.

Find the derivative of the function:

We select the intervals of constancy of the derivative: at. ODZ here. Thus, we have three intervals of constancy of the derivative and three sections of monotonicity of the original function. Let us determine the signs of the derivative on each interval. When the derivative is positive, the function increases; when the derivative is negative, the function decreases. In this case, the point is the minimum, since the derivative changes sign from minus to plus; on the contrary, the maximum point.

 

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