Function graph. Sketch of a graph of a function (on the example of a fractional-quadratic function) Benefits of plotting online

Build a function

We bring to your attention a service for plotting function graphs online, all rights to which belong to the company Desmos. Use the left column to enter functions. 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))

With us it is easy to build graphs of varying complexity online. The construction is done instantly. 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.

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; наименьшее значение функция 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 approximate view function graph 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 with 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.

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

Theme: Repetition

Lesson: Sketching a function graph (using a fractional quadratic function as an example)

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

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

The sketching technique is as follows:

1. Select the intervals of constancy of sign and determine the sign of the function on each (Figure 1)

We considered 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 discontinuity points of the ODZ.

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

Let's find the roots:

Let's single out intervals of sign constancy. We have found the roots of the function and the break points of the domain of definition - the roots of the denominator. It is important to note that within each interval the function retains its sign.

Rice. 1. INTERVALS OF CONSTANT SIGN OF A FUNCTION

To determine the sign of a function on each interval, you can take any point that belongs 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 trial point and conclude that the function will have the same sign over the entire chosen interval.

However, it is possible to set the signs automatically, without calculating the values of the function, to do 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 :

Rice. 2. Graph in the vicinity of the roots

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

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

Rice. 3. Graph of the function in the vicinity of the discontinuity points of the ODZ

When or the denominator of a fraction is practically equal to zero, then when the value of the argument approaches these numbers, the value of the fraction approaches 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 exits from minus infinity. Around the four, on the contrary, the function tends to minus infinity on the left, and exits from plus infinity on the right.

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

Rice. 4. Sketch of the graph of the function

Consider the following important task - to construct a sketch of the graph of a function in the vicinity of infinitely distant points, i.e. when the argument tends to 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:

Rice. 5. Sketch of the graph of a function in the vicinity of points at infinity

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

Example 1 - sketch a function graph:

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

We determine the signs of the function on 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, then the curve is first above the axis, then passes through zero and then is located under the x-axis. When or the denominator of a fraction is practically equal to zero, then when the value of the argument approaches these numbers, the value of the fraction approaches 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 exits plus infinity. Around two is the same.

Let's find the derivative of the function:

It is clear that the derivative is always less than zero, therefore, the function decreases on all segments. So, in the area from minus infinity to minus two, the function decreases from zero to minus infinity; in the area 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 area from two to plus infinity, the function decreases from plus infinity to zero.

Let's illustrate:

Rice. 6. Sketch of the graph of the function for example 1

Example 2 - sketch a graph of a function:

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

First, we examine the given function:

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

Note that the given function is odd.

We determine the signs of the function on 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, then the curve is first 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 tends to plus or minus infinity. In this case, the constant terms can be neglected. We have:

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

Let's find the derivative of the function:

We single out the intervals of constant sign of the derivative: at . ODZ is here. Thus, we have three intervals of constancy of the derivative and three segments of monotonicity of the original function. Let us determine the signs of the derivative on each interval. When the derivative is positive, the function is increasing; when the derivative is negative, the function is decreasing. In this case, the minimum point, because the derivative changes sign from minus to plus; on the contrary, the maximum point.

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