C 3 graph function. Graphs and basic properties of elementary functions. Power function with irrational exponent

Let us choose a rectangular coordinate system on the plane and plot the values of the argument on the abscissa axis NS, 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 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 NS, at which satisfy the relation y = f (x).

In fig. 45 and 46 are graphs of functions y = 2x + 1 and y = 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 = a belongs to the domain of the function y = f (x), then to find the number f (a)(i.e., the values of the function at the point x = a) you should do this. It is necessary through a point with an abscissa x = a draw a straight line parallel to the ordinate; 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.

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

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

The table looks like this:

Having compiled such a table, we can outline several points of 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).

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 taken points remains unknown.

Example 1... To plot the 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? 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 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 a straight line at all (it is shown in Fig. 49). Another example is the function y = x + l + sinπx; its values are also described in the table above.

These examples show that the pure multi-point charting method is unreliable. 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 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.

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

The graph of the function y = | f (x) |.

Often you have to plot a function y = | 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 = | f (x) | can be obtained from the graph, function y = f (x) as follows: all points of the graph of the function y = f (x) for which the ordinates are non-negative should be left unchanged; further, instead of the points of the graph of the function y = f (x) with negative coordinates, you should build the corresponding points of 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 NS, should be symmetrically reflected about the axis NS).

Example 2. Plot function y = | x |.

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

Example 3... Plot 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 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 it is this part of the graph reflect symmetrically about the abscissa 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 plotting the function y = f (x) + g (x). if function graphs are given y = f (x) and y = g (x).

Note that the domain 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 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 = 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 the function graphs y = f (x)... and y = g (x) replacing each point ( x n, y 1) function graphics y = f (x) point (x n, y 1 + y 2), where y 2 = g (x n), i.e., by the shift of 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 NS 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 the addition of the graphs of the functions y = f (x) and y = g (x)

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

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

First, try to find the scope of the function:

Did you manage? Let's compare the answers:

Is that correct? Well done!

Now let's try to find the range of values of the function:

Found? Compare:

Did it come together? 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's values.

How to find both the domain and the domain of a function (advanced)

Here's what happened:

With the graphs, I think you figured it out. Now let's try, in accordance with the formulas, to find the scope of the function definition (if you do not know how to do this, read the section on):

Did you manage? Verify the answers:

, since the radical expression must be greater than or equal to zero.
, since you cannot divide by zero and the radical expression cannot be negative.
, since, respectively, for all.
, since you cannot divide by zero.

However, we still have one more not analyzed moment ...

I will repeat the definition again and emphasize it:

Did you notice? The word "only" is a very, very important element of our definition. I will try to explain it to you on my 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 compile a table of different values and plot a given function to be sure.

"Look! - you say, - "" occurs twice! " So maybe a parabola is not a function? No, it is!

The fact that "" occurs twice is not a reason to blame the parabola for ambiguity!

The fact is that, when calculating for, we got one game. And when calculating with, we got one game. So that's right, a parabola is a function. Look at the graph:

Understood? If not, here's a real life example so 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 possible that several guys live in one city, but it is impossible for one person to live in several cities at the same time. This is like a logical representation of our "parabola" - several different Xs correspond to the same game.

Now let's come up with an example where the dependency is not a function. Let's say the same guys told what specialties they applied for:

Here we have a completely different situation: one person can easily submit documents for both one and several directions. That is one element set is put into correspondence multiple items sets. Respectively, it is not a function.

Let's put your knowledge to the test.

Determine from the pictures what is a function and what is not:

Understood? Here comes the answers:

The function is - B, E.
A function is not - A, B, D, D.

Why do you ask? Here's why:

In all figures except V) and E) there are several for one!

I am sure that now you can easily distinguish a function from a non-function, you will tell what an argument is and what a dependent variable is, as well as define the range of valid values of the argument and the range of definition of the function. Moving on to the next section - how do you 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 we are talking about in this case. And explain so that everyone understands you correctly and the graphs of functions drawn by people according to your explanation are 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, is using the 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 X turns into a game.

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, which everyone forgets, in connection with which the question "how else can you set a function?" is baffling. Let's figure it out in order, and start with the analytical method.

Analytical way of defining a function

The analytical way is to define a function using a formula. This is the most versatile and comprehensive and unambiguous way. If you have a formula, then you know absolutely everything about a function - you can make a table of values based 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, an expression in parentheses is called an argument. And this argument can be any expression, not necessarily just. Accordingly, whatever the argument (expression in brackets), we will write it instead of in the expression.

In our example, it will look like this:

Let's consider another task related to the analytical way of setting a function that you will have on the exam.

Find the value of the expression, when.

I am sure that at first, you were scared when you saw such an expression, but there is absolutely nothing wrong with it!

Everything is the same as in the previous example: whatever the argument (expression in brackets), we will write it instead of in the expression. For example, for a function.

What needs to 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 a function having the form

Even in our examples, we define a function in exactly this way, but analytically, you can define a function implicitly, for example.

Try to build this function yourself.

Did you manage?

This is how I built it.

What equation did we derive in the end?

Right! Linear, which means that the graph will be a straight line. Let's make a plate to determine which points belong to our line:

This is exactly what we talked 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 happened to you?

"Because several values correspond to one value!"

What conclusion can we draw from this?

That's right, a function cannot always be expressed explicitly, and not always what is "disguised" as a function is a function!

Tabular way of defining a function

As the name suggests, this method is a simple sign. Yes Yes. Like the one that you and I have already made up. For example:

Here you immediately noticed a pattern - the game is three times more than the X. And now the task for "thinking very well": do you think a function given in the form of a table is equivalent to a function?

We will not argue for a long time, but we will draw!

So. We draw a function specified by the wallpaper in the following ways:

Do you see the difference? The point is not at all about the marked points! Take a closer look:

Did you see it now? When we set the function in a tabular way, we reflect on the chart 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 analytically, 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 the game 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 were talking about at the very beginning - not every "squiggle" drawn in the coordinate system is a function! Remembered? Just in case, I'll copy the definition here for what a function is:

As a rule, people usually name exactly those three ways of defining a function that we have analyzed - analytical (using a formula), tabular and graphical, completely forgetting that the function can be described verbally. Like this? It's very simple!

Functional description

How do you 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. You, of course, will object - "there are such complex functions that it is simply impossible to set verbally!" Yes, there are some, but there are functions that are easier to describe verbally than using 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 record is taken as the decreasing one." Now let's see how our verbal description of the function is implemented in practice:

The largest digit in a given number is, accordingly, the decreasing, 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 / are working and will work in the course of school and college mathematics, that is, we will get to know them, so to speak, and give them a brief description. Read more about each function in the corresponding section.

Linear function

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.

The position of the straight line on the coordinate plane depends on the slope.

The scope of the function (aka the scope of valid argument values) is.

Range of values -.

Quadratic function

Function of the form, where

The graph of the function is a parabola, when the branches of the parabola are directed downward, when - upward.

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 point of the apex of the parabola) and the coefficient (the direction of the branches of the parabola)

Inverse proportion

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 - .

Range of values -.

SUMMARY AND BASIC FORMULAS

1. A function is a rule according to which each element of a set is associated with a single element of the set.

is a formula that denotes a function, that is, the dependence of one variable on another;
- variable, or, argument;
- dependent quantity - changes when the argument changes, that is, according to a certain formula reflecting the dependence of one quantity on another.

2. Valid argument values, or the domain of a function is that which is related to the possible, in which the function makes sense.

3. Range of values of the function- this is what values it takes, given the acceptable values.

4. There are 4 ways to define a function:

analytical (using formulas);
tabular;
graphic
verbal description.

5. The main types of functions:

:, where, - real numbers;
: , where;
: , where.

Build function

We bring to your attention a service for the construction of graphs of functions 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 entered functions
Building very complex graphs
Creation of graphs given implicitly (for example, ellipse x ^ 2/9 + y ^ 2/16 = 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 to build charts of varying complexity online with us. 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.

A function graph is a visual representation of the behavior of a function on a coordinate plane. Graphs help you understand various aspects of a function that cannot be identified from the function itself. You can plot graphs of many functions, and each of them will be given by a certain formula. The graph of any function is built according to a certain algorithm (if you have forgotten the exact process of plotting a graph of a specific function).

Steps

Plotting a Linear Function

Determine if the function is linear. The linear function is given by a formula of the form F (x) = k x + b (\ displaystyle F (x) = kx + b) or y = k x + b (\ displaystyle y = kx + b)(for example), and its graph is a straight line. Thus, the formula includes one variable and one constant (constant) without any exponents, root signs, and the like. Given a function of a similar type, it is quite easy to plot such a function. Here are other examples of linear functions:

Use a constant to mark a point on the Y axis. Constant (b) is the “y” coordinate of the point of intersection of the graph with the y-axis. That is, it is the point whose “x” coordinate is 0. Thus, if you substitute x = 0 in the formula, then y = b (constant). In our example y = 2 x + 5 (\ displaystyle y = 2x + 5) the constant is 5, that is, the y-intercept has coordinates (0.5). Draw this point on the coordinate plane.

Find the slope of the line. It is equal to the multiplier of the variable. In our example y = 2 x + 5 (\ displaystyle y = 2x + 5) at the variable "x" there is a factor of 2; thus, the slope is 2. The slope determines the angle of inclination of the straight line to the X-axis, that is, the larger the slope, the faster the function increases or decreases.

Write down the slope as a fraction. The slope is equal to the tangent of the slope, that is, the ratio of the vertical distance (between two points on a straight line) to the horizontal distance (between the same points). In our example, the slope is 2, so we can state that the vertical distance is 2 and the horizontal distance is 1. Write this as a fraction: 2 1 (\ displaystyle (\ frac (2) (1))).

If the slope is negative, the function is decreasing.

From the intersection of the line with the Y-axis, draw a second point using the vertical and horizontal distances. A linear function graph can be plotted from two points. In our example, the y-intercept has coordinates (0.5); from this point, move 2 divisions up, and then 1 division to the right. Mark the point; it will have coordinates (1,7). Now you can draw a straight line.

Use a ruler to draw a straight line through two points. Find the third point to avoid mistakes, but in most cases you can plot a graph from two points. Thus, you have plotted a linear function.

Placing points on the coordinate plane
1. Define a function. The function is denoted as f (x). All possible values of the variable "y" are called the range of values of the function, and all possible values of the variable "x" are called the range of the function. For example, consider the function y = x + 2, namely f (x) = x + 2.
  
  Draw two intersecting perpendicular lines. The horizontal line is the X-axis. The vertical line is the Y-axis.
  
  Label the coordinate axes. Divide each axis into equal segments and number them. The point of intersection of the axes is 0. For the X-axis, positive numbers are plotted to the right (from 0), and negative numbers to the left. For the Y-axis: positive numbers are plotted above (from 0), and negative numbers below.
  
  Find the y-values from the x-values. In our example, f (x) = x + 2. Plug in the specific x-values into this formula to calculate the corresponding y-values. If you have a complex function, simplify it by isolating the "y" on one side of the equation.
  - -1: -1 + 2 = 1
  - 0: 0 +2 = 2
  - 1: 1 + 2 = 3
2. Draw points on the coordinate plane. For each pair of coordinates, do the following: find the corresponding value on the X-axis and draw a vertical line (dotted line); find the corresponding value on the Y-axis and draw a horizontal line (dotted line). Mark the point of intersection of the two dashed lines; thus you have plotted a point on the graph.
  
  Erase the dotted lines. Do this after plotting all the points of the graph on the coordinate plane. Note: the graph of the function f (x) = x is a straight line passing through the center of coordinates [point with coordinates (0,0)]; the graph f (x) = x + 2 is a straight line parallel to the straight line f (x) = x, but shifted two units up and therefore passing through the point with coordinates (0,2) (because the constant is 2).
Plotting a Complex Function