Online trigonometry graphing. Function graph. Lesson on the topic: "Graph and properties of the function $y=x3$. Examples of plotting graphs"

Graphing functions is one of Excel's capabilities. In this article we will look at the process of plotting some mathematical functions: linear, quadratic and inverse proportionality.

A function is a set of points (x, y) satisfying the expression y=f(x). Therefore, we need to fill in an array of such points, and Excel will build a function graph based on them.

1) Consider an example of plotting linear function: y=5x-2

The graph of a linear function is a straight line that can be constructed from two points. Let's create a sign

In our case y=5x-2. To the cell with the first value y let's introduce the formula: =5*D4-2. You can enter the formula in another cell in the same way (by changing D4 on D5) or use the autocomplete marker.

As a result, we will get a plate:

Now you can start creating a graph.

Select: INSERT -> SOT -> SOT WITH SMOOTH CURVES AND MARKERS (I recommend using this type of diagram)

An empty chart area will appear. Click the SELECT DATA button

Let's select the data: the range of cells on the x-axis (x) and ordinate (y) axis. As the name of the series, we can enter the function itself in quotes “y=5x-2” or something else. Here's what happened:

Click OK. We have a graph of a linear function.

2) Consider the process of plotting quadratic function— parabolas y=2x 2 -2

It is no longer possible to construct a parabola from two points, unlike a straight line.

Set the interval on the axis x, on which our parabola will be built. I'll choose [-5; 5].

I'll take a step. The smaller the step, the more accurate the constructed graph will be. I'll choose 0,2 .

Filling out the column with values X using the autocomplete marker to the value x=5.

Value Column at calculated by the formula: =2*B4^2-2. Using the autocomplete marker, we calculate the values at for the rest X.

Select: INSERT -> POINT -> POINT WITH SMOOTH CURVES AND MARKERS and proceed similarly to constructing a graph of a linear function.

To avoid points on the graph, change the chart type to DOT WITH SMOOTH CURVES.

Any other graphics continuous functions are built similarly.

3) If the function is piecewise, then it is necessary to combine each “piece” of the graph in one area of ​​the diagrams.

Let's look at this using the function example y=1/x.

The function is defined on the intervals (- infinite;0) and (0; +infinite)

Let's create a graph of the function on the intervals: [-4;0) and (0; 4].

Let's prepare two tables where x changes in steps 0,2 :

Finding the function values ​​from each argument X similar to the examples above.

You must add two rows to the diagram - for the first and second plates, respectively

We get the graph of the function y=1/x

In addition, I provide a video showing the procedure described above.

In the next article I will tell you how to create 3-dimensional graphs in Excel.

Thank you for your attention!

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, it will turn out broken line, not a curve. Therefore, in this case, the computer is an 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 in given 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 contains only the necessary tools, such as zooming, moving around coordinate plane and viewing the coordinates of the point at which the mouse is pointing.

Instructions:

1. Enter your equation in the field after the "=" sign.
2. 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:

1. Select the desired method for setting the schedule.
2. Enter your equation.
3. Set the interval.
4. 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:

1. Find the function you need from the list.
2. Left click on it
3. If necessary, enter coefficients in the field "Function:".
4. 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:

1. In the left column, right-click on an empty line.
2. In the lower left corner, click on the keyboard icon.
3. In the panel that appears, enter the required equation (to write the names of functions, go to the “A B C” section).
4. 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 to satisfy the requirements of any mathematician, young or old. Good luck to you in comprehending the “queen of sciences”!

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

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Properties of the function $y=x^3$

Let's describe the properties of this function:

1. x is an independent variable, y is a dependent variable.

2. Domain of definition: it is obvious that for any value of the argument (x) the value of the function (y) can be calculated. Accordingly, the domain of definition of this function is the entire number line.

3. Range of values: y can be anything. Accordingly, the range of values ​​is also the entire number line.

4. If x= 0, then y= 0.

Graph of the function $y=x^3$

1. Let's create 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 for negative values ​​of x the function $y=x^3$ has opposite meanings, then the graph of the function is symmetrical about 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. On a small ship it was completely over fresh water. It is necessary to bring a sufficient amount of 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 I order so as not to overpay for an extra cube and completely fill the tank? It is known that the tank has same length, width and height, which are equal to 1.5 m. Let's solve this problem without performing calculations.

Solution:

1. Let's plot the function $y=x^3$.
2. Find point A, x coordinate, which is equal to 1.5. We see that the coordinate of the function is between values ​​3 and 4 (see Fig. 2). So you need to order 4 cubes.

Constructing graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, everything is not so bad. It is enough to remember a few algorithms for solving such problems, and you can easily build a graph of even the most seemingly complex function. Let's figure out what kind of algorithms these are.

1. Plotting a graph of the function y = |f(x)|

Note that the set of function values ​​y = |f(x)| : y ≥ 0. Thus, the graphs of such functions are always located entirely in the upper half-plane.

Plotting a graph of the function y = |f(x)| consists of the following simple four steps.

1) Carefully and carefully construct a graph of the function y = f(x).

2) Leave unchanged all points on the graph that are above or on the 0x axis.

3) Display the part of the graph that lies below the 0x axis symmetrically relative to the 0x axis.

Example 1. Draw a graph of the function y = |x 2 – 4x + 3|

1) We build a graph of the function y = x 2 – 4x + 3. Obviously, the graph of this function is a parabola. Let's find the coordinates of all points of intersection of the parabola with the coordinate axes and the coordinates of the vertex of the parabola.

x 2 – 4x + 3 = 0.

x 1 = 3, x 2 = 1.

Therefore, the parabola intersects the 0x axis at points (3, 0) and (1, 0).

y = 0 2 – 4 0 + 3 = 3.

Therefore, the parabola intersects the 0y axis at the point (0, 3).

Parabola vertex coordinates:

x in = -(-4/2) = 2, y in = 2 2 – 4 2 + 3 = -1.

Therefore, point (2, -1) is the vertex of this parabola.

Draw a parabola using the data obtained (Fig. 1)

2) The part of the graph lying below the 0x axis is displayed symmetrically relative to the 0x axis.

3) We get a graph of the original function ( rice. 2, shown in dotted line).

2. Graphing the function y = f(|x|)

Note that functions of the form y = f(|x|) are even:

y(-x) = f(|-x|) = f(|x|) = y(x). This means that the graphs of such functions are symmetrical about the 0y axis.

Plotting a graph of the function y = f(|x|) consists of the following simple chain of actions.

1) Graph the function y = f(x).

2) Leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the part of the graph specified in point (2) symmetrically to the 0y axis.

4) As the final graph, select the union of the curves obtained in points (2) and (3).

Example 2. Draw a graph of the function y = x 2 – 4 · |x| + 3

Since x 2 = |x| 2, then the original function can be rewritten in the following form: y = |x| 2 – 4 · |x| + 3. Now we can apply the algorithm proposed above.

1) We carefully and carefully build a graph of the function y = x 2 – 4 x + 3 (see also rice. 1).

2) We leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display right side graphics are symmetrical to the 0y axis.

(Fig. 3).

Example 3. Draw a graph of the function y = log 2 |x|

We apply the scheme given above.

1) Graph the function y = log 2 x (Fig. 4).

3. Plotting the function y = |f(|x|)|

Note that functions of the form y = |f(|x|)| are also even. Indeed, y(-x) = y = |f(|-x|)| = y = |f(|x|)| = y(x), and therefore, their graphs are symmetrical about the 0y axis. The set of values ​​of such functions: y ≥ 0. This means that the graphs of such functions are located entirely in the upper half-plane.

To plot the function y = |f(|x|)|, you need to:

1) Carefully construct a graph of the function y = f(|x|).

2) Leave unchanged the part of the graph that is above or on the 0x axis.

3) Display the part of the graph located below the 0x axis symmetrically relative to the 0x axis.

4) As the final graph, select the union of the curves obtained in points (2) and (3).

Example 4. Draw a graph of the function y = |-x 2 + 2|x| – 1|.

1) Note that x 2 = |x| 2. This means that instead of the original function y = -x 2 + 2|x| – 1

you can use the function y = -|x| 2 + 2|x| – 1, since their graphs coincide.

We build a graph y = -|x| 2 + 2|x| – 1. For this we use algorithm 2.

a) Graph the function y = -x 2 + 2x – 1 (Fig. 6).

b) We leave that part of the graph that is located in the right half-plane.

c) We display the resulting part of the graph symmetrically to the 0y axis.

d) The resulting graph is shown in the dotted line in the figure (Fig. 7).

2) There are no points above the 0x axis; we leave the points on the 0x axis unchanged.

3) The part of the graph located below the 0x axis is displayed symmetrically relative to 0x.

4) The resulting graph is shown in the figure with a dotted line (Fig. 8).

Example 5. Graph the function y = |(2|x| – 4) / (|x| + 3)|

1) First you need to plot the function y = (2|x| – 4) / (|x| + 3). To do this, we return to Algorithm 2.

a) Carefully plot the function y = (2x – 4) / (x + 3) (Fig. 9).

Note that this function is fractional linear and its graph is a hyperbola. To plot a curve, you first need to find the asymptotes of the graph. Horizontal – y = 2/1 (the ratio of the coefficients of x in the numerator and denominator of the fraction), vertical – x = -3.

2) We will leave that part of the graph that is above the 0x axis or on it unchanged.

3) The part of the graph located below the 0x axis will be displayed symmetrically relative to 0x.

4) The final graph is shown in the figure (Fig. 11).

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First, try to find the domain of the function:

Did you manage? Let's compare the answers:

Is everything right? Well done!

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

Found it? Let's compare:

Got it? Well done!

Let's work with graphs again, only now it will be a little more complicated - find both the domain of definition of the function and the range of values ​​of the function.

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

Here's what happened:

I think you've figured out the graphs. Now let’s try to find the domain of definition of a function in accordance with the formulas (if you don’t know how to do this, read the section about):

Did you manage? Let's check answers:

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

However, we still have one more unanswered point...

I will repeat the definition once again and emphasize it:

Did you notice? The word “single” is a very, very important element of our definition. I'll try to explain it to you with my fingers.

Let's say we have a function defined by a straight line. . At, we substitute this value into our “rule” and get that. One value corresponds to one value. We can even make a table of the different values ​​and graph this function to see for ourselves.

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

The fact that “ ” appears twice is not a reason to accuse the parabola of ambiguity!

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

Got it? If not, here is a life example that is very far from mathematics!

Let's say we have a group of applicants who met while submitting documents, each of whom told in a conversation where he lives:

Agree, it is quite possible for several guys to 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 X's correspond to the same game.

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

Here we have a completely different situation: one person can easily submit documents for one or several directions. That is one element sets are put into correspondence several elements multitudes. Respectively, this is not a function.

Let's test your knowledge in practice.

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

Got it? And here it is answers:

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

You ask why? Yes, here's why:

In all pictures except IN) And E) There are several for one!

I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the range of permissible values ​​of an argument and the range of definition of a function. Let's move on to the next section - how to set a function?

Methods for specifying a function

What do you think the words mean? "set function"? That's right, this means explaining to everyone what the function is in this case. we're talking about. And explain it in such a way that everyone understands you correctly and the function graphs drawn by people based on your explanation are the same.

How can this be done? How to set a function? The simplest method, 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 by which it becomes clear to us and to another person how an X turns into a Y.

Usually, this is exactly what they do - in tasks we see ready-made functions specified by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” baffles. Let's understand everything in order, and let's start with the analytical method.

Analytical method of specifying a function

The analytical method is to specify a function using a formula. This is the most universal, comprehensive and unambiguous method. If you have a formula, then you know absolutely everything about a function - you can make a table of values ​​​​from it, you can build a graph, determine where the function increases and where it decreases, in general, study it in full.

Let's consider the function. What's the difference?

"What does it mean?" - you ask. I'll explain now.

Let me remind you that in the notation the expression in brackets is called an argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (the expression in brackets) is, we will write it instead in the expression.

In our example it will look like this:

Let's consider another task related to the analytical method of specifying a function, which you will have on the exam.

Find the value of the expression at.

I'm sure that at first you were scared when you saw such an expression, but there is absolutely nothing scary about it!

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

What needs to be done in our example? Instead you need to write, and instead -:

shorten the resulting expression:

That's it!

Independent work

Now try to find the meaning of the following expressions yourself:

1. , If
2. , If

Did you manage? Let's compare our answers: We are used to the fact that the function has the form

Even in our examples, we define the function in exactly this way, but analytically it is possible to define the function in an implicit form, for example.

Try building this function yourself.

Did you manage?

This is how I built it.

What equation did we finally derive?

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

This is exactly what we were talking 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 the help of a drawing. What did you get?

“Because one value corresponds to several values!”

What conclusion can we draw from this?

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

Tabular method of specifying a function

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

Here you immediately noticed a pattern - the Y is three times larger than the X. And now the task to “think very carefully”: do you think that a function given in the form of a table is equivalent to a function?

Let's not talk for a long time, but let's draw!

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

Do you see the difference? It's not all about the marked points! Take a closer look:

Have you seen it now? When we define a function tabular method, we reflect on the graph 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. This is the peculiarity. Remember!

Graphical method of constructing a function

The graphical method of constructing a function is no less convenient. We draw our function, and another interested person can find what y 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 talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Do you remember? Just in case, I’ll copy here the definition of what a function is:

As a rule, people usually name exactly the three ways of specifying a function that we have discussed - analytical (using a formula), tabular and graphical, completely forgetting that a function can be described verbally. How is this? Yes, very simple!

Verbal description of the function

How to describe a function verbally? Let's take our recent example - . This function can be described as “for every real value of x there corresponds its triple value.” That's it. Nothing complicated. You, of course, will object - “there are so complex functions, which are simply impossible to ask verbally!” Yes, there are such, but there are functions that are easier to describe verbally than to define with a formula. For example: “each natural value of x corresponds to the difference between the digits of which it consists, while the minuend is taken to be the largest digit contained in the notation of the number.” Now let's look at how our verbal description of the function is implemented in practice:

The highest figure in given number- , respectively, is a minuend, then:

Main types of functions

Now let's move on to the most interesting part - let's look at the main types of functions with which you have worked/are working and will work in the course of school and college mathematics, that is, let's get to know them, so to speak, and give them brief description. Read more about each function in the corresponding section.

Linear function

Function of the form, where, - real numbers.

The graph of this function is a straight line, so constructing a linear function comes down to finding the coordinates of two points.

The position of the straight line on the coordinate plane depends on the angular coefficient.

The scope of a 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 downwards, when the branches are directed upwards.

Many properties of a quadratic function depend on the value of the discriminant. The discriminant is calculated using the formula

The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:

Domain of definition

The range of values ​​depends on the extremum of the given function (vertex point of the parabola) and the coefficient (direction of the branches of the parabola)

Inverse proportionality

The function given by the formula, where

The number is called the coefficient of inverse proportionality. Depending on the value, the branches of the hyperbola are in different squares:

Scope of definition - .

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.

• - this is a formula denoting a function, that is, the dependence of one variable on another;
• - variable value, or argument;
• - dependent quantity - changes when the argument changes, that is, according to any specific formula reflecting the dependence of one quantity on another.

2. Valid argument values, or the domain of a function, is what is associated with the possibilities in which the function makes sense.

3. Function range- this is what values ​​it takes, given acceptable values.

4. There are 4 ways to set a function:

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

5. Main types of functions:

• : , where, are real numbers;
• : , Where;
• : , Where.