"Natural logarithm" - 0.1. Natural logarithms... 4. "Logarithmic darts". 0.04. 7.121.

"Grade 9 power function" - U. Cubic parabola. Y = x3. Grade 9 teacher Ladoshkina I.A. Y = x2. Hyperbola. 0.Y = xn, y = x-n where n is a given natural number... X. Indicator - an even natural number (2n).

"Quadratic function" - 1 Definition quadratic function 2 Properties of the function 3 Graphs of the function 4 Quadratic inequalities 5 Conclusion. Properties: Inequalities: Prepared by grade 8A student Andrey Gorlitz. Plan: Graph: - Monotonic intervals for a> 0 for a< 0. Квадратичная функция. Квадратичные функции используются уже много лет.

"Quadratic function and its graph" - Decision.y = 4x A (0.5: 1) 1 = 1 A-belongs. For a = 1, the formula y = ax takes the form.

"Grade 8 quadratic function" - 1) Construct the vertex of the parabola. Plotting a quadratic function. x. -7. Plot the function. Algebra Grade 8 Teacher of School 496 Bovina T.V. -1. Build plan. 2) Construct the axis of symmetry x = -1. y.

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 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:

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 not analyzed moment ...

I will repeat the definition again and emphasize it:

Did you notice? The word "only" is very, very important element 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 given value into our "rule" and we get that. One value corresponds to one value. We can even compile a table of different values ​​and graph this 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 to you life example 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? And here it is 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, 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?

Methods for setting a function

What do you think the words mean "Set function"? That's right, it means explaining to everyone what function in this case we are talking about. 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 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, 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'm 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:

1. , if
2. , 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 instance:

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 for 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 is so much complex functions, which is simply impossible to ask 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 figure in this number-, respectively, - diminished, 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 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, therefore the construction linear function is reduced to finding the coordinates of two points.

Straight line position on 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. Allowed 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.

Into the golden age information technologies few people will buy graph paper and spend hours drawing a function or an arbitrary data set, and why bother doing such dreary work when you can plot a function online. In addition, it is almost impossible and difficult to calculate millions of values ​​of an expression for correct display, and despite all the efforts, it will turn out to be a broken line, not a curve. Therefore, the computer in this case is irreplaceable assistant.

What is a graph of functions

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

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

Services for plotting functions online

Let's take a look at some of the most popular and best-performing services that allow you to quickly draw a graph of a function.

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

Instructions:

1. Enter your equation in the box 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 a module, trigonometric, exponential - is shown right below the graph. Also, if necessary, you can define the equation parametrically or plot graphs in a polar coordinate system.

Yotx has all the functions of the previous service, but at the same time it contains such interesting innovations as creating an interval for displaying a function, 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 presents a service with the ability to select the one you need from the list with one click of the mouse.

Instructions:

1. Find the function you need in the list.
2. Left-click on it
3. If necessary, enter the 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 completely delete it.

Desmos is by far the most sophisticated equation building service online. Moving the cursor with the left mouse button pressed along the graph, you can see in detail all the solutions to the equation with an accuracy of 0.001. The built-in keyboard allows you to quickly write exponents and fractions. The most important plus is the ability to write the equation in any state, without leading to the form: y = f (x).

Instructions:

1. In the left column, right-click on a free line.
2. In the lower left corner, click on the keyboard icon.
3. On the panel that appears, type the required equation (to write the names of functions, go to the section "A B C").
4. The graph is plotted in real time.

The visualization is just perfect, adaptive, you can see that the designers worked on the application. On the plus side, there is a huge abundance of opportunities, for the development of which you can see examples in the menu in the upper left corner.

There are a great many sites for plotting functions, but everyone is free to choose for themselves based on the required functionality and personal preferences. The list of the best has been formed to satisfy the requirements of any mathematician, young and old. I wish you success in comprehending the "queen of sciences"!

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 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 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 a consideration of Fig. 46 it is clear that the function y = x 2 - 2x takes positive values ​​at X< 0 and at x> 2, negative - at 0< x < 2; smallest value function y = x 2 - 2x takes at x = 1.

To plot the 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 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 X give a finite number of values ​​- say, x 1, x 2, x 3, ..., x k and make up a table containing 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 points taken remains unknown.

Example 1... To plot the function y = 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 ​​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 + 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 most simple and often 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 definition absolute value numbers can be written

This means that the graph of the function y = | f (x) | can be obtained from 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 X, should be symmetrically reflected about the axis X).

Example 2. Plot function y = | x |.

We 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 reflect about the axis X... As a result, we get the graph of the function y = | x |(Fig. 50, b).

Example 3... Plot 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 upwards, the apex 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, that is, 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 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 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 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.

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

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Properties of the function $ y = x ^ 3 $

Let's describe the properties of this function:

1.x is the independent variable, y is the 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 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 x the function $ y = x ^ 3 $ has opposite meanings, then the graph of the function is symmetric about the origin.

Now let's mark points on the coordinate plane and build a graph (see Fig. 1).

This curve is called a cubic parabola.

Examples of

I. The small ship is completely over fresh water... It is necessary to bring enough water from the city. Water is ordered in advance and is paid for a full cube, even if you fill it with a little less. How many cubes do you need to order so as not to overpay for an extra cubic meter and completely fill the tank? It is known that the tank has the same length, width and height, which are equal to 1.5 m. Let's solve this problem without performing any calculations.

Solution:

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