Power function and roots - definition, properties and formulas. Function y = square root of x, its properties and graph

Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"

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Definition of a power function - cube root

Guys, we continue to study power functions. Today we will talk about the "Cubic root of x" function.
What is a cube root?
The number y is called a cubic root of x (root of the third degree) if the equality $y^3=x$ holds.
Denoted as $\sqrt(x)$, where x is a radical number, 3 is an exponent.
$\sqrt(27)=3$; $3^3=$27.
$\sqrt((-8))=-2$; $(-2)^3=-8$.
As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers.
The third root of a negative number is negative number. When raised to an odd power, the sign is preserved; the third power is odd.

Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. Using the notation for roots we obtain the desired identity.

Properties of cubic roots

a) $\sqrt(a*b)=\sqrt(a)*\sqrt(6)$.
b) $\sqrt(\frac(a)(b))=\frac(\sqrt(a))(\sqrt(b))$.

Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's build a graph of our function.
1) The domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, from an arbitrarily large number we can calculate the third root, and we can move upward indefinitely, finding ever larger values ​​of the argument.
5) For $x≥0$ the smallest value is 0. This property is obvious.
Let's build a graph of the function by points at x≥0.




Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd.

Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.

7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).

Examples of solving power functions

Examples
1. Solve the equation $\sqrt(x)=x$.
Solution. Let's build two graphs on one coordinate plane$y=\sqrt(x)$ and $y=x$.

As you can see, our graphs intersect at three points.
Answer: (-1;-1), (0;0), (1;1).

2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, by parallel translation two units to the right and three units down.

3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function.
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) Greatest value No. Lowest value equals minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).

Problems to solve independently

1. Solve the equation $\sqrt(x)=2-x$.
2. Construct a graph of the function $y=\sqrt((x+1))+1$.
3.Plot a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.

Instead of introducing

The use of modern technologies (CTE) and teaching aids (multimedia board) in lessons helps the teacher plan and conduct effective lessons, create conditions for students to consciously understand, memorize and practice skills.

The lesson turns out to be dynamic and interesting if you combine various forms of teaching during the training session.

In modern didactics, there are four general organizational forms of training:

  • individually mediated;
  • steam room;
  • group;

collective (in shift pairs). (Dyachenko V.K. Modern didactics. - M.: Public education, 2005).

In a traditional lesson, as a rule, only the first three organizational forms of teaching listed above are used. The collective form of teaching (work in pairs in shifts) is practically not used by the teacher. However, this organizational form of training makes it possible for the team to train everyone and everyone to actively participate in the training of others. The collective form of training is leading in CSR technology.

One of the most common methods of collective learning technology is the “Mutual Training” technique.

This “magic” technique is good in any subject and in any lesson. The purpose is training.

Training is the heir to self-control; it helps the student to establish contact with the subject of study, making it easier to find the right steps and actions. Through training in the acquisition, consolidation, regrouping, revision, and application of knowledge, a person’s cognitive abilities develop. (Yanovitskaya E.V. How to teach and study in a lesson so that you want to learn. Reference album. - St. Petersburg: Educational Projects, M.: Publisher A.M. Kushnir, 2009.-P.14; 131)

It will help you quickly repeat a rule, remember the answers to the questions you have studied, and consolidate the necessary skill. The optimal time to work using the method is 5-10 minutes. As a rule, work on training cards is carried out during oral calculation, that is, at the beginning of the lesson, but at the discretion of the teacher it can be carried out at any stage of the lesson, depending on its goals and structure. A training card can contain from 5 to 10 simple examples (questions, tasks). Each student in the class receives a card. The cards are different for everyone or different for everyone in the “combined squad” (children sitting on the same row). A combined detachment (group) is a temporary cooperation of students formed to perform a specific educational task. (Yalovets T.V. Technology of a collective method of teaching in teacher training: Educational and methodological manual. - Novokuznetsk: IPK Publishing House, 2005. - P. 122)

Lesson project on the topic “Function y=, its properties and graph”

In the lesson project, the topic of which is: “ Function y=, its properties and graph” The use of mutual training techniques in combination with the use of traditional and multimedia teaching tools is presented.

Lesson topic: “ Function y=, its properties and graph”

Goals:

  • preparation for the test;
  • testing knowledge of all properties of a function and the ability to build graphs of functions and read their properties.

Tasks: subject level:

supra-subject level:

  • learn to analyze graphic information;
  • practice the ability to conduct dialogue;
  • develop the ability to work with an interactive whiteboard using the example of working with graphs.
Lesson structure Time
1. Teacher Information Input (TII) 5 min.
2. Updating basic knowledge: work in shift pairs according to the methodology “Mutual training” 8 min.
3. Introduction to the topic “Function y=, its properties and graph”: teacher presentation 8 min.
4. Consolidation of newly learned and already covered material on the topic “Function”: using an interactive whiteboard 15 min.
5. Self-control : in the form of a test 7 min.
6. Summing up, recording homework. 2 min.

Let us reveal in more detail the content of each stage.

1. Teacher Information Input (TII) includes organizational moment; articulating the topic, purpose and lesson plan; showing a sample of pair work using the mutual training method.

Demonstration of a sample of work in pairs by students at this stage of the lesson is advisable for repeating the algorithm of work of the methodology we need, because At the next stage of the lesson, the work of the entire class team is planned on it. At the same time, you can name the errors in working with the algorithm (if there were any), as well as evaluate the work of these students.

2. Updating of basic knowledge is carried out in shift pairs using the mutual training method.

The methodology algorithm includes individual, pair (static pairs) and collective (shift pairs) organizational forms of training.

Individual: everyone who receives the card gets acquainted with its contents (reads the questions and answers on the back of the card).

  • first(in the role of the “trainee”) reads the task and answers the questions on the partner’s card;
  • second(in the role of “coach”) – checks the correctness of the answers on the back of the card;
  • work similarly on another card, changing roles;
  • make a mark on an individual sheet and exchange cards;
  • move to a new couple.

Collective:

  • in the new pair they work like in the first; transition to a new pair, etc.

The number of transitions depends on the time allocated by the teacher for this stage of the lesson, on the diligence and speed of comprehension of each student and on the partners in joint work.

After working in pairs, students make marks on their record sheets, and the teacher conducts a quantitative and qualitative analysis of the work.

The accounting sheet may look like this:

Ivanov Petya 7 “b” grade

Date Card number Number of errors Who did you work with?
20.12.09 №7 0 Sidorov K.
№3 2 Petrova M.
№2 1 Samoilova Z.

3. Introduction to the topic “Function y=, its properties and graph” is carried out by the teacher in the form of a presentation using multimedia learning tools (Appendix 4). On the one hand, this is a version of clarity that is understandable to modern students, on the other hand, it saves time on explaining new material.

4. Consolidation of newly learned and already covered material on the topic “Function ” organized in two versions, using traditional teaching tools (blackboard, textbook) and innovative ones (interactive whiteboard).

First, several tasks from the textbook are offered to consolidate the newly learned material. The textbook used for teaching is used. Work is carried out simultaneously with the whole class. In this case, one student completes task “a” - on a traditional board; the other is task “b” on interactive whiteboard, the rest of the students write down the solutions to the same tasks in a notebook and compare their solution with the solution presented on the boards. Next, the teacher evaluates the students’ work at the board.

Then, to more quickly consolidate the studied material on the topic “Function”, frontal work with an interactive whiteboard is proposed, which can be organized as follows:

  • the task and schedule appear on the interactive board;
  • a student who wants to answer goes to the board, performs the necessary constructions and voices the answer;
  • a new task and a new schedule appear on the board;
  • Another student comes out to answer.

Thus, in a short period of time, it is possible to solve quite a lot of tasks and evaluate student answers. Some tasks of interest (similar to tasks from the upcoming test) can be recorded in a notebook.

5. At the self-control stage, students are offered a test followed by self-test (Appendix 3).

Literature

  1. Dyachenko, V.K. Modern didactics [Text] / V.K. Dyachenko - M.: Public education, 2005.
  2. Yalovets, T.V. Technology of a collective method of teaching in teacher training: Educational and methodological manual [Text] / T.V. Yalovets. – Novokuznetsk: IPK Publishing House, 2005.
  3. Yanovitskaya, E.V. How to teach and learn in a lesson so that you want to learn. Reference album [Text] / E.V. Yanovitskaya. – St. Petersburg: Educational projects, M.: Publisher A.M. Kushnir, 2009.

Topic "Root of a degree" n"It is advisable to split it into two lessons. In the first lesson, consider the cube root, compare its properties with the arithmetic square root and consider the graph of this Cube root function. Then in the second lesson, students will better understand the concept of a crown n-th degree. Comparing the two types of roots will help you avoid “typical” errors in the presence of values ​​from negative expressions under the root sign.

View document contents
"Cubic root"

Lesson topic: Cube root

Zhikharev Sergey Alekseevich, mathematics teacher, MKOU “Pozhilinskaya Secondary School No. 13”


Lesson objectives:

  • introduce the concept of cube root;
  • develop skills in calculating cube roots;
  • repeat and generalize knowledge about the arithmetic square root;
  • continue preparing for the State Examination.

Checking the d.z.






One of the numbers below is marked on the coordinate line with a dot A. Enter this number.



What concept are the last three tasks related to?

What is the square root of a number? A ?

What is the arithmetic square root of a number? A ?

What values ​​can the square root take?

Can a radical expression be a negative number?


Among these geometric bodies, name a cube

What properties does a cube have?


How to find the volume of a cube?

Find the volume of a cube if its sides are equal:


Let's solve the problem

The volume of the cube is 125 cm³. Find the side of the cube.

Let the edge of the cube be X cm, then the volume of the cube is X³ cm³. By condition X³ = 125.

Hence, X= 5 cm.


Number X= 5 is the root of the equation X³ = 125. This number is called cube root or third root from number 125.


Definition.

The third root of the number A this number is called b, the third power of which is equal to A .

Designation.


Another approach to introducing the concept of cube root

For a given cubic function value A, you can find the value of the argument of the cubic function at this point. It will be equal, since extracting the root is the inverse action of raising to a power.




Square roots.

Definition. The square root of a name the number whose square is equal to A .

Definition. Arithmetic square root of a is a non-negative number whose square is equal to A .

Use the designation:

At A

Cube roots.

Definition. cube root from number a name the number whose cube is equal to A .

Use the designation:

"The cube root of A", or

"The 3rd root of A »

The expression makes sense for any A .





Launch the MyTestStudent program.

Open the “9th grade lesson” test.


A minute of rest

In what lessons or

you met in life

with the concept of root?



"Equation"

When you solve an equation, my friend,

You must find him spine.

The meaning of a letter is easy to check,

Put it into the equation carefully.

If you achieve true equality,

That root call the meaning immediately.




How do you understand Kozma Prutkov’s statement “Look to the root.”

When is this expression used?


In literature and philosophy there is the concept “Root of Evil”.

How do you understand this expression?

In what sense is this expression used?


Think about it, is it always easy and accurate to extract the cube root?

How can you find approximate cube root values?


Using the graph of a function at = X³, you can approximately calculate the cube roots of some numbers.

Using the graph of a function

at = X³ orally find the approximate meaning of the roots.



Do functions belong to the graph?

dots: A(8;2); In (216;–6)?


Can the radical expression of a cube root be negative?

What is the difference between a cube root and a square root?

Can the cube root be negative?

Define a root of the third degree.


Which is equal to a. In other words, this is the solution to the equation x^3 = a(usually real solutions are meant).

Real root

Demonstrative form

The root of complex numbers can be defined as follows:

x^(1/3) = \exp (\tfrac13 \ln(x))

If you imagine x How

x = r\exp(i\theta)

then the formula for a cubic number is:

\sqrt(x) = \sqrt(r)\exp (\tfrac13 i\theta).

This geometrically means that in polar coordinates we take the cube root of the radius and divide the polar angle by three to determine the cube root. So if x complex, then \sqrt(-8) will mean not -2, but there will be 1 + i\sqrt(3).

At a constant density of matter, the dimensions of two similar bodies are related to each other as the cube roots of their masses. So, if one watermelon weighs twice as much as another, then its diameter (as well as its circumference) will be only a little more than a quarter (26%) larger than the first; and to the eye it will seem that the difference in weight is not so significant. Therefore, in the absence of scales (sale by eye), it is usually more profitable to buy a larger fruit.

Calculation methods

Column

Before starting, you need to divide the number into triplets (the integer part - from right to left, the fractional part - from left to right). When you reach the decimal point, you must add a decimal point at the end of the result.

The algorithm is as follows:

  1. Find a number whose cube is smaller than the first group of digits, but when it increases by 1 it becomes larger. Write down the number you find to the right of the given number. Write the number 3 below it.
  2. Write the cube of the number found under the first group of numbers and subtract. Write the result after subtraction under the subtrahend. Next, take down the next group of numbers.
  3. Next, we replace the found intermediate answer with the letter a. Calculate using the formula such a number x that its result is less than the lower number, but when increased by 1 it becomes larger. Write down what you find x to the right of the answer. If the required accuracy is achieved, stop calculations.
  4. Write down the result of the calculation under the bottom number using the formula 300\times a^2\times x+30\times a\times x^2+x^3 and do the subtraction. Go to step 3.

See also

Write a review about the article "Cubic root"

Literature

  • Korn G., Korn T. 1.3-3. Representation of sum, product and quotient. Powers and roots // Handbook of mathematics. - 4th edition. - M.: Nauka, 1978. - P. 32-33.

An excerpt characterizing the cube root

By nine o'clock in the morning, when the troops had already moved through Moscow, no one else came to ask the count's orders. Everyone who could go did so of their own accord; those who remained decided with themselves what they had to do.
The count ordered the horses to be brought in to go to Sokolniki, and, frowning, yellow and silent, with folded hands, he sat in his office.
In calm, not stormy times, it seems to every administrator that it is only through his efforts that the entire population under his control moves, and in this consciousness of his necessity, every administrator feels the main reward for his labors and efforts. It is clear that as long as the historical sea is calm, the ruler-administrator, with his fragile boat resting his pole against the ship of the people and himself moving, must seem to him that through his efforts the ship he is resting against is moving. But as soon as a storm arises, the sea becomes agitated and the ship itself moves, then delusion is impossible. The ship moves with its enormous, independent speed, the pole does not reach the moving ship, and the ruler suddenly goes from the position of a ruler, a source of strength, into an insignificant, useless and weak person.
Rastopchin felt this, and it irritated him. The police chief, who was stopped by the crowd, together with the adjutant, who came to report that the horses were ready, entered the count. Both were pale, and the police chief, reporting the execution of his assignment, said that in the count’s courtyard there was a huge crowd of people who wanted to see him.
Rastopchin, without answering a word, stood up and quickly walked into his luxurious, bright living room, walked up to the balcony door, grabbed the handle, left it and moved to the window, from which the whole crowd could be seen more clearly. A tall fellow stood in the front rows and with a stern face, waving his hand, said something. The bloody blacksmith stood next to him with a gloomy look. The hum of voices could be heard through the closed windows.
- Is the crew ready? - said Rastopchin, moving away from the window.
“Ready, your Excellency,” said the adjutant.
Rastopchin again approached the balcony door.
- What do they want? - he asked the police chief.
- Your Excellency, they say that they were going to go against the French on your orders, they shouted something about treason. But a violent crowd, your Excellency. I left by force. Your Excellency, I dare to suggest...
“If you please, go, I know what to do without you,” Rostopchin shouted angrily. He stood at the balcony door, looking out at the crowd. “This is what they did to Russia! This is what they did to me!” - thought Rostopchin, feeling an uncontrollable anger rising in his soul against someone who could be attributed to the cause of everything that happened. As often happens with hot-tempered people, anger was already possessing him, but he was looking for another subject for it. “La voila la populace, la lie du peuple,” he thought, looking at the crowd, “la plebe qu"ils ont soulevee par leur sottise. Il leur faut une victime, [“Here he is, people, these scum of the population, the plebeians, whom they raised with their stupidity! They need a victim.”] - it came to his mind, looking at the tall fellow waving his hand. And for the same reason it came to his mind that he himself needed this victim, this object for his anger.
- Is the crew ready? – he asked another time.
- Ready, Your Excellency. What do you order about Vereshchagin? “He’s waiting at the porch,” answered the adjutant.
- A! - Rostopchin cried out, as if struck by some unexpected memory.
And, quickly opening the door, he stepped out onto the balcony with decisive steps. The conversation suddenly stopped, hats and caps were taken off, and all eyes rose to the count who had come out.
- Hello, guys! - the count said quickly and loudly. - Thank you for coming. I’ll come out to you now, but first of all we need to deal with the villain. We need to punish the villain who killed Moscow. Wait for me! “And the count just as quickly returned to his chambers, slamming the door firmly.
A murmur of pleasure ran through the crowd. “That means he will control all the villains! And you say French... he’ll give you the whole distance!” - people said, as if reproaching each other for their lack of faith.

Guys, we continue to study power functions. The topic of today's lesson will be the function - the cubic root of x. What is a cube root? The number y is called a cube root of x (root of the third degree) if the equality is satisfied Denote:, where x is the radical number, 3 is the exponent.


As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers. The third root of a negative number is equal to a negative number. When raised to an odd power, the sign is preserved; the third power is odd. Let's check the equality: Let. Let us raise both expressions to the third power. Then or In the notation of roots we obtain the desired identity.




Guys, let's now build a graph of our function. 1) The domain of definition is the set of real numbers. 2) The function is odd, since Next we will consider our function at x 0, then we will display the graph relative to the origin. 3) The function increases as x 0. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase. 4) The function is not limited from above. In fact, from an arbitrarily large number we can calculate the third root, and we can move upward indefinitely, finding ever larger values ​​of the argument. 5) When x 0 the smallest value is 0. This property is obvious.




Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd. Properties of the function: 1) D(y)=(-;+) 2) Odd function. 3) Increases by (-;+) 4) Unlimited. 5) There is no minimum or maximum value. 6) The function is continuous on the entire number line. 7) E(y)= (-;+). 8) Convex downward by (-;0), convex upward by (0;+).






Example. Draw a graph of the function and read it. Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For x-1 we build a graph of the cubic root, and for x-1 we build a graph of a linear function. 1) D(y)=(-;+) 2) The function is neither even nor odd. 3) Decreases by (-;-1), increases by (-1;+) 4) Unbounded from above, limited from below. 5) There is no greatest value. The smallest value is minus one. 6) The function is continuous on the entire number line. 7) E(y)= (-1;+)