Graph of the function cubic root of x 1. Function y = third root of x, its properties and graph

Topic "The root of the degree P"It is advisable to break 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 the crown P-th degree. Comparison of two types of roots will help to avoid "typical" errors for the presence of values ​​from negative expressions that are under the sign of the root.

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"Cube Root"

Lesson topic: cube root

Zhikharev Sergey Alekseevich, teacher of mathematics, MKOU "Pozhilinskaya School No. 13"

Lesson Objectives:

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

Checking d.z.

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

What is the concept of the last three tasks?

What is the square root of a number a ?

What is the arithmetic square root of a number a ?

What values ​​can Square root?

Can the root expression be a negative number?

Name a cube among these geometric bodies

What are the properties of a cube?

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.

Consequently, X= 5 cm.

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

Definition.

Third root of number a this number is called b, whose third power is equal to a .

Designation.

Another approach to introducing the concept of a cube root

Given the value of the cubic function a, you can find the value of the cubic function argument at that point. It will be equal, since extracting a root is the opposite 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 .

The notation is used:

At a

cube roots.

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

The notation is used:

"cube root of a", or

"3rd root of a »

The expression makes sense for any a .

Launch the MyTestStudent program.

Open the test "Grade 9 lesson".

Minute of rest

What lessons or

you met in your life

with the concept of a root?

"The equation"

When you solve the equation, my friend,

You must find him spine.

The meaning of the letter is easy to check,

Put it into the equation carefully.

If you get the right equality,

That root call the value immediately.

How do you understand the saying of Kozma Prutkov "Look at the root."

When is this expression used?

In literature and philosophy there is the concept of "The Root of Evil".

How do you understand this expression?

In what sense is this expression used?

Think about whether the cube root is always easily and accurately extracted?

What can be used to find approximate values ​​of the cube root?

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

Using the function graph

at = X³ verbally find the approximate value of the roots.

Do the functions belong to the graph

points: A(8;2); In (216;–6)?

Can the subradical 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 third root.

The main properties are given power function, including formulas and properties of roots. The derivative, integral, power series expansion and representation by means of complex numbers of the power function are presented.

Definition

Definition
Power function with exponent p is the function f (x) = xp, whose value at the point x is equal to the value of the exponential function with base x at the point p .
In addition, f (0) = 0 p = 0 for p > 0 .

For natural values ​​of the exponent , the power function is the product of n numbers equal to x :
.
It is defined for all real .

For positive rational values ​​of the exponent , the power function is the product of n roots of degree m from the number x:
.
For odd m , it is defined for all real x . For even m , the power function is defined for non-negative .

For negative , the power function is defined by the formula:
.
Therefore, it is not defined at the point .

For irrational values ​​of the exponent p, the exponential function is determined by the formula:
,
where a is an arbitrary positive number, not equal to one: .
For , it is defined for .
For , the power function is defined for .

Continuity. A power function is continuous on its domain of definition.

Properties and formulas of the power function for x ≥ 0

Here we consider the properties of the power function for not negative values argument x . As mentioned above, for some values ​​of the exponent p , the exponential function is also defined for negative values ​​of x . In this case, its properties can be obtained from the properties at , using even or odd parity. These cases are discussed and illustrated in detail on the page "".

A power function, y = x p , with exponent p has the following properties:
(1.1) defined and continuous on the set
at ,
at ;
(1.2) has many meanings
at ,
at ;
(1.3) strictly increases at ,
strictly decreases at ;
(1.4) at ;
at ;
(1.5) ;
(1.5*) ;
(1.6) ;
(1.7) ;
(1.7*) ;
(1.8) ;
(1.9) .

The proof of the properties is given on the Power Function (Proof of Continuity and Properties) page.

Roots - definition, formulas, properties

Definition
Root of x to the power of n is the number whose raising to the power n gives x:
.
Here n = 2, 3, 4, ... - natural number, greater than one.

You can also say that the root of the number x of degree n is the root (that is, the solution) of the equation
.
Note that the function is the inverse of the function .

The square root of x is a root of degree 2: .

Cube root of x is a root of degree 3: .

Even degree

For even powers n = 2 m, the root is defined for x ≥ 0 . A frequently used formula is valid for both positive and negative x :
.
For square root:
.

The order in which the operations are performed is important here - that is, squaring is performed first, resulting in a non-negative number, and then the root is extracted from it (from a non-negative number, you can extract the square root). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.

odd degree

For odd powers, the root is defined for all x:
;
.

Properties and formulas of roots

The root of x is a power function:
.
For x ≥ 0 the following formulas hold:
;
;
, ;
.

These formulas can also be applied for negative values ​​of the variables. It is only necessary to ensure that the radical expression of even powers is not negative.

Private values

The root of 0 is 0: .
The root of 1 is 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .

Example. Root from roots

Consider the example of the square root of roots:
.
Convert the internal square root using the above formulas:
.
Now let's transform the original root:
.
So,
.

y = x p for different values ​​of the exponent p .

Here are the graphs of the function for non-negative values ​​of the x argument. Graphs of the power function defined for negative values ​​of x are given on the page "Power function, its properties and graphs"

Inverse function

The inverse of a power function with exponent p is a power function with exponent 1/p .

If , then .

Power function derivative

Derivative of the nth order:
;

Derivation of formulas > > >

Integral of a power function

P≠- 1 ;
.

Power series expansion

At - 1 < x < 1 the following decomposition takes place:

Expressions in terms of complex numbers

Consider a function of a complex variable z :
f (z) = z t.
We express the complex variable z in terms of the modulus r and the argument φ (r = |z| ):
z = r e i φ .
We represent the complex number t as real and imaginary parts:
t = p + i q .
We have:

Further, we take into account that the argument φ is not uniquely defined:
,

Consider the case when q = 0 , that is, the exponent is a real number, t = p. Then
.

If p is an integer, then kp is also an integer. Then, due to the periodicity of trigonometric functions:
.
That is exponential function with an integer exponent, for a given z, has only one value and is therefore single-valued.

If p is irrational, then the products of kp do not give an integer for any k. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the argument z is incremented 2 pi(one turn), we move to a new branch of the function.

If p is rational, then it can be represented as:
, where m,n are integers with no common divisors. Then
.
First n values, for k = k 0 = 0, 1, 2, ... n-1, give n different meanings kp :
.
However, subsequent values ​​give values ​​that differ from the previous ones by an integer. For example, for k = k 0+n we have:
.
Trigonometric functions, whose arguments differ by multiples of 2 pi, have equal values. Therefore, with a further increase in k, we obtain the same values ​​of z p as for k = k 0 = 0, 1, 2, ... n-1.

Thus, the exponential function with rational indicator degree is multivalued and has n values ​​(branches). Whenever the argument z is incremented 2 pi(one turn), we move to a new branch of the function. After n such turns, we return to the first branch from which the countdown began.

In particular, a root of degree n has n values. As an example, consider the nth root of a real positive number z = x. In this case φ 0 = 0 , z = r = |z| = x, .
.
So, for the square root, n = 2 ,
.
For even k, (- 1 ) k = 1. For odd k, (- 1 ) k = - 1.
That is, the square root has two meanings: + and -.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

Instead of an introduction

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

The lesson turns out to be dynamic and interesting if you combine different forms of learning during the lesson.

In modern didactics, there are four general organizational forms learning:

• individually mediated;
• steam room;
• group;

collective (in pairs of interchangeable composition). (Dyachenko V.K. Modern didactics. - M .: National education, 2005).

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

One of the most common methods of the technology of the collective way of learning is the method of "Mutual training".

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

Training is the successor of self-control, it helps the student to establish his contact with the subject of study, making it easier to find the right steps-actions. Through training in the acquisition, consolidation, regrouping, revision, application of knowledge, the development of human cognitive abilities occurs. (Yanovitskaya E.V. How to teach and learn in the classroom so that you want to learn. Reference book. - St. Petersburg: Educational projects, M.: Publisher A.M. Kushnir, 2009.-p.14;131)

It will help to quickly repeat any rule, remember the answers to the studied questions, consolidate the necessary skill. The optimal time for working according to the method is 5-10 minutes. As a rule, work on training cards is carried out during oral counting, 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. In the training card there can be 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 “consolidated squad” (children sitting on the same row). A consolidated detachment (group) is a temporary cooperation of students formed to perform a specific educational task. (Yalovets T.V. The technology of a collective method of teaching in advanced training of a teacher: Educational and methodological manual. - Novokuznetsk: IPC Publishing House, 2005. - P. 122)

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

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

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

Goals:

• preparation for control work;
• checking the knowledge of all the properties of a function and the ability to plot function graphs and read their properties.

Tasks: subject level:

oversubject level:

• learn to analyze graphic information;
• develop the ability to conduct a dialogue;
• develop the ability and skill of working with an interactive whiteboard using the example of working with graphs.

Lesson structure	Time
1. Information input of the teacher (ITI)	5 minutes.
2. Actualization of basic knowledge: work in pairs of shifts according to the methodology “Mutual training”	8 min.
3. Acquaintance with the topic “Function y=, its properties and graph”: teacher presentation	8 min.
4. Consolidation of the newly studied and already passed material on the topic “Function”: using an interactive whiteboard	15 minutes.
5. Self-control : in the form of a test	7 min.
6. Summing up, recording homework.	2 minutes.

Let's take a closer look at the content of each stage.

1. Teacher information input (ITI) includes Organizing time; voicing the topic, purpose and lesson plan; showing a sample of work in pairs according to the method of mutual training.

Demonstration of a sample of work in pairs by students at this stage of the lesson is advisable to repeat the algorithm of the work of the technique 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 the work according to the algorithm (if any), as well as evaluate the work of these students.

2. Actualization of reference knowledge is carried out in pairs of shift composition according to the method of mutual training.

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

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

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

Collective:

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

The number of transitions depends on the time allotted by the teacher for this stage lesson, from the diligence and speed of understanding of each student and from partners in collaboration.

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

The listing might look like this:

Ivanov Petya 7 "b" class

the date	Card number	Number of mistakes	Who did you work with
20.12.09	№7	0	Sidorov K.
	№3	2	Petrova M.
	№2	1	Samoilova Z.

3. Acquaintance with 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 visualization option that is understandable to modern students, on the other hand, it saves time on explaining new material.

4. Consolidation of the newly studied and already passed material on the topic “Function ” organized in two versions, using traditional teaching aids (board, textbook) and innovative (interactive whiteboard).

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

Then, in order 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 whiteboard;
• 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, to evaluate the answers of students. Some tasks of interest (similar to tasks from the upcoming control work), can be recorded in a notebook.

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

Literature

1. Dyachenko, V.K. Modern didactics [Text] / V.K. Dyachenko - M.: Public education, 2005.
2. Yalovets, T.V. The technology of the collective method of teaching in the professional development of the teacher: Educational and methodological manual [Text] / T.V. Yalovets. - Novokuznetsk: IPC Publishing House, 2005.
3. Yanovitskaya, E.V. How to teach and learn in the classroom so that you want to learn. Reference book [Text] / E.V. Yanovitskaya. - St. Petersburg: Educational projects, M.: Publisher A.M. Kushnir, 2009.

Basic goals:

1) to form an idea of ​​the expediency of a generalized study of the dependences of real quantities on the example of quantities, related relationship y=

2) to form the ability to plot y= and its properties;

3) repeat and consolidate the methods of oral and written calculations, squaring, extracting the square root.

Equipment, demonstration material: handout.

1. Algorithm:

2. Sample for completing the task in groups:

3.Sample for self-test of independent work:

4. Card for the reflection stage:

1) I figured out how to graph the function y=.

2) I can list its properties according to the schedule.

3) I did not make mistakes in my independent work.

4) I made mistakes in independent work (list these mistakes and indicate their reason).

During the classes

1. Self-determination to learning activities

Purpose of the stage:

1) include students in learning activities;

2) determine the content of the lesson: we continue to work with real numbers.

Organization educational process at step 1:

What did we study in the last lesson? (We have studied many real numbers, actions with them, built an algorithm for describing the properties of a function, repeated the functions studied in grade 7).

– Today we will continue to work with the set of real numbers, a function.

2. Updating knowledge and fixing difficulties in activities

Purpose of the stage:

1) update the educational content necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs

y \u003d kx + m, y \u003d kx, y \u003d c, y \u003d x 2, y \u003d - x 2,

2) to update the mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;

3) fix all repeated concepts and algorithms in the form of schemes and symbols;

4) to fix an individual difficulty in activity, demonstrating the insufficiency of existing knowledge at a personally significant level.

Organization of the educational process at stage 2:

1. Let's remember how you can set the dependencies between the quantities? (Via text, formula, table, graph)

2. What is called a function? (The relationship between two quantities, where each value of one variable corresponds to a single value of the other variable y = f(x)).

What is x called? (Independent variable - argument)

What is the name of u? (Dependent variable).

3. Did we learn functions in 7th grade? (y = kx + m, y = kx, y =c, y =x 2 , y = - x 2 , ).

Individual task:

What is the graph of functions y = kx + m, y =x 2 , y = ?

3. Identification of the causes of difficulties and setting the goal of the activity

Purpose of the stage:

1) organize communicative interaction, during which the distinguishing feature tasks that caused difficulty in educational activities;

2) agree on the purpose and topic of the lesson.

Organization of the educational process at stage 3:

What is special about this task? (The dependence is given by the formula y = which we have not met yet).

- What is the purpose of the lesson? (Get acquainted with the function y \u003d, its properties and graph. The function in the table determines the type of dependence, build a formula and graph.)

- Can you guess the topic of the lesson? (Function y=, its properties and graph).

- Write the topic in your notebook.

4. Building a project for getting out of a difficulty

Purpose of the stage:

1) organize communicative interaction to build a new mode of action that eliminates the cause of the identified difficulty;

2) fix new way actions in a sign, verbal form and with the help of a standard.

Organization of the educational process at stage 4:

The work at the stage can be organized into groups by inviting the groups to plot y = , then analyze the results. Also, groups can be offered to describe the properties of this function according to the algorithm.

5. Primary consolidation in external speech

The purpose of the stage: to fix the studied educational content in external speech.

Organization of the educational process at stage 5:

Build a graph y= - and describe its properties.

Properties y= - .

1.Scope of function definition.

2.Scope of function values.

3. y=0, y>0, y<0.

y=0 if x=0.

y<0, если х(0;+)

4.Increase, decrease function.

The function is decreasing at x.

Let's plot y=.

Let's select its part on the segment . Let us note that at Naim. = 1 for x = 1, and y max. \u003d 3 for x \u003d 9.

Answer: naim. = 1, at the max. =3

6. Independent work with self-test according to the standard

The purpose of the stage: to test your ability to apply the new learning content in typical conditions based on comparing your solution with a standard for self-testing.

Organization of the educational process at stage 6:

Students perform the task on their own, conduct a self-test according to the standard, analyze, correct errors.

Let's plot y=.

Using the graph, find the smallest and largest values ​​​​of the function on the segment.

7. Inclusion in the knowledge system and repetition

The purpose of the stage: to train the skills of using new content in conjunction with previously studied: 2) repeat the educational content that will be required in the following lessons.

Organization of the educational process at stage 7:

Solve graphically the equation: \u003d x - 6.

One student at the blackboard, the rest in notebooks.

8. Reflection of activity

Purpose of the stage:

1) fix the new content learned in the lesson;

2) evaluate their own activities in the lesson;

3) thank classmates who helped to get the result of the lesson;

4) fix unresolved difficulties as directions for future learning activities;

5) Discuss and write down homework.

Organization of the educational process at stage 8:

- Guys, what was the goal for us today? (Study the function y \u003d, its properties and graph).

- What knowledge helped us achieve the goal? (The ability to look for patterns, the ability to read graphs.)

- Review your activities in class. (Reflection cards)

Homework

item 13 (up to example 2) № 13.3, 13.4

Solve graphically the equation:

Draw a function graph and describe its properties.

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

Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions! All materials are checked by an antivirus program.

Teaching aids and simulators in the online store "Integral" for grade 9
Educational complex 1C: "Algebraic problems with parameters, grades 9-11" Software environment "1C: Mathematical constructor 6.0"

Definition of a power function - cube root

Guys, we continue to study power functions. Today we are going to talk about the Cube Root of x function.
What is a cube root?
A number y is called a cube root of x (third degree root) if $y^3=x$ is true.
They are denoted as $\sqrt(x)$, where x is the root number, 3 is the 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 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: $\sqrt((-x))$=-$\sqrt(x)$.
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 $a=-b$. In the notation of the roots, we obtain the desired identity.

Properties of cube 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 $\sqrt(\frac(a)(b))$ in the cube is equal to $\frac(a)(b)$ and then it is equal to $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's plot our function graph.
1) The domain of definition is the set of real numbers.
2) The function is odd because $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then reflect the graph relative to the origin.
3) The function increases for $х≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increasing.
4) The function is not limited from above. In fact, from an arbitrarily large number, you can calculate the root of the third degree, and we can move up to infinity, 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 for x≥0.

Let's build our graph of the function on 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 downwards by (-∞;0), convex upwards by (0;+∞).

Examples of solving power functions

Examples
1. Solve the equation $\sqrt(x)=x$.
Solution. Let's build two graphs on the same 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. Build 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 shifting two units to the right and three units down.

3. Build a function graph and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's build two graphs of functions on the same coordinate plane, taking into account our conditions. For $х≥-1$ we build a graph of a cubic root, for $х≤-1$ 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) There is no maximum value. The smallest value is minus one.
6) The function is continuous on the entire real line.
7) E(y)= (-1;+∞).

Tasks for independent solution

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