Municipal educational institution

secondary school No. 1

Art. Bryukhovetskaya

municipal formation Bryukhovetsky district

Mathematic teacher

Guchenko Angela Viktorovna

year 2014

Function y =
, its properties and graph

Lesson type: learning new material

Lesson objectives:

Problems solved in the lesson:

teach students to work independently;

make assumptions and guesses;

be able to generalize the factors being studied.

Equipment: board, chalk, multimedia projector, handouts

Timing of the lesson.

Determining the topic of the lesson together with students -1 min.

Determining the goals and objectives of the lesson together with students -1 min.

Updating knowledge (frontal survey) –3 min.

Oral work -3 min.

Explanation of new material based on creating problem situations -7min.

Fizminutka –2 minutes.

Plotting a graph together with the class, drawing up the construction in notebooks and determining the properties of a function, working with a textbook -10 min.

Consolidating acquired knowledge and practicing graph transformation skills –9min .

Summing up the lesson, providing feedback -3 min.

Homework -1 min.

Total 40 minutes.

During the classes.

Determining the topic of the lesson together with students (1 min).

The topic of the lesson is determined by students using guiding questions:

function- work performed by an organ, the organism as a whole.

function- possibility, option, skill of a program or device.

function- duty, range of activities.

function character in a literary work.

function- type of subroutine in computer science

function in mathematics - the law of dependence of one quantity on another.

Determining the goals and objectives of the lesson together with students (1 min).

The teacher, with the help of students, formulates and pronounces the goals and objectives of this lesson.

Updating knowledge (frontal survey – 3 min).

Oral work – 3 min.

Frontal work.

(A and B belong, C does not)

Explanation of new material (based on creating problem situations – 7 min).

Problem situation: describe the properties of an unknown function.

Divide the class into teams of 4-5 people, distribute forms for answering the questions asked.

Form No. 1

y=0, with x=?

The scope of the function.

Set of function values.

One of the team representatives answers each question, the rest of the teams vote “for” or “against” with signal cards and, if necessary, complement the answers of their classmates.

Together with the class, draw a conclusion about the domain of definition, the set of values, and the zeros of the function y=.

Problem situation : try to build a graph of an unknown function (there is a discussion in teams, searching for a solution).

The teacher recalls the algorithm for constructing function graphs. Students in teams try to depict the graph of the function y= on forms, then exchange forms with each other for self- and mutual testing.

Fizminutka (Clowning)

Constructing a graph together with the class with the design in notebooks – 10 min.

After a general discussion, the task of constructing a graph of the function y= is completed individually by each student in a notebook. At this time, the teacher provides differentiated assistance to students. After students complete the task, the graph of the function is shown on the board and students are asked to answer the following questions:

Conclusion: Together with the students, draw a conclusion about the properties of the function and read them from the textbook:

Consolidating acquired knowledge and practicing graph transformation skills – 9 min.

Students work on their card (according to the options), then change and check each other. Afterwards, graphs are shown on the board, and students evaluate their work by comparing it with the board.

Card No. 1

Card No. 2

Conclusion: about graph transformations

1) parallel transfer along the op-amp axis

2) shift along the OX axis.

9. Summing up the lesson, providing feedback – 3 min.

SLIDES – insert missing words

The domain of definition of this function, all numbers except ...(negative).

The graph of the function is located in... (I) quarters.

When the argument x = 0, the value... (functions) y = ... (0).

The greatest value of the function... (does not exist), smallest value - …(equals 0)

10. Homework (with comments – 1 min).

According to the textbook- §13

According to the problem book– No. 13.3, No. 74 (repetition of incomplete quadratic equations)

8th grade

Teacher: Melnikova T.V.

Lesson objectives:

Equipment:

Computer, interactive whiteboard, handouts.

Presentation for the lesson.

DURING THE CLASSES

Lesson plan.

Teacher's opening speech.

Repetition of previously studied material.

Learning new material (group work).

Function study. Chart properties.

Discussion of the schedule (front work).

Game of math cards.

Lesson summary.

I. Updating of basic knowledge.

Greeting from the teacher.

Teacher :

The dependence of one variable on another is called a function. So far you have studied the functions y = kx + b; y =k/x, y=x 2. Today we will continue to study functions. In today's lesson you will learn what a graph of a square root function looks like, and learn how to build graphs of square root functions yourself.

Write down the topic of the lesson (slide1).

2. Repetition of the studied material.

1. What are the names of the functions specified by the formulas:

a) y=2x+3; b) y=5/x; c) y = -1/2x+4; d) y=2x; e) y = -6/x f) y = x 2?

2. What is their graph? How is it located? Indicate the domain of definition and domain of value of each of these functions ( in Fig. graphs of functions given by these formulas are shown; for each function, indicate its type) (slide2).

3. What is the graph of each function, how are these graphs constructed?

(Slide 3, schematic graphs of functions are constructed).

3. Studying new material.

Teacher:

So today we are studying the function
and her schedule.

We know that the graph of the function y=x2 is a parabola. What will be the graph of the function y=x2 if we take only x ≥ 0 ? Part of the parabola is its right branch. Let us now plot the function
.

Let us repeat the algorithm for constructing graphs of functions ( slide 4, with algorithm)

Question : Looking at the analytical notation of the function, do you think we can say what values X acceptable? (Yes, x≥0). Since the expression
makes sense for all x greater than or equal to 0.

Teacher: In natural phenomena and human activity, dependencies between two quantities are often encountered. How can this relationship be represented by a graph? ( group work)

The class is divided into groups. Each group receives a task: build a graph of the function
on graph paper, performing all points of the algorithm. Then a representative from each group comes out and shows the group's work. (Slad 5 opens, a check is carried out, then the schedule is built in notebooks)

4. Study of the function (work in groups continues)

Teacher:

find the domain of the function;

find the range of the function;

determine the intervals of decrease (increase) of the function;

y>0, y<0.

Write down the results for you (slide 6).

Teacher: Let's analyze the graph. The graph of a function is a branch of a parabola.

Question : Tell me, have you seen this graph somewhere before?

Look at the graph and tell me if it intersects the line OX? (No) OU? (No). Look at the graph and tell me whether the graph has a center of symmetry? Axis of symmetry?

Let's summarize:

Now let’s see how we learned a new topic and repeated the material we covered. A game of mathematical cards. (rules of the game: each group of 5 people is offered a set of cards (25 cards). Each player receives 5 cards with questions written on them. The first student gives one of the cards to the second student, who must answer the question from the card If the student answers the question, then the card is broken, if not, then the student takes the card for himself and moves on, etc. for a total of 5 moves. If the student has no cards left, then the score is -5, 1 card remains - score 4, 2 cards – score 3, 3 cards – score 2)

5. Lesson summary.(students are graded on checklists)

Homework assignment.

Study paragraph 8.

Solve No. 172, No. 179, No. 183.

Prepare reports on the topic “Application of functions in various fields of science and literature.”

Reflection.

Show your mood with pictures on your desk.

Today's lesson

I like it.

I did not like.

Lesson material I ( understood, did not understand).

The basic properties of the power function are given, including formulas and properties of the roots. The derivative, integral, power series expansion, and complex number representation of a power function are presented.

Definition

Definition
Power function with exponent p is the function f (x) = xp, the value of which at point x is equal to the value of the exponential function with base x at 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 valid .

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

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

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

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

Properties and formulas of power functions for x ≥ 0

Here we will consider the properties of the power function for non-negative values ​​of the argument x. As stated above, for certain values ​​of the exponent p, the power function is also defined for negative values ​​of x. In this case, its properties can be obtained from the properties of , using even or odd. 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 with ,
strictly decreases as ;
(1.4) at ;
at ;
(1.5) ;
(1.5*) ;
(1.6) ;
(1.7) ;
(1.7*) ;
(1.8) ;
(1.9) .

Proof of properties is given on the page “Power function (proof of continuity and properties)”

Roots - definition, formulas, properties

Definition
Root of a number x of degree n is the number that when raised to the power n gives x:
.
Here n = 2, 3, 4, ... - a natural number greater than one.

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

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 formula that is often used is valid for both positive and negative x:
.
For square root:
.

The order in which the operations are performed is important here - that is, first the square is performed, resulting in a non-negative number, and then the root is taken from it (the square root can be taken from a non-negative number). 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:
.
When x ≥ 0 the following formulas apply:
;
;
, ;
.

These formulas can also be applied for negative values ​​of variables. You just need to make sure that the radical expression of even powers is not negative.

Private values

The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .

Example. Root of roots

Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.

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

Here are graphs of the function for non-negative values ​​of the argument x. Graphs of a 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.

Derivative of a power function

Derivative of nth order:
;

Deriving formulas > > >

Integral of a power function

P ≠ - 1 ;
.

Power series expansion

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

Expressions using complex numbers

Consider the function of the complex variable z:
f (z) = z t.
Let us 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 in the form of real and imaginary parts:
t = p + i q .
We have:

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

Let's 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 an integer. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous.

If p is irrational, then the products kp for any k do not produce an integer. 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 2π(one turn), we move to a new branch of the function.

If p is rational, then it can be represented as:
, Where m, n- integers that do not contain common divisors. Then
.
First n values, with k = k 0 = 0, 1, 2, ... n-1, give n different values ​​of kp:
.
However, subsequent values ​​give values ​​that differ from the previous ones by an integer. For example, when k = k 0+n we have:
.
Trigonometric functions whose arguments differ by multiples of 2π, 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, an exponential function with a rational exponent is multivalued and has n values ​​(branches). Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function. After n such revolutions 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 a 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 college students, “Lan”, 2009.

Function graph and properties at = │Oh│ (module)

Consider the function at = │Oh│, where A- a certain number.

Domain of definition functions at = │Oh│, is the set of all real numbers. The figure shows respectively function graphs at = │X│, at = │ 2x │, at = │X/2│.

You can notice that the graph of the function at = | Oh| obtained from the graph of the function at = Oh, if the negative part of the function graph at = Oh(it is located below the O axis X), reflect symmetrically this axis.

It's easy to see from the graph properties functions at = │ Oh │.

At X= 0, we get at= 0, that is, the graph of the function belongs to the origin; at X= 0, we get at> 0, that is, all other points of the graph lie above the O axis X.

For opposite values X, values at will be the same; O axis at this is the axis of symmetry of the graph.

For example, you can plot the function at = │X 3 │. To compare features at = │X 3 │and at = X 3, let's make a table of their values ​​with the same values ​​of the arguments.

From the table we see that in order to plot a function graph at = │X 3 │, you can start by plotting the function at = X 3. After this it stands symmetrically to the O axis X display that part of it that is below this axis. As a result, we get the graph shown in the figure.

Function graph and properties at = x 1/2 (root)

Consider the function at = x 1/2 .

Domain of definition this function is the set of non-negative real numbers, since the expression x 1/2 only matters when X > 0.

Let's build a graph. To compile a table of its values, we use a microcalculator, rounding the function values ​​to tenths.

After drawing points on the coordinate plane and smoothly connecting them, we get graph of a function at = x 1/2 .

The constructed graph allows us to formulate some properties functions at = x 1/2 .

At X= 0, we get at= 0; at X> 0, we get at> 0; the graph passes through the origin; the remaining points of the graph are located in the first coordinate quarter.

Theorem. Graph of a function at = x 1/2 is symmetrical to the graph of the function at = X 2 where X> 0, relatively straight at = X.

Proof. Function graph at = X 2 where X> 0, is the branch of the parabola located in the first coordinate quadrant. Let the point R (A; b) is an arbitrary point of this graph. Then the equality is true b = A 2. Since by condition the number A non-negative, then the equality is also true A= b 1/2. This means that the coordinates of the point Q (b; A) transform the formula at = x 1/2 to true equality, or otherwise, period Q (b; A at= x 1/2 .

It is also proved that if the point M (With; d) belongs to the graph of the function at = x 1/2 then point N (d; With) belongs to the graph at = X 2 where X > 0.

It turns out that each point R(A; b) function graph at = X 2 where X> 0, corresponds to a single point Q (b; A) function graph at = x 1/2 and vice versa.

It remains to prove that the points R (A; b) And Q (b; A) are symmetrical about a straight line at = X. Dropping perpendiculars to the coordinate axes of points R And Q, we get points on these axes E(A; 0), D (0; b), F (b; 0), WITH (0; A). Dot R intersections of perpendiculars RE And QC has coordinates ( A; A) and therefore belongs to the line at = X. Triangle PRQ is isosceles, since its sides R.P. And RQ equal │ b – A│ each. Straight at = X bisects like an angle DOF, and the angle PRQ and intersects the segment PQ at a certain point S. Therefore the segment R.S. is the bisector of the triangle PRQ. Since the bisector of an isosceles triangle is its altitude and median, then PQ ┴ R.S. And PS = QS. And this means that the points R (A; b) And Q (b; A) symmetrical about a straight line at = X.

Since the graph of the function at = x 1/2 is symmetrical to the graph of the function at = X 2 where X> 0, relatively straight at= X, then the graph of the function at = x 1/2 is the branch of the parabola.

Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

Below are some examples of the types of personal information we may collect and how we may use such information.

What personal information do we collect:

• When you submit an application on the site, we may collect various information, including your name, telephone number, email address, etc.

How we use your personal information:

• The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send important notices and communications.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.

Disclosure of information to third parties

We do not disclose the information received from you to third parties.

Exceptions:

• If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government authorities in the territory of the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
• In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.

Respecting your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.