Graph of the function 1 root. Power function and roots - definition, properties and formulas

Square root as an elementary function.

Square root- This elementary function and a special case power function at . The arithmetic square root is smooth at , and at zero it is right continuous but not differentiable.

As a function, a complex variable root is a two-valued function whose leaves converge at zero.

Graphing the square root function.

1. Filling out the data table:

X
at

2. We plot the points that we received on the coordinate plane.

3. Connect these points and get a graph of the function square root:

Transforming the graph of a square root function.

Let us determine what function transformations need to be made in order to construct function graphs. Let's define the types of transformations.

	Conversion type	Conversion
		Transferring a function along an axis OY for 4 units up.
	internal	Transferring a function along an axis OX for 1 unit to the right.
	internal	The graph approaches the axis OY 3 times and compresses along the axis OH.
		The graph moves away from the axis OX OY.
	internal	The graph moves away from the axis OY 2 times and stretched along the axis OH.

Often, function transformations are combined.

For example, you need to plot the function . This is a square root plot that needs to be moved one unit down the axis OY and one unit to the right along the axis OH and at the same time stretching it 3 times along the axis OY.

It happens that immediately before constructing a graph of a function, preliminary identical transformations or simplifications of functions are needed.

Basic goals:

1) form an idea of ​​the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=

2) to develop the ability to construct a graph y= and its properties;

3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.

Equipment, demonstration material: handouts.

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 understood how to graph the function y=.

2) I can list its properties using a graph.

3) I did not make mistakes in independent work.

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

During the classes

1. Self-determination for educational activities

Purpose of the stage:

1) include students in educational activities;

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

Organization educational process at stage 1:

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

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

2. Updating knowledge and recording difficulties in activities

Purpose of the stage:

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

y = kx + m, y = kx, y =c, y =x 2, y = - x 2,

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

3) record all repeated concepts and algorithms in the form of diagrams and symbols;

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

Organization of the educational process at stage 2:

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

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

What is the name of x? (Independent variable - argument)

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

3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).

Individual task:

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

3. Identifying the causes of difficulties and setting goals for activities

Purpose of the stage:

1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;

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

Organization of the educational process at stage 3:

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

– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)

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

– Write the topic in your notebook.

4. Construction of a project for getting out of a difficulty

Purpose of the stage:

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

2) fix new way actions in a symbolic, verbal form and using a standard.

Organization of the educational process at stage 4:

Work at this stage can be organized in groups, asking the groups to build a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.

5. Primary consolidation in external speech

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

Organization of the educational process at stage 5:

Construct a graph of y= - and describe its properties.

Properties y= - .

1.Domain of definition of a function.

2. Range of values ​​of the function.

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

y =0 if x = 0.

y<0, если х(0;+)

4.Increasing, decreasing functions.

The function decreases as x.

Let's build a graph of y=.

Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.

Answer: at our name. = 1, y max. =3

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

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

Organization of the educational process at stage 6:

Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.

Let's build a graph of y=.

Using a 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 together with previously studied: 2) repeat the educational content that will be required in the next lessons.

Organization of the educational process at stage 7:

Solve the equation graphically: = x – 6.

One student is at the blackboard, the rest are in notebooks.

8. Reflection of activity

Purpose of the stage:

1) record new content learned in the lesson;

2) evaluate your own activities in the lesson;

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

4) record unresolved difficulties as directions for future educational activities;

5) discuss and write down your homework.

Organization of the educational process at stage 8:

- Guys, what was our goal today? (Study the function y=, its properties and graph).

– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)

– Analyze your activities in class. (Cards with reflection)

Homework

paragraph 13 (before example 2) № 13.3, 13.4

Solve the equation graphically.

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).