O. The “nth root” function is a function of the form T. Graphs. Mathematics lesson “Function y = √x, its properties and graph Thank you for the lesson

Sections: Mathematics

Goals: consolidate knowledge of the properties of a function when performing exercises, test the skills and abilities of students and the degree of their assimilation of the studied material during independent work, repeat previously studied material.

Tasks: encourage students to self-control, mutual control, and self-analysis of their educational activities. Develop creative and mental thinking.

Method of work in the lesson:

Students work in pairs. Each desk is a separate option. It is advisable to seat the children next to the weaker student and the stronger one.

An envelope with 1) an assessment sheet, 2) a sheet for oral work, 3) a “Loto” task + a rebus is distributed to each desk.

In the previous lesson, you can assign independent homework according to the following options:

Task 1. Construct a figure bounded by the graphs of functions.

Option 1.
Option 2.

Stage 1. Organizational moment (3 min) Greeting. Report topic. State the lesson plan. The work consists of three stages. Students record the results of each stage on individual assessment sheets. (distribute the assessment sheet from Appendix 2)

Stage 2. Checking homework (5 min)

Students exchange their notebooks with the next desk.

1 student at the board shows solution No. 350 Slide 3

Checking homework No. 1. Slide 4

We calculate the number of points: for correctly completed number 350 - 1 point, for correctly completed independent work we set points as follows: for each correctly constructed graph 1 point, 1 point for a correctly designated figure. Result – 5 points for completing 2 tasks correctly. We put points on the score sheet. Slide 6

Stage 3. Oral work (Repetition of theory) (5 min) Slide 6

Distribute to students a sheet with a task for oral work (see Appendix 2)

2 minutes . For checking. Verification with mutual control (we change answers again). Slide 7

Stage 4. Practical part (20 min) Slide 10-13

Goal: to be able to determine the identity of a point without constructing a graph, compare numbers using the properties of a function graph, promote teamwork and develop the cognitive process with the help of puzzles.

On their desks, students have a card with a task, an envelope with answer options (9 cards with different answers, but 3 have correct ones) and a blank card with the task number for composing a rebus.

The tasks are designed in such a way that the first two letters are solved by one student, and the second two letters are solved by the second student, and only No. 3 is solved together.

“Loto” – differentiated independent work(performed according to options and in pairs)

Exercise 1. Solve 3 tasks from the option written on the card, find cards with the correct answers and cover the corresponding tasks with them, then you will get a rebus on the top side of them.

Task 2. Solve the puzzle by answering the question.

IN 1. What is another name for the arithmetic square root?

AT 2. What mathematician once remarked that: “A mathematical theory can be considered perfect only when you have made it so clear that you undertake to explain its content to the first person you meet?

Answer card:

2. Write down differentiated homework

“3” – 357
“4” – 357 + 351 (b, d)
“5” – 357 + 351 (b, d) + 456

Individual homework for strong students:

Construct graphs of functions in one coordinate system and draw conclusions about what happens to the graph of the function. (graph conversion has not been studied yet).

Hello!

Today we have an unusual activity. We will conduct a math lesson on health.

Along with “consolidating” mathematical knowledge, we will remember the main secrets of health.

And the epigraph of the lesson will be the words "The Great Book of Health is Written in Mathematical Symbols"

How do you understand these words?

Without mathematical knowledge, no science is possible, even such as the science of health. And we will see this today.

So, in the last lesson we got acquainted with the function

, its properties and schedule.

Write the date and topic of the lesson.

I suggest that during the survey process, you determine what knowledge you need to remember and apply today?

2. Updating theoretical knowledge (frontal survey) (5 min.)

Task: Complete the phrases.

A) The arithmetic square root of a is called...

IN) The expression makes no sense when...

WITH) The graph of a function is...

D) The function has distinctive…

E) From the graph of the function you can determine...

What tasks will we set for ourselves?

Objectives: improve the ability to graph a function of the form y=
, repeat the properties of this function, check your mastery of the material by finding square roots, through solving expressions and equations.

As you noticed, the letters denoting the sequence of phrases are capital Latin. In medicine, this is what vitamins are called. This list presents a group of vitamins that are present in many foods and help you see well and be resistant to colds and stressful situations.

That's why, The first rule of health is healthy and proper nutrition.

- To discover the second secret of health, let's sit down correctly and play mathematical lotto together.

Computational warm-up. (8 min.)

Game "Mathematical Lotto"

Calculate

Calculate, indicate the correct answer

What integer is included between
And

That more ,
; 3,2 ?

Find the largest value of the function y= on the interval from 1 to 25

Solve the equation
=4

Find the largest root of the equation x2 = 4

Calculate

Calculate
+

Calculate

Find the side of a square if its area is 64 cm2

Find the perimeter of a square if its area is 9 cm2

-The second secret of health is the daily routine. This is the right combination and alternation of work, activities and rest. In the section “This is interesting!” we learn about the daily routine of the famous mathematician.

4. This is interesting! (3 min.)

Pythagoras is perhaps the most popular scientist in the entire history of mankind. Mathematician, mechanic, musician, Olympic champion of antiquity, the name of no scientist is repeated so often. He established his own school, the students of the school were called Pythagoreans. It was very difficult to get into the Pythagorean school. Pythagoras developed a special daily routine for himself and his students. Rising before sunrise, the Pythagoreans went to the seashore to greet the dawn, did gymnastic exercises, and had breakfast. At the end of the day they took walks together, sea swimming and had dinner, and after dinner they prayed to the gods and read.

And you and I will not violate the regime and rest a little. Let's sit comfortably and watch the puck with our eyes.

5. Physical exercise for the eyes (2 min.)

This physical exercise gives a hint about third secret of health. Which one?

- Playing sports, constantly moving.

And now we will arrange a kind of mathematical competition between pairs to test your knowledge on the topic of the lesson.

6. Development of knowledge, abilities, skills (10 min.)

1. Work in pairs (forming 3 pairs).

Task: find inaccuracy in the proposed properties of the function
, mark the selected option with the checkbox of your pair, if possible first, and be sure to give the correct wording of the property, otherwise the answer goes to the next pair:

The domain of definition of a function is the set of non-negative numbers (x≥0).

The range of values ​​of the function is the set Z.

3. Function increases.

4. y=0 at x=0; y<0 при x<0; y>0 at x>0

5. There is no greatest and least value of a function.

6. The graph of the function is symmetrical to the graph of the function y = x², where x≥0 relative to the straight line y = x.

7. Practical application of knowledge (10 min.)

Assignment in textbook No. 357 p. 84:

Solve the equation graphically by one student at the board with an oral explanation of the solution steps.

8. Reflection (3 min.)

Our lesson ends, let's summarize.

Were you interested?

What knowledge and skills should you have used in the lesson?

What new things did you discover during the lesson?

How are you feeling? Does mood affect health? That's the last secret is “good mood”.

Positive emotions are also necessary for a healthy lifestyle. Today in class you experienced the joy of learning, satisfaction with your successes, and goodwill in communication. Health is an invaluable asset not only for each individual person, but also for the entire society.

Let's look at each other, smile and take this positive charge of emotion with us to the next lesson.

Take care of yourself and your health, and then mathematical problems will be solved faster and easier.

9. Homework (1 min.)

paragraph 15 No. 365; No. 367;
No. 344(a).

Thank you for the lesson!

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)

Function

its properties and schedule.

Oral work.

Find errors: Explain the answer.

Correct answers:

does not exist

Use the template to graph the function and list its properties.

at

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X

Function properties

• D - ?
• E - ?
• When x = 0, ____; and for x 0, _______. Therefore, the graph is located in the ___ quarter.
• Increasing, decreasing.
• The largest and smallest value of a function.
• Continuity of function.

X

U

X ≥ 0

1 2 3 4 5 6 7 8 9 10 11

Tasks for independent work:

• List properties of a function

• Determine whether the points belong to the graph of the function.

Self-test. Function properties

• If x = 0, then y = 0; and if x 0, then y 0. Therefore, the graph is located in the 4th quarter.
• The function decreases on the interval
• The maximum value of the function is 0, achieved at y = 0.
• The functions are continuous.

Self-test:

• A(81; -9). x = 81, y = - 9.

Answer: yes

2) B(-25; 625). x = -25; y = 625.

Answer: no.

Answer: yes

Solve the equation graphically:

Let's construct graphs of functions in one coordinate system:

0 1 2 3 4 5 6 9

X

U

y= x-6

X

U

Let's find the abscissa of the intersection points of the graphs

X =9

ANSWER:

• ANSWERS:
• a) 1; b) 1.

• ANSWERS:
• a) (4; - 2); b) (0; 0); (4; - 2).

• Horizontally:

• The action used to find the square root.
• The quarter in which the graph of the function is located
• Square root of 144.
• Endless fraction with repeating digits.
• Dependence of one variable on another.
• A rational number is the ……… of a whole number to a natural number.

• Vertically:

• The name of the expression containing the roots.
• Ancient Greek mathematician who proved that he is not a rational number.
• Arithmetic root.
• Graph of a function y = x 2

A trigger is used. When you click on the red numbers, the answers are horizontal. When you click on the blue numbers, the answers are vertical.

Ancient Greek mathematician Euclid

• Date of birth: around 325 BC
• Place of birth: or Athens, or Shooting Range
• Scientific field: mathematics
• The main work is “Beginnings”.
• Known as: "The Father of Geometry".
• Author of works on astronomy, optics, music, etc.

• Homework:
• Paragraph 13, No. 9, No. 11.

“Definition of a numerical function” - Graphical method. Definition of a numerical function. Y=f(x). Analytical method. It is convenient to describe graphs by matrices. The function is given in a table. Verbal formulation. The function y=f(x) is given. The function is given graphically. The scope of the function. Express each variable in terms of the other two. Numerical set X and rule f.

““Functions” algebra” - The function F is called the antiderivative of the function f. “Integral from a to b ef from x de x.” Let's find one of the antiderivatives for the function. Let's make a table. Derivative of trigonometric functions. Intersections with Ou. Interval method. The largest and smallest value of a function. We are building a schedule. Derivative of a complex function.

“Elementary functions” - Power function with a natural exponent. Elementary functions. Formula for transition between logarithms. Arc cosine. Mathematics. Formulas. Basic properties of degrees. Inverse trigonometric functions. Function properties. Exponential function. Basic values ​​of arcsine and arccosine. Basic properties of logarithms.

The value of y at which x=3. Check: Student at the blackboard. Using the graph, determine: - The value of x at which f(x)=0. Study of functions. Student at the blackboard. Reinforcing the material covered. Warm up. Within the scope of the school curriculum. - Determine the properties of this function. Methodological topic. 2. Is the function given by the formula linear and indicate K and B:

“Numerical functions” - The simplest examples of such interdependencies are provided by geometry. Function graph. The set X is called the domain of assignment or the domain of definition of the function f and is denoted by D (f). Introduction. Example 1. A paratrooper jumps from a hovering helicopter. Just one number. Definition. Definition Let X be a number set.

“Problems on functions” - Variable. Functions. Some number. Meanings. Variable dependence. Dependent variable. A bunch of. Independent variable. Instructions for using the simulator. Independent variable values. Argument values.

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