Medicine      03/27/2020

Direct proportionality and its graph. Study of the direct proportionality function $f(x)=kx$ and its graph

Goals:

  • give the foundations of a healthy lifestyle,
  • learn to value and maintain their own health,
  • develop horizons and cognitive activity.

Equipment: drawings of children, felt-tip pens, sheets of paper, drawing - a poster of a neatly combed person and false drawings of bad habits

The little son came to his father.
The little one asked:
"What is good
And what is bad?

If you are charging
If you eat salad
And you don't like chocolate
You will find a health treasure.

If you don't want to wash your ears
You don't go to the bathroom
You are friends with a cigarette -
So you won't get health.

You need to practice in the mornings and evenings.

Wash, temper, boldly engage in sports,

Try to be healthy.

We only need this.

1. Introductory conversation.

For centuries, people's ideas about what it means to be healthy have not changed. Health, as a natural state of man, is the same in the West and in the East. Doctors and philosophers of antiquity quite rightly believed that a person is an inseparable part of the surrounding world.

Even in Russia, Catherine II understood how important it was for the younger generation to learn how to take care of their health. She approved the city folk schools, where students were already taught how to take care of their health.

What do you think health is?

In 1783, the author of the textbook wrote: “We call the health of our body that state when our body is free from all shortcomings and diseases.” In modern dictionaries, health means - this or that physical state person.

What expressions do you know related to the word health? (I started for health, but brought it to rest, I wish you good health, good health, etc.)

What affects our health and what does it depend on?

2. Game-competition.

Before you is the image of a sloppy sick person. Now we will try to find out what bad habits led him to this appearance. I ask you questions if you answer correctly bad habit removed and gradually the person takes on a normal healthy appearance.

Why can't you chew on the tip of a pencil or pen? (Teeth will be uneven)

Why can't you smoke? (Smell from the mouth, yellow teeth, cough, lameness)

How does drinking alcohol affect health? (Dilation of blood vessels, red nose, diseases of the liver, stomach, poor memory)

How often and why should you wash your hair? (To keep the hair clean, lice do not start, wash once every 5-6 days).

Why can't you bite your nails? (Under the nails are pathogens of various diseases, helminth eggs).

What can the habit of taking inedible objects in the mouth lead to? (You can swallow an object, pierce the sky or cheek, get sick)

Why can't you change clothes and shoes, wear other people's hats? (you can get skin infections)

Why can't you watch TV for hours and sit at the computer? (Vision deteriorates)

Conclusion:

Don't take other people's dresses
Don't bite your nails either.
Don't be friends with a cigarette
So health is not found.

3. Creative task.

At home, you were given the task to draw one picture each on the topic: “We are for a healthy lifestyle.” Everyone has to explain the meaning of their drawing. (Discussion of drawings)

4. Quiz.

1. What common insect acts as a carrier of germs, contaminating unsealed food? (Fly)

2. Hand washing is an elementary rule of personal hygiene. Name four times when hand washing is necessary to ensure food safety. (Before eating, after using the toilet, after touching pets, after handling raw meat).

3. What plants cause allergies? (poplar, alder, bird cherry, wormwood)

4. This vegetable is very cool,

Kills everything around.

Phytoncides help,

And from smallpox and plague.

And we can also cure a cold for them. (Onion)

5. Name the means of hardening. (Sun, air, water)

6. Why can't pet other people's cats and dogs? (Possible infection with eggs of worms, fleas, deprive)

7. Why can't you chew on the tip of a pencil or pen? (Teeth will be uneven, germs may enter)

8. Why can't you change clothes, shoes, hats? (can be infected with skin and infectious diseases).

9. What can squeezing blackheads and pimples lead to? (to the emergence and spread of infections, up to death from blood poisoning)

5. Final word.

Maintaining health is a regular job. Now you are young and healthy. But do not flatter yourself - you should always remember about maintaining health. And in conclusion, we will try to draw up the main provisions of the health code.

What are the rules to follow to be healthy? (I give out codes to children)

Health Code:

Do not smoke, do not drink alcoholic beverages.

Exercise.

Eat more fish, vegetables, fruits.

Breathe fresh air.

Drink water, milk, juices, tea.

Walk as much as possible.

Sleep enough.

Show kindness.

Smile more often.

To love life.

Try to follow this health code and you will long years keep youth and beauty.

6. Closing remarks by students.

1st. We are born to live in the world for a long time:

2nd. Sad and sing, laugh and love.

3rd. But to make all dreams possible,

5th. Ask yourself: are you ready to work -

6th. Move actively and eat and drink in moderation?

7th. Throw away the cigarette? Throw away a cigarette butt?

9th. Take a look around: beautiful Nature

10th. We are called to live in peace with her.

11th. Give me a hand, friend! Let's help you

Class hour on the topic:

"I choose a healthy lifestyle" (lesson - competition)

Purpose of the competition : to form in students the need for a healthy lifestyle.

Tasks:

to help students realize the importance of a reasonable attitude to their health;

promote the health of children;

develop group work skills, communication skills, attention, imagination, ingenuity, Creative skills, speech;

to cultivate a culture of behavior and communication when working in groups.

Cabinet layout:

a poster with the name of the competition program "It's great to be healthy!";

drawings of students on the board;

along the perimeter of the office are family wall newspapers "We are for a healthy lifestyle!";

tables for two teams.

Equipment: computer; sheets with letters З, Д, О, Р, О, В, b, Е; cards with proverbs; chest and items for it; sheets for tasks; markers; letters.

Teacher : Hello, Dear Guys! I am glad to see you at our health holiday with the name "It's great to be healthy!".

Together: Hello! We wish everyone good health!

Teacher: Human health is the main value in life. Money can't buy health. Being sick, you will not be able to realize your dreams, you will not be able to solve vital tasks. We all want to grow up strong and healthy. Being healthy is a natural human desire, sooner or later everyone thinks about their health. Each of us must be aware of what a priceless treasure this is. Let's think together about what health, a healthy lifestyle is.

Boy:

For us to be beautiful

To not be whiny,

So that in the hands of any business

Argued amicably, burned ...

Girl:

To sing songs louder

To make our life more interesting...

Boy and girl:You need to be strong and healthy.

Teacher: Our competition is called "It's healthy to be great!" And today we will try to prove it to each other and to ourselves. The game is played by 2 teams. The first game table is the team _____________________________________;

the second game table is the team.

Our competition will be monitored by a strict jury composed of ____________________________________________________________________________________________________________________________________.

So, we begin our competitive program. The first competition is called "Health".

Competition "Health"

For each letter in the word "health" you need to choose other words that are related to health, a healthy lifestyle. Each word will bring the team one point. You have one minute to complete the task. (Music plays during commands.)

Contest "Folk Wisdom Says"

Teams receive cards with unfinished proverbs. The task of the participants is to complete the proverbs about health. At the end of the competition, team representatives read out their options for ending the proverbs. For each correct answer, the team receives one point.

Purity - ___________________________________________.

(Answer: a guarantee of health.)

Health is in order - _______________________________.

(Answer: thanks to the charger.)

If you want to be healthy, _______________________________.

(Answer: temper.)

In a healthy body - ____________________________________.

(Answer: a healthy mind.)

Teacher: In the meantime, the teams are answering, let's play with the fans. Answer together in chorus "It's me, it's me, it's all my friends" if you agree with me. If this is not about you, then be silent, do not make noise.

Game "It's me, it's me, it's all my friends" (for fans)

Questions for students:

which of you is always ready to live life without doctors;

who does not want to be healthy, vigorous, slim and cheerful;

which of you does not walk gloomy, loves sports and physical education;

who is not afraid of frost, flies on skates like a bird;

well, who will start lunch with a chewing gum with a couple of sweets;

who loves tomatoes, fruits, vegetables, lemons;

who ate and brushed their teeth regularly twice a day;

which of you, from the kids, walks dirty to the ears;

who, according to the schedule, performs physical exercises;

who, I want to know from you, likes to sing and relax?

Competition "Health can not be bought - his mind gives"

Teams are given questions that need to be answered in the affirmative or negative. For each correct answer, the team receives one point. Questions:

1. Do you agree that exercise is a source of vitality and health?

(Answer: yes.)

2. Is it true that chewing gum saves teeth? (Answer: no.)

3. Is it true that teeth should be brushed once a day? (Answer: no.)

4. Is it true that bananas cheer you up? (Answer: yes.)

5. Is it true that carrots slow down the aging of the body? (Answer: yes.)

6. Is it easy to quit smoking? (Answer: no.)

7. Is it true that the lack of sun causes a bad mood? (Answer: yes.)

8. Is it true that in the summer you can stock up on vitamins for the whole year? (Answer: no.)

9. Is it true that you should drink two glasses of milk every day? (Answer: yes.)

10. Is it true that a bad mood affects health? (Answer: yes.)

Teacher: To be healthy, you need to know and be able to do a lot. During the next competition, you will also need to answer questions.

For each correct answer, the team will receive 2 points.

Competition "Knowledge is power"

There are questions to be answered. Questions:

1. What is the name of the result of exposure to frost on the human body? (Answer: frostbite.)

2. Who can infect a person with rabies? (Answer: animals.)

3. What are the names of drinks that destroy human health? (Answer: alcohol.)

4. What is the name of the dressing material? (Answer: bandage.)

5. What is the name of the injury caused by fire? (Answer: burn.)

6. What contributes to the hardening of the body? (Answer: sun, air, water.)

Competition "Mysteries about the daily routine"

Puzzles:

1. You decided to become healthy, so observe ... (answer: regimen);

2. In the morning at seven, our cheerful friend rings persistently ... (answer: alarm clock);

3. All our friendly people got up to exercise ... (answer: family);

4. Of course, I won’t break the regime - I wash under cold ... (answer: shower);

5. After a shower and exercise, a hot one is waiting for me ... (answer: breakfast);

6. I always wash my hands with soap, no need to call us ... (answer: Moidodyr);

7. After dinner, you can have a sweet sleep, or you can also in the yard ... (answer: play);

8. After dinner, fun - we take dumbbells in our hands, we do sports with dad, our mom ... (answer: smile);

9. The moon looks out the window to us, which means that it’s been a long time to sleep ... (answer: it’s time).

For fans

Teacher:

Friends, I have poems for you,

But you can help read them.

As soon as I raise my hand,

Everyone says the word "health"!

We must know the laws of ____________ for sure!

Take care of your _________ and protect it!

Means a lot!

All the more important!

Captains Competition.

Teacher: Now answer the riddles. You will be given a verbal prompt and you must guess what it is. Listen carefully to the prompt:

1. These plants with a characteristic pungent odor are a good remedy in the prevention of colds (answer: onion, garlic);

2. Substances that must be consumed if you are taking strong medications (answer: vitamins);

4. Liquid, not water, white, not snow (answer: milk).

Competition "Healthy lifestyle - it's stylish!

Students should write down the rules on sheets of paper. healthy lifestyle life. Teams work to the music, then read out their rules. For each rule, the team receives one point.

Girl: And now let's rest a little. I ask everyone to stand up. Align your socks, clasp your hands behind your back, sit down several times. Close your eyes, open your eyes (5 times). Raise your shoulders in turn (5 times). Hands on the belt, tilts to the right, to the left. How can you call what we are doing now? That's right, warm-up, exercise, that is, we lead an active lifestyle that promotes health.

(The jury summarizes the results of the competition and awards the teams with certificates.)

Boy:

Man was born

Get on your feet and go!

Made friends with the wind, the sun,

To breathe well!

Girl:

Trained myself to order

He got up early in the morning.

Cheerfully he did exercises,

I took a cold shower.

Boy:

Every day he ran, jumped,

I swam a lot, played ball,

Gaining strength for life

And he did not whine, he did not get sick.

Girl:

Went to bed at eight-thirty

I fell asleep very quickly.

Interested in learning

And I got five.

Boy:

Get up early in the morning,

Take a cool shower

Get on the charger

Refresh yourself with porridge with butter!

Teacher: Health is an invaluable wealth in the life of any person. Each of us has an inherent desire to be strong and healthy, to maintain mobility, vigor, energy as long as possible and to achieve longevity. I hope that today's game was not in vain and you learned a lot from it. No wonder they say: "If you are healthy, you will get everything." So be healthy everyone, goodbye!


Definition of Direct Proportionality

First, let's recall the following definition:

Definition

Two quantities are called directly proportional if their ratio is equal to a specific non-zero number, that is:

\[\frac(y)(x)=k\]

From here we see that $y=kx$.

Definition

A function of the form $y=kx$ is called direct proportionality.

Direct proportionality is a special case of linear functions$y=kx+b$ for $b=0$. The number $k$ is called the coefficient of proportionality.

An example of direct proportionality is Newton's second law : Acceleration body is directly proportional to the force applied to it:

Here the mass is the coefficient of proportionality.

Study of the direct proportionality function $f(x)=kx$ and its graph

First, consider the function $f\left(x\right)=kx$, where $k > 0$.

  1. $f"\left(x\right)=(\left(kx\right))"=k>0$. Therefore, this function increases over the entire domain of definition. There are no extreme points.
  2. $(\mathop(lim)_(x\to -\infty ) kx\ )=-\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=+\infty $
  3. Graph (Fig. 1).

Rice. 1. Graph of the function $y=kx$, for $k>0$

Now consider the function $f\left(x\right)=kx$, where $k

  1. The scope is all numbers.
  2. The scope is all numbers.
  3. $f\left(-x\right)=-kx=-f(x)$. The direct proportionality function is odd.
  4. The function passes through the origin.
  5. $f"\left(x\right)=(\left(kx\right))"=k
  6. $f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
  7. $(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
  8. Graph (Fig. 2).

Rice. 2. Graph of the function $y=kx$, for $k

Important: to plot the function $y=kx$, it is enough to find one that is different from the beginning coordinate point$\left(x_0,\ y_0\right)$ and draw a line through this point and the origin.

Let's build a graph of the function, given formulay = 0.5x.

1. The domain of this function is the set of all numbers.

2. Let's find some corresponding values ​​of variables X And at.

If x = -4, then y = -2.
If x = -3, then y = -1.5.
If x = -2, then y = -1.
If x = -1, then y = -0.5.
If x = 0, then y = 0.
If x = 1, then y = 0.5.
If x = 2, then y = 1.
If x = 3, then y = 1.5.
If x = 4, then y = 2.

3. We mark in the coordinate plane the points whose coordinates we determined in paragraph 2. We note that the constructed points belong to some straight line.

4. Let us determine whether other points of the graph of the function belong to this line. To do this, we find the coordinates of several more points on the graph.

If x = -3.5, then y = -1.75.
If x = -2.5, then y = -1.25.
If x = -1.5, then y = -0.75.
If x = -0.5, then y = -0.25.
If x = 0.5, then y = 0.25.
If x = 1.5, then y = 0.75.
If x = 2.5, then y = 1.25.
If x = 3.5, then y = 1.75.

Having constructed new points of the function graph, we notice that they belong to the same line.

If we decrease the step of our values ​​(take, for example, the values X through 0,1; through 0,01 etc.), we will get other points of the graph that belong to the same line and are located closer and closer to each other. The set of all points in the graph of a given function is a straight line passing through the origin.

Thus, the graph of the function given by the formula y = kх, where k ≠ 0, is a line passing through the origin.

If the domain of the function given by the formula y = kх, where k ≠ 0, does not consist of all numbers, then its graph is a subset of points on a straight line (for example, a ray, a segment, individual points).

To construct a straight line, it is enough to know the position of two of its points. Therefore, a graph of direct proportionality given on the set of all numbers can be built using any two of its points (it is convenient to take the origin as one of them).

Let, for example, it is required to build a graph of the function given by the formula y = -1.5x. Let's choose some value X, not equal 0 , and calculate the corresponding value at.

If x = 2, then y = -3.

Mark a point on the coordinate plane with coordinates (2; -3) . Draw a straight line through this point and the origin. This straight line is the desired graph.

Based on this example, it can be shown that any straight line passing through the origin and not coinciding with the axes is a graph of direct proportionality.

Proof.

Let some straight line passing through the origin of coordinates and not coinciding with the axes be given. Take a point on it with abscissa 1. Denote the ordinate of this point by k. It is obvious that k ≠ 0. Let us prove that this line is a direct proportionality graph with coefficient k.

Indeed, from the formula y = kx it follows that if x = 0, then y = 0, if x = 1, then y = k, i.e. the graph of the function given by the formula y \u003d kx, where k ≠ 0, is a straight line passing through the points (0; 0) and (1; k).

Because only one straight line can be drawn through two points, then this straight line coincides with the graph of the function given by the formula y = kх, where k ≠ 0, which was to be proved.

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