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Synopsis for the lesson "Rivnyannya z dvoma zminnim that yogo graph". The topic of the lesson: "Rivnyannya z two zmіnnim and yogo graph"

META: 1) Oznayomiti scholars with the understandings "rivnyannya with two wintry";

2) Navchit to start the steps of the family with two winter ones;

3) Navchit to start for a given function, like a figure є graph

given іvnyannya;

4) To look at the transformation of graphs with two different ones;

a given ryvnyannnyu with two zminnymi, vicoristovuchi program Agrapher;

6) Development of logical misinterpretations of scholars.

I. New material - an explanatory lecture with the elements of conversation.

(The lectures are conducted on the basis of the author's slides;

У: When vivchennі linіy, there are two defaults:

Behind the geometric powers of the given line to know the її її ryvnyannya;

Zvorotniy zvdannya: according to the given line of predictions of geometrical power.

First, we looked at the geometries in the course of the geometrical circle and straight line.

This year we will look at the final problem.

Clear view:

a) x (x-y) = 4; b) 2y-x 2 =-2 ; v) x (x + y 2 ) = X +1.

- tse put the priests with two wives.

Rivnyannya with two wintry houses NSі at ma viglyad f (x, y) = (x, y), de fі - virazi with wintry NSі at.

Yaksho in Rivnyanny x (x-y) = 4 present a change NSїї value is -1, and vice versa at- the value is 3, then we see the correct parity: 1 * (- 1-3) = 4,

Pair (-1; 3) value of the change NSі atє rіshennyam rіvnyannya x (x-y) = 4.

Tobto solutions for rivnyannya they call them two bezl_ch well-ordered couples are the meaning of the winners, which set the price in the correct parity.

Rivnyannya with two winter ma, which, as a rule, is not endlessly rich in solution. vignettes warehouse, for example, such rivnyannya, yak NS 2 + (Y 2 - 4) 2 = 0 abo

2x 2 + at 2 = 0 .

First of them there are two solutions (0; -2) and (0; 2), the other - one solution (0; 0).

Rivnyannya x 4 + y 4 +3 = 0 vzagal is not a big deal. Tsіkavim є, if the meanings of the winners in the ryvnyannі serve the whole number. Virishuchi is like a pair of winners, there is a bet of a whole number of numbers. In such vipadkah it seems that the r_vnyannya is shown in whole numbers.

Two rivnyannya, which may be one і also bezlіch decision, I call with strong equals... For example, equal x (x + y 2) = x + 1 є equal to the third step, so that it can be converted into equal xy 2 + x 2 - x-1 = 0, the right part of which is a polynomial of the standard third step.

The step equal to the two steps represented by the view F (x, y) = 0, de F (x, y) -polynomial of standard form, is called the step of the polynomial F (x, y).

As all solutions are drawn with two different points in the coordinate area, then the graph is shown with two small ones.

grafik Rivnyannya with two wines are called pointless points, the coordinates of which serve as solutions of the rivnyannya.

So, grafik rivnyannya ax + by + c = 0 I am straightforward, if I would like to have one a abo b not expensive to zero (fig. 1)... yaksho a = b = c = 0, That is the graph of tsyo rivnyannya є coordinate area (Fig. 2), well a = b = 0, a c0, Then grafik є empty lot (fig. 3).

Graph Rivnyannya y = a x 2 + By + c is a parabola (Fig. 4), graph xy = k (k0) – hyperbola (fig. 5)... grafikom rivnyannya NS 2 + y 2 = r, De x і y - change, r - positive number, є circumference centered on the cob of coordinates and radius equal r(Fig. 6). Graph Rivnyannya є elips, de aі b- large and small pivosi elipsa (Fig. 7).

Pobudova graphykiv deyakikh pryvnyany lie down to vikorstannya їkh revision. Clear re-creation of graphs of rivnyan with two winter I will formulate the rules, for those who are vikonuyutsya simplest re-implementation of the graph

1) Graph F (-x, y) = 0 Enter from graph F (x, y) = 0 for an additional symmetry on the axis at.

2) Graph F (x, -y) = 0 Enter from graph F (x, y) = 0 for an additional symmetry on the axis NS.

3) Graph F (-x, -y) = 0 Enter from graph F (x, y) = 0 behind an additional central symmetry of coordinates.

4) Graph F (x-a, y) = 0 Enter from graph F (x, y) = 0 for an additional move parallel to the x-axis on | a | odinit (right, yaksho a> 0, і вліво, якщо a < 0).

5) Graph F (x, y-b) = 0 Enter from graph F (x, y) = 0 for additional transfer to | b | one parallel to the axis at(Upra, yaksho b> 0, i down, where b < 0).

6) Graph F (ax, y) = 0 Enter from graph F (x, y) = 0 for additional compression to the axis y and a time, if a> 1, і for an additional extension from the axis at times, where 0< a < 1.

7) Graph F (x, by) = 0 Enter from graph F (x, y) = 0 b times, yaksho b> 1, і for additional stretching from the axis x times, where 0 < b < 1.

If the graph of the deyakogo ryvnyannya is turned to the deyaky kut close to the cob of coordinates, then the new graph will be the graph of the first line. It is important to turn to the kuti 90 0 і 45 0.

8) The graph of the equal F (x, y) = 0 as a result of the rotation close to the cob of coordinates on the corner 90 0 for the year arrow go to the graph of the equal F (-y, x) = 0, and the opposite arrow - in the graph of the equal F (y , -x) = 0.

9) The graph of the equal F (x, y) = 0 as a result of turning close to the cob of coordinates to the corner 45 0 for the year-old arrow go to the graph of the equal F = 0, and the opposite arrow - into the graph of the equal F = 0.

With the rules we have seen for the conversion of graphs, it is easy to follow the rules for the conversion of graphs of functions.

Butt 1. It will be shown that graph NS 2 + y 2 +2 x - 8Y + 8 = 0є circle (fig. 17).

Rivnyannya recast in such a rank:

1) zgrupuєmo dodanka, scho to avenge the change NS and revenge the change at, The first obvious cutaneous group of dodanks in the viglyad of the main square of the trinomial: (x 2 + 2x + 1) + (y 2 -2 * 4 * y + 16) + 8 - 1 - 16 = 0;

2) it is written in the view of the square of the sum (ізніні) two virazіv otrimanі Trinomial: (x + 1) 2 + (y - 4) 2 - 9 = 0;

3) pro-analysis, based on the rules for rewriting the graphs of rivnyans with two different ones, rivnyannya (x + 1) 2 + (y - 4) 2 = 3 2: the graph of this equal є circle centered at point (-1; 3) and radius single.

Appendix 2. Wake up the graph of rivnyannya NS 2 + 4y 2 = 9 .

Uyavimo 4y 2 from viglyadі (2y) 2, mо rіvnyannya х 2 + (2у) 2 = 9, the graph of which can be trimmed from number х 2 + у 2 = 9 compressed to axes x 2 times.

A circle with a center on the cob of coordinates and a radius of 3 units is permissible.

The skin point from the X axis is changed 2 times, and the graphical rivnyannya is recognized

x 2 + (2y) 2 = 9.

We trimmed the figure behind the help of squeezing the circumference to one of three diameters (up to the diameter, which lies on the X axis). Taku figuru is called elips (Fig. 18).

Appendix 3. Z'yasuєmo, which is a graph of rivnyannya x 2 - y 2 = 8.

Skoristaєmosya formula F = 0.

In accordance with the law, the substitute X and the deputy Y, are recognized as follows:

U: Who is the graph of rivnyannya y =?

D: Graph of rivnyannya y = є hyperbole.

U: We have re-created the equal form x 2 - y 2 = 8 into equal y =.

Yaka line will be the graph of this ryvnyannya?

D: So, і graph іvnyannya х 2 - у 2 = 8 є hіperbola.

У: Які straight є asymptotes of the hyperbola у =.

D: asymptotes of the hyperbola y = є straight y = 0 and x = 0.

Y: With a viconious turn, go straight to straight = 0 and = 0, so in straight y = x and y = - x. (Fig. 19).

Appendix 4: Z'yasuєmo, which is a kind of reception equal to y = x 2 parabols when turning the cob of coordinates on the cube 90 0 behind the year line.

Vikoristovuchi formula F (-y; x) = 0, substitute in equal y = x 2 change x to - y, and change y to x. Otrimaєmo rivnyannya x = (-y) 2, i.e., X = y 2 (Fig. 20).

We looked at the butts of the graphs of the ryvnyans of the other step with two z'yasuvali, but the graphs of such rivnyans can be a parabola, hyperbola, elips (zokrema circle). In addition, the graph of the level of another step can be a pair of straight lines (transverse or parallel). So the graph is equal x 2 - at 2 = 0 є a pair of intersecting straight lines (Fig. 21), and the graph is equal to x 2 - 5x + 6 + 0y = 0- parallel straight lines.

II Closing.

(Scientists see “Card-Instructions” as well as displaying the motives of the graphs of the village with two children in the Agrapher program (Dodatok 2) and the cards “Practical Zavdannya” (Dodatok 3) for the form of the slides of the employees in the 4-5 children’s graphs ).

Manager 1. Yaki z pairs (5; 4), (1; 0), (-5; -4) і (-1; -)

a) x 2 - y 2 = 0, b) x 3 - 1 = x 2 y + 6Y?

Decision:

Having submitted a pair in the assignment, according to the coordinates of the given points of the change, the pair is given not to the solutions of the equal x 2 - y 2 = 0, but to the solutions of the equal x 3 - 1 = x 2 y + 6U є bet (5; 4), 1; 0) і (-1; -).

125 - 1 = 100 + 24 (I)

1 - 1 = 0 + 0 (I)

125 - 1 = -100 - 24 (L)

1 - 1 = - - (I)

as follows: a); b) (5; 4), (1; 0), (-1; -).

Head 2. Know this decision rivnyannya xy 2 - x 2 y = 12, in which values NS door 3.

Decision: 1) Substitute X in tasks of equal value 3.

2.)

3y 2 - 9y = 12.

4) Virishimo tse rivnyannya:

3y 2 - 9y - 12 = 0

D = 81 + 144 = 225

View: bet (3, 4) і (3; -1) є xy 2 - x 2 y = 12

Assignment3. Enter the steps of the ryvnyannya:

a) 2y 2 - 3x 3 + 4x = 2; c) (3 x 2 + x) (4x - y 2) = x;

b) 5y 2 - 3y 2 x 2 + 2x 3 = 0; d) (2y - x 2) 2 = x (x 2 + 4xy + 1).

View: a) 3; b) 5; at 4; d) 4.

Assignment 4. Yaka figura є grafikom rivnyannya:

a) 2x = 5 + 3y; b) 6 x 2 - 5x = y - 1; c) 2 (x + 1) = x 2 - y;

d) (x - 1.5) (x - 4) = 0; e) xy - 1.2 = 0; f) x 2 + y 2 = 9.

Assignment5. Write an ryvnyannya, a graph which is symmetric, a graph of a rivnyannya x 2 - xy + 3 = 0 (Fig. 24): a) a) NS; b) axis at; c) straight y = x; d) straight y = -x.

Assignment6. Store the rivnyannya, the graph of how to expand the graph, the graph of the rivnyannya y = x 2 -3 (Fig. 25):

a) from axes x 2 times; b) from axis y 3 times.

Check the correctness of the application for the help of the Agrapher program.

View: a) y - x 2 + 3 = 0 (Fig. 25); b) y- (x) 2 + 3 = 0 (pic.25b).

b) straight parallel, shifting parallel to the x axis by 1 unit to the right and parallel to the y axis by 3 units down (pic.26b);

c) change straight, symmetrically displayed along the x axis (pic.26c);

d) change straight, symmetrically imaged according to the y axis (pic.26d);

e) straight parallel, symmetrically displayed according to a cob of coordinates (pic.26e);

f) straighten, rotate close to the cob of coordinates by 90 by the year's arrow and symmetrically displayed along the x axis (Fig. 26).

III. Self-operation of the robot initial character.

(Scientists see the cards "Self-styled robot" and "Sound table of results of self-styled robots", in which they write down their views and for self-revision, according to the proponated scheme to evaluate the robot) Dodatok 4 ..

I. variant.

a) 5x 3 -3x 2 y 2 + 8 = 0; b) (x + y + 1) 2 - (x-y) 2 = 2 (x + y).

a) x 3 + y 3 -5x 2 = 0; b) x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 = 1.

x 4 + y 4 -8x 2 + 16 = 0.

a) (x + 1) 2 + (y-1) 2 = 4;

b) x 2 -y 2 = 1;

c) x - y 2 = 9.

x 2 - 2x + y 2 - 4y = 20.

Specify the coordinates for the center of the stake і її radius.

6. Yak slid on the coordinate area to change the hyperbola y =, did the rivnyannya take the form x 2 - y 2 = 16?

Revise your view, showing the graphical prompts, the victorious programs of the Agrapher.

7. Yak slid on the coordinate area, move the parabola y = x 2, and the line took the form x = y 2 - 1

II option.

1.Define the step of the family:

a) 3xy = (y-x 3) (x 2 + y); b) 2y 3 + 5x 2 y 2 - 7 = 0.

2. Chi є a pair of numbers (-2; 3) for the solution:

a) x 2 -y 2 -3x = 1; b) 8x 3 + 12x 2 y + 6x 2 + y 3 = -1.

3. Know the simple answer:

x 2 + y 2-2x - 8Y + 17 = 0.

4. Yaky crooked (hyperbola, circle, parabola)

a) (x-2) 2 + (y + 2) 2 = 9

b) y 2 - x 2 = 1

c) x = y 2 - 1.

(Reverse for the additional Agrapher programs the correctness of the application)

5. Indulge, the victorious program of the Artist, grafik rivnyannya:

x 2 + y 2 - 6x + 10y = 2.

6. Yak slid on the coordinate area to change the hyperbola y =, so the rivnyannya took the form x 2 - y 2 = 28?

7. On the coordinate area, change the parabola y = x 2, and the line took the form x = y 2 + 9.

At the end of the day, I can see in detail the prompting of the graphs of rivnyan. It’s possible to guess that it’s also a more rational and ineffectual decision, so it’s a graph of rivnyannya. The graph of the line-of-sight and power of the line function, which can be read in the graph, can be seen in detail. The graph of the square function and the power of the quadratic function is far away. The hyperbolic function and the graph and the graph of the circle are clearly visible. We will go over to the prompting of that vyvchennya sukupnostі graphіkіv.

Topic: Systemi Rivnyan

Lesson: Graphs of Rivnyans

We look at the rational mind and systems and the rational mind

We said that the skin of the ryvnyannya in the whole system is maє svіy graph, which is very special є rіshennya ryvnyan. We looked at the little picture of the graphs of the rural people.

Infection is systematically discernible from the skin of the people we see grafikami rivnyan.

1. Lіnіine rіvnyannya with two wintry

x, y - in the first step; a, b, c - specific numbers.

butt:

The graph of the rivnyannya is straight line.

Miyyali were equally powerful re-enactments - they were abandoned in the place, everything else was moved to the other side with the opposite signs. Pochatkov and і іtrimanе іvnyаnnya іvnyаnnya, in order to think one and the same without a solution. The graph of the rivnyannya mi vmієmo buduvati, and the technique of yo motivate this: it is known that the points will cross with the coordinate axes, and along them it will be straight.

In this vipadku

Know the graph of rivnyannya, we can tell a lot about the decision of the vicious rivnyannya, but itself: yakscho if

The function of growth is high, so that it grows x to grow y. We have rendered two private decisions, but how can we write down all the decisions?

If the point is abscissa x, then the ordinate of the point

Mean numbers

We had a lot of ryvnyannya, we had a graph, we knew the decision. Without all the couples - skilki їkh? Power is strong.

Tse ratsіonalne іvnyannya,

We know y, with strong perversions obsessed

It is satisfactory and irreconcilable to a quadratic function;

Butt: Get a graph of a rational rivnyannya.

The graph is a parabola, the heads are straight up the hill.

Know the root of the rivnyannya:

Schematically depictable graph ( Small. 2).

Behind the additional graph we will recognize all the views and about the functions, and about the solutions of the rational competition. We have started the intervals of constancy, now we know the coordinates of the vertex of the parabola.

Rivnyannya has no problem, no couples, no less happy couples, happy rivnyannya, but everything And how can you be x? Be yakim!

If it is possible to put x, then we can make a point

Rishennyam vikhіdnogo ryvnyannya є without couples

3. Stay with the graph of rivnyannya

Need to see y. Two options are available.

By the graph of the function є the hyperbola, the function is not assigned when

The function is declining.

If we look at the point from the abscissa, then the ordinate will be

Rishennyam vikhіdnogo ryvnyannya є without couples

A hyperbola can be induced to destroy a number of coordinate axes.

For example, graph of functions - also hyperbole - there will be a change to one uphill along the ordinate axis.

4. Rivnyannya stake

Tse ratsіonalne ryvnyannya with two wintry. Bezlichchyu solution є points of a circle. Center at point radius road R (Fig. 4).

Concrete butt is clearly visible.

Pointing to the standard square of the circle, for which you can see the square sumi:

- we got the rіvnyannya cola with the center in .

I'll stay with the graph of rivnyannya (Fig. 5).

b. Stay in the grafik rivnyannya

Sgadaimo, if you want to go to zero, todi and only todi, if one of the multipliers goes to zero, and the other one is.

The graph of a given rivnyannya warehouse is based on the number of graphs of the first and other rivnians, so two straight.

Let's stay yogo (Fig. 6).

As soon as possible the graph of the function The straight line will pass through the point (0; -1). Ale yak won’t pass - will you be slow or fall? Visually, the price can be helped by the kutovy kofіtsієnt, the kutovy kofіtsіnt at x, the win is negative, the price to inform about those ubuvans. We know the cross-point point from the ox, the point (-1; 0).

Similarly, there will be a graph of another rivnyannya. Pass straight through the point (0; 1), ale of growth, so the kutoviy kofіtsієnt is positive.

The coordinates of all points are two prompted straight lines and solutions.

From the same time, we analyzed the graphs of the most popular rational races, the stench will be victorious both in the graphical method and in the illustration of the latest methods in the verification of systems of races.

1. Mordkovich A.G. that in. Algebra 9th grade: Textbook. For zalnoosvit. Institutions. - 4th ed. - M .: Mnemosyna, 2002.-192 p.: Il.

2. Mordkovich A.G. that in. Algebra 9th grade: Problem book for scholars in the field of education / A. G. Mordkovich, T. N. Mishustina і ін. - 4th ed. - M .: Mnemosina, 2002.-143 p.: Il.

3. Makarich Yu. M. Algebra. 9th class: navch. for scientists zalnoosvit. ustanovka / Yu.M. Makarichєv, N.G. Mindyuk, K.I. Neshkov, I. E. Feoktistov. - 7th ed., Rev. і add. - M .: Mnemosina, 2008.

4. Alimov Sh.A., Kolyagin Yu.M., Sidorov Yu.V. Algebra. Class 9 16th ed. - M., 2011 .-- 287 p.

5. Mordkovich A. G. Algebra. Class 9 At 2 o'clock, Part 1. Pidruchnik for scholars of the educational establishment / A. G. Mordkovich, P. V. Semenov. - 12th ed., Ster. - M .: 2010 .-- 224 p.: Il.

6. Algebra. Class 9 At 2 hours, Part 2. Problem book for scholars of the educational setting / A. G. Mordkovich, L. A. Aleksandrova, T. N. Mishustina and I.; Pid ed. A. G. Mordkovich. - 12th ed., Rev. - M.: 2010.-223 p.: Il.

1. Rozdil College.ru from mathematics ().

2. Internet project "Zavdannya" ().

3. lighting portal"VIRISHU ЄDI" ().

1. Mordkovich A.G. that in. Algebra 9th grade: Problem book for scholars in the field of education / A. G. Mordkovich, T. N. Mishustina i ін. - 4th ed. - M .: Mnemozina, 2002.-143 p.: Il. No. 95-102.

Topic: Systemi Rivnyan

Lesson: Graphs of Rivnyans

We look at the rational mind and systems and the rational mind

We said that the skin of the ryvnyannya in the whole system is maє svіy graph, which is very special є rіshennya ryvnyan. We looked at the little picture of the graphs of the rural people.

Infection is systematically discernible from the skin of the people we see grafikami rivnyan.

1. Lіnіine rіvnyannya with two wintry

x, y - in the first step; a, b, c - specific numbers.

butt:

The graph of the rivnyannya is straight line.

In this vipadku

Know the graph of rivnyannya, we can tell a lot about the decision of the vicious rivnyannya, but itself: yakscho if

The function of growth is high, so that it grows x to grow y. We have rendered two private decisions, but how can we write down all the decisions?

If the point is abscissa x, then the ordinate of the point

Mean numbers

We had a lot of ryvnyannya, we had a graph, we knew the decision. Without all the couples - skilki їkh? Power is strong.

Tse ratsіonalne іvnyannya,

We know y, with strong perversions obsessed

It is satisfactory and irreconcilable to a quadratic function;

Butt: Get a graph of a rational rivnyannya.

The graph is a parabola, the heads are straight up the hill.

Know the root of the rivnyannya:

Schematically depictable graph ( Small. 2).

Rivnyannya has no problem, no couples, no less happy couples, happy rivnyannya, but everything And how can you be x? Be yakim!

If it is possible to put x, then we can make a point

Rishennyam vikhіdnogo ryvnyannya є without couples

3. Stay with the graph of rivnyannya

Need to see y. Two options are available.

By the graph of the function є the hyperbola, the function is not assigned when

The function is declining.

If we look at the point from the abscissa, then the ordinate will be

Rishennyam vikhіdnogo ryvnyannya є without couples

A hyperbola can be induced to destroy a number of coordinate axes.

For example, graph of functions - also hyperbole - there will be a change to one uphill along the ordinate axis.

4. Rivnyannya stake

Tse ratsіonalne ryvnyannya with two wintry. Bezlichchyu solution є points of a circle. Center at point radius road R (Fig. 4).

Concrete butt is clearly visible.

Pointing to the standard square of the circle, for which you can see the square sumi:

- we got the rіvnyannya cola with the center in .

I'll stay with the graph of rivnyannya (Fig. 5).

b. Stay in the grafik rivnyannya

Sgadaimo, if you want to go to zero, todi and only todi, if one of the multipliers goes to zero, and the other one is.

The graph of a given rivnyannya warehouse is based on the number of graphs of the first and other rivnians, so two straight.

Let's stay yogo (Fig. 6).

Similarly, there will be a graph of another rivnyannya. Pass straight through the point (0; 1), ale of growth, so the kutoviy kofіtsієnt is positive.

The coordinates of all points are two prompted straight lines and solutions.

1. Mordkovich A.G. that in. Algebra 9th grade: Textbook. For zalnoosvit. Institutions. - 4th ed. - M .: Mnemosyna, 2002.-192 p.: Il.

2. Mordkovich A.G. that in. Algebra 9th grade: Problem book for scholars in the field of education / A. G. Mordkovich, T. N. Mishustina і ін. - 4th ed. - M .: Mnemosina, 2002.-143 p.: Il.

3. Makarich Yu. M. Algebra. 9th class: navch. for scientists zalnoosvit. ustanovka / Yu.M. Makarichєv, N.G. Mindyuk, K.I. Neshkov, I. E. Feoktistov. - 7th ed., Rev. і add. - M .: Mnemosina, 2008.

4. Alimov Sh.A., Kolyagin Yu.M., Sidorov Yu.V. Algebra. Class 9 16th ed. - M., 2011 .-- 287 p.

5. Mordkovich A. G. Algebra. Class 9 At 2 o'clock, Part 1. Pidruchnik for scholars of the educational establishment / A. G. Mordkovich, P. V. Semenov. - 12th ed., Ster. - M .: 2010 .-- 224 p.: Il.

1. Rozdil College.ru from mathematics ().

2. Internet project "Zavdannya" ().

3. Osvіtnіy portal "VIRISHU ЄDI" ().

The straight-sided coordinate system is a pair of perpendicular coordinate lines, called the coordinate axes, which are displaced so that the stench changes into the cob.

The designation of the coordinate axes by the letters x and y є is readily accepted, protectors can be-yaki. As soon as it is victorious it is lithi x and y, then the area is called xy-area... In older children, there may be stasis in the form of letters x and y letters, and it is shown in the lower little ones, є uv-areaі ts-area.

ordered pair

From an ordered pair of valid numbers, there are two valid numbers in a singing order. The skin point P in the coordinate area can be tied with a unique, ordered pair of real numbers by drawing two straight lines through the point P: one perpendicular to the X axis, and іnshu - perpendicular to the y axis.

For example, if we are (a, b) = (4,3), then on the coordinate

When you reach a point P (a, b), it means to find a point with coordinates (a, b) on the coordinate area. For example, the points are prompted to the little thing below.

In rectangular coordinate systems, the axes of coordinates extend the area into several areas, called quadrants. The stench is numbered against the year of the year in Roman numerals, yak is shown on the little one

viznachennya graph

grafik rivnyannya with two different x and y, are called without points on the xy-area, the coordinates of which є members are without a decision

Butt: namaluvati graph y = x 2

Through those that do not have a value of 1 / x, if x = 0, we can only have points for which x ≠ 0

Butt: Know all the crosswalks with the axles
(A) 3x + 2y = 6
(B) x = y 2 -2y
(C) y = 1 / x

Drive y = 0, todі 3x = 6 or x = 2

є shukanoy point to cross the x axis.

Having inserted, uh0 x = 0, we know that by a point the cross-point of the y axis є is the point y = 3.

With such its rank, you can see the rivnyannya (b), and the solution for (c) is pointed below

x-peretin

Drive y = 0

1 / x = 0 => x cannot be assigned, so that it is

Come on x = 0

y = 1/0 => y also does not have a value, => does not change from the point of y

At the bottom of the little one, the points (x, y), (-x, y), (x, -y) і (-x, -y) denote the kuti of the rectum.

The graph is symmetric on the axis x, for the skin point (x, y) the graph, the point (x, -y) is also a point on the graph.

The graph is symmetrical on the y-axis, but for the skin point of the graph (x, y) the point (-x, y) can also be plotted.

The graph is symmetric with respect to the center of coordinates, as for the skin point (x, y) the graph, the point (-x, -y) can also be shown in the graph.

value:

Graph functions on the coordinate area, the yak graph is starting y = f (x)

Induce graph f (x) = x + 2

Application 2. Get the graph f (x) = | x |

The graph is displayed from the line y = x for x > 0 і s lіnієу y = -x

for x< 0 .

graph of f (x) = -x

Two graphs together, we will be able to

graph f (x) = | x |

Butt 3. Stay with the graph

t (x) = (x 2 - 4) / (x - 2) =

= ((X - 2) (x + 2) / (x - 2)) =

= (X + 2) x ≠ 2

Otzhe, the whole function can be recorded at the viglyad

y = x + 2 x ≠ 2

Graph h (x) = x 2 - 4 Or x - 2

graph y = x + 2 x ≠ 2

Appendix 4. Stay with the graph

Graphs of functions with changes

Suppose that the graph of the function f (x) views

Todi we can know the graphics

y = f (x) + c - graph of the function f (x), displacements

VGORU on c value

y = f (x) - c - graph of function f (x), displacements

DOWN by c value

y = f (x + c) - graph of the function f (x), displacements

Left to c value

y = f (x - c) - graph of the function f (x), displacements

Right c value

Appendix 5. Stay tuned

graph y = f (x) = | x - 3 | + 2

Change graph y = | x | 3 digits RIGHT, schob trim the graph

Change graph y = | x - 3 | by 2 values VGORU, schob otrimati graphіk y = | x - 3 | + 2

encourage graph

y = x 2 - 4x + 5

Rivnyannya was re-set by the offensive rank, adding to both parts 4:

y + 4 = (x 2 - 4x + 5) + 4 y = (x 2 - 4x + 4) + 5 - 4

y = (x - 2) 2 + 1

Here, bachimo, the graph can be changed to the shift of the graph y = x 2 to the right by 2 values, so x - 2, and up by 1 value, so +1.

y = x 2 - 4x + 5

visualization

(-X, y) є in the images (x, y) along the y axis

(X, -y) є in the images (x, y) along the axis x

Graphs y = f (x) і y = f (-x) є in the images of one of the same axis y

Graphs y = f (x) і y = -f (x) є in the images of one of the same axis x

The graph can be rejected for images and changes:

paint the graph

We know the image of the y axis, and the graphic

Changeable graph to the right for 2 values і і

Axis shukaniy graphіk

If f (x) is multiplied by a positive constant c, then

graph f (x) squeeze vertically, like 0< c < 1

graph f (x) extend vertically, if c> 1

A curve is not a graph y = f (x) for a function f

Hey, given Rivnyannya with two winter F (x; y)... You have also learned about the ways in which you can relate such priests in an analytical way. The lack of solution of such people can be presented in a graphical view.

The graph of the equal F (x; y) is called the pointless points of the coordinate area xOy, the coordinates of which are satisfactory.

To induce the graph of the family with two winters, there is a chance to play in the rural life through the change x.

Singingly, even in the same place, there will be different graphs of the equal with two different ones: ax + b = c - straight line, yx = k - hyperbole, (x - a) 2 + (y - b) 2 = R 2 - circumference, radius of the road R, and the center is located at point O (a; b).

Butt 1.

Pobuduvati graph іvnyannya x 2 - 9y 2 = 0.

Decision.

Can be spread on multipliers for a part of a rivnyannya.

(X - 3y) (x + 3y) = 0, so y = x / 3 or y = -x / 3.

View: baby 1.

Especially the place of the loan is the establishment of the figures on the square of the rivnyannyi, where the sign absolute value, On which we will write in detail. A clear step to induce the graphs of rivnyan to mind | y | = F (x) i | y | = | F (x) |.

Perse rivnyannya equivalent to systems

(F (x) ≥ 0,
(Y = f (x) or y = -f (x).

So this graph is stored in two graphs: y = f (x) і y = -f (x), de f (x) ≥ 0.

To induce the graph of another equal, there will be graphs of two functions: y = f (x) і y = -f (x).

Butt 2.

Pobuduvati graphіk рівняння | y | = 2 + x.

Decision.

The set is equal to the system

(X + 2 ≥ 0,
(Y = x + 2 or y = -x - 2.

There will be no points.

View: baby 2.

Butt 3.

Pobuduvati graphіk рівняння | y - x | = 1.

Decision.

If y ≥ x, then y = x + 1, if y ≤ x, then y = x - 1.

View: baby 3.

When prompted by the graphs of the ryvnyan, to revenge the change under the sign of the module, manually and rationally area method, Appointments on the rosbit of the coordinate area on the part, in which the skin of the modular viraz has its own sign.

Butt 4.

Pobuduvati grafik rivnyannya x + | x | + Y + | y | = 2.

Decision.

In this attachment, the sign of skin pidmodular bends is located as a coordinate screw.

1) For the first coordinate space x ≥ 0 and y ≥ 0. To open the module, the following parameters are specified:

2x + 2y = 2, and if forgiven x + y = 1.

2) The other four, de x< 0, а y ≥ 0, уравнение будет иметь вид: 0 + 2y = 2 или y = 1.

3) In the third four x< 0, y < 0 будем иметь: x – x + y – y = 2. Перепишем этот результат в виде уравнения 0 · x + 0 · y = 2.

4) In the fourth four, for x ≥ 0, and y< 0 получим, что x = 1.

The graph of this ryvnyannya will be buuvati by heart.

View: baby 4.

Butt 5.

Draw free points, at which coordinates are satisfactory | x - 1 | + | Y - 1 | = 1.

Decision.

Zero submodular virases x = 1 and y = 1 break the coordinate area into some areas. Development of modules by regions. It can be issued at the table view.

region	Sign of submodular bends	Otrimane rivnyannya pislya module opening
I	x ≥ 1 і y ≥ 1	x + y = 3
II	x< 1 и y ≥ 1	-x + y = 1
III	x< 1 и y < 1	x + y = 1
IV	x ≥ 1 і y< 1	x - y = 1

View: baby 5.

On the coordinate area of the figuri you can set irregularities.

graph of irregularities Two of them are called the powerless of all points of the coordinate area, the coordinates of which are the solutions of the chain of irregularities.

Clear Algorithm to induce models to solve irregularities with two changes:

Write down your rivnyannya, in case of irregularities.
Pobuduvati graph іvnyannya z point 1.
Vibrate to a point in one area. Revision, how satisfied are the coordinates of the reverse point of this irregularity.
To depict graphically without the need for all the solutions of the nerves.

Clearly, first for everything, the irregularity ax + bx + c> 0. Equal ax + bx + c = 0 is set straight, breaking the area into two squares. They have the function f (x) = ax + bx + c zberіgaє sign. To make a mark, finish taking a point, how to establish a point, and enumerate the meaning of a function in a point. As a sign of function is to be a sign of irregularity, then it will be a sign of irregularity.

Put it on clearly graphic solution the most often misbehavior is caused by two wintry ones.

1) ax + bx + c ≥ 0. baby 6.

2) | X | ≤ a, a> 0. babies 7.

3) x 2 + y 2 ≤ a, a> 0. babies 8.

4) y ≥ x 2. Malunok 9.

5) xy ≤ 1. Malunok 10.

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