Abscissa coordinate system. What is an ordinate? There are also problems to determine the length of a segment

The word "ordinate" comes from the Latin "ordinatus" - "arranged in order." Ordinate is a purely mathematical term used to denote the coordinate of a point in a rectangular coordinate system.

Let's look a little more closely at what an ordinate is.

Abscissa, ordinate and applicate

In a rectangular two-dimensional coordinate system, the abscissa and ordinate are used to accurately determine the coordinates of a particular point or segment. The abscissa is the coordinate of the point along the OX axis, the ordinate is the coordinate along the OY axis. To determine the abscissa and ordinate value of the point of interest in a rectangular coordinate system, it is necessary to draw perpendiculars from this point to the OX and OY axes, respectively. The values ​​on the axes and will be the abscissa and ordinate values ​​of the point.

If the point is located in a three-dimensional coordinate system, the concept of “applicate” is also added - this is the value of the point along the OZ axis.

How to mark a point and plot a graph using abscissa and ordinate

Just as, having a point in a rectangular coordinate system, you can find its abscissa and ordinate, and knowing the values ​​of the abscissa and ordinate, you can mark a point in the coordinate system. The coordinates of a point are usually indicated in the following format - A (2; 5), with the abscissa value indicated first, that is, the value of the point along the OX axis, and then the ordinate value - the value along the OY axis.

An abscissa and ordinate can define a point, a pair of abscissas and ordinates can define a straight segment, and to construct, for example, a parabola, you will need to know three abscissas and ordinates.

To construct a particular graph, the dependence of the ordinate values ​​on the abscissa is used. For example: y = 2x + 8. To build a graph, you need to go through different x values ​​and mark the corresponding y values ​​on the coordinate system.

CHAPTER VIII

COORDINATES AND SIMPLE GRAPHICS

§ 41. Coordinate axes. Abscissa and ordinate of a point on a plane.

1258. Construct a rectangular coordinate system and mark points having the following coordinates:

1) X = 5, at = 3; 2) X = - 4, at = 6;

3) X = - 3, at =- 4; 4) X = 5, y = -2.

1259. Construct points with the following coordinates:

1) X = 8 1 / 2 , at = - 5 1 / 2 2) X = - 6,5, at = 4,5;

3) X = -2,8, at =-3,2; 4) X = 7,3, at =8,4;

5) A (-3 3 / 4; 5 1 / 2); "6) V (-0.8; - l.4). ,

1260. 1) Using these coordinates, construct points and indicate under what conditions the points are located on the axis X -ov or on the axis Y -s.

1) X = 4, at = 0;

2) X =- 2, at = 0\

3) X = 0, at = 3;

4) X = 0, at =-4;

5) X = 0, at = 0.

2) Determine and record the coordinates of each point indicated on drawing 35.

1261. Construct a straight line connecting two points with coordinates:

1) A(5; 4) and B (-3;-2); 2) C (-4; 2) and D (5; - 3).

1262. 1) Construct a triangle using the coordinates of its vertices A, B and C:

A (4; 5); B (8; 2); C (- 6; 3).

2) Construct a quadrilateral according to the coordinates of its vertices A, B, C and D:

A (- 3; 8); B (10; 6); C (5; -5); D (-7; -4).

1263. 1) Given point A (4; 6). Construct point B symmetrical to point A relative to the x-axis OH , and find the coordinates of this point.

2) Construct several more points located symmetrically relative to the x-axis.

3) Show that if points A and B are symmetrical about the abscissa axis, then their abscissas are equal, and their ordinates differ only in signs.

1264. 1) Construct point A(4; 6) and point B, symmetrical to point A relative to the ordinate axis. What is the difference between the abscissa and ordinate of these points?

2) Construct several pairs of points symmetrical about the ordinate axis OY , find their coordinates and show that if points A and B are symmetrical about the ordinate axis, then their ordinates are equal, and the abscissas differ only in signs.

1265. 1) Construct point A (3; 7) and point B, symmetrical to point A relative to the origin. What is the difference between the abscissa and ordinate of these points?

2) Construct several pairs of points that are symmetrical with respect to the origin of coordinates and show that the coordinates of each pair of such points differ only in sign.

1266. The points on the plane are:

A(1; 3); B(2; 5); C(1; -3); D(-2; -5); E(-1; 3).

Determine which pairs of these points are symmetrical with respect to: 1) the abscissa axis; 2) ordinate axes; 3) the origin of coordinates.

1267. 1) Construct a quadrilateral using the following coordinates of its vertices: "

A(0; 0); B(1; 3); C (8; 5); D(9; 1).

Note. Take 1 cm as a scale unit.

2) From vertex A, draw the diagonal of the quadrilateral and, by directly measuring the base and heights of the resulting triangles (with an accuracy of 0.1 cm), calculate their area and the area of ​​the entire quadrilateral.

3) Draw from the vertex to the second diagonal and again find the area of ​​the quadrilateral by performing the appropriate measurements and calculations.

4) Calculate the arithmetic mean of the two results obtained and round the answer to two significant figures.

5) Find the absolute and relative errors of the resulting answer, knowing that the area of ​​this quadrilateral is 28 cm 2 .

1268. The results of air temperature measurements during the day are recorded in the following table:

1) Using the table data, construct a graph of changes in air temperature during the day.

2) Determine the air temperature according to the schedule: at 3 o’clock; at 9 o'clock; at 13 o'clock; at 21 o'clock

3) Find from the graph at what time the air temperature was equal to: -1°; -4°; + 2°; +5°.

4) Establish according to the graph in what period of time the temperature rose and fell.

5) Find from the graph when during the day the temperature was the highest and lowest.

1269. When a body is in free fall, the speed at any time is determined by the formula v = gt , Where v - speed in meters per second, g ≈ 9.81 m/sec 2 , t - time in seconds.

Draw a graph of changes in the speed of a falling body depending on the time of fall.

1270. From observations of changes in water temperature with increasing depth in the equatorial Pacific Ocean, the following data were obtained:

1) Draw a graph of changes in water temperature with changes in depth.

2) Determine at what depth the water temperature decreases most quickly? slowest?

1271. When heating began, the water in the boiler had a temperature of 8°. When heated, the water temperature increased by 2° every minute.

1).Write a formula expressing the change in temperature of water depending on time t heating it.

2) Make a table of values at for a time from 1 minute to 10 minutes.

3) Plot a graph of changes in water temperature depending on changes in heating time.i

4) Find from the graph with an accuracy of 1: the temperature of the water 14 minutes after heating; How many minutes after heating starts will the water temperature reach 20°? 35°? Check by calculating using the formula.

If you are at some zero point and are wondering how many units of distance you need to go straight ahead and then straight to the right to get to some other point, then you are already using a rectangular Cartesian coordinate system on the plane. And if the point is located above the plane on which you stand, and to your calculations you add an ascent to the point along the stairs strictly upward also by a certain number of distance units, then you are already using a rectangular Cartesian coordinate system in space.

An ordered system of two or three intersecting axes perpendicular to each other with a common origin (origin of coordinates) and a common unit of length is called rectangular Cartesian coordinate system .

The name of the French mathematician René Descartes (1596-1662) is associated primarily with a coordinate system in which a common unit of length is measured on all axes and the axes are straight. In addition to the rectangular one, there is general Cartesian coordinate system (affine coordinate system). It may also include axes that are not necessarily perpendicular. If the axes are perpendicular, then the coordinate system is rectangular.

Rectangular Cartesian coordinate system on a plane has two axes and rectangular Cartesian coordinate system in space - three axes. Each point on a plane or in space is defined by an ordered set of coordinates - numbers corresponding to the unit of length of the coordinate system.

Note that, as follows from the definition, there is a Cartesian coordinate system on a straight line, that is, in one dimension. The introduction of Cartesian coordinates on a line is one of the ways by which any point on a line is associated with a well-defined real number, that is, a coordinate.

The coordinate method, which arose in the works of Rene Descartes, marked a revolutionary restructuring of all mathematics. It became possible to interpret algebraic equations (or inequalities) in the form of geometric images (graphs) and, conversely, to look for solutions to geometric problems using analytical formulas and systems of equations. Yes, inequality z < 3 геометрически означает полупространство, лежащее ниже плоскости, параллельной координатной плоскости xOy and located above this plane by 3 units.

Using the Cartesian coordinate system, the membership of a point on a given curve corresponds to the fact that the numbers x And y satisfy some equation. Thus, the coordinates of a point on a circle with a center at a given point ( a; b) satisfy the equation (x - a)² + ( y - b)² = R² .

Rectangular Cartesian coordinate system on a plane

Two perpendicular axes on a plane with a common origin and the same scale unit form Cartesian rectangular coordinate system on the plane . One of these axes is called the axis Ox, or x-axis , the other - the axis Oy, or y-axis . These axes are also called coordinate axes. Let us denote by Mx And My respectively, the projection of an arbitrary point M on the axis Ox And Oy. How to get projections? Let's go through the point M Ox. This straight line intersects the axis Ox at the point Mx. Let's go through the point M straight line perpendicular to the axis Oy. This straight line intersects the axis Oy at the point My. This is shown in the picture below.

x And y points M we will call the values ​​of the directed segments accordingly OMx And OMy. The values ​​of these directed segments are calculated accordingly as x = x0 - 0 And y = y0 - 0 . Cartesian coordinates x And y points M abscissa And ordinate . The fact that the point M has coordinates x And y, is denoted as follows: M(x, y) .

Coordinate axes divide the plane into four quadrant , the numbering of which is shown in the figure below. It also shows the arrangement of signs for the coordinates of points depending on their location in a particular quadrant.

In addition to Cartesian rectangular coordinates on a plane, the polar coordinate system is also often considered. About the method of transition from one coordinate system to another - in the lesson polar coordinate system .

Rectangular Cartesian coordinate system in space

Cartesian coordinates in space are introduced in complete analogy with Cartesian coordinates in the plane.

Three mutually perpendicular axes in space (coordinate axes) with a common origin O and with the same scale unit they form Cartesian rectangular coordinate system in space .

One of these axes is called an axis Ox, or x-axis , the other - the axis Oy, or y-axis , the third - axis Oz, or axis applicate . Let Mx, My Mz- projections of an arbitrary point M space on the axis Ox , Oy And Oz respectively.

Let's go through the point M OxOx at the point Mx. Let's go through the point M plane perpendicular to the axis Oy. This plane intersects the axis Oy at the point My. Let's go through the point M plane perpendicular to the axis Oz. This plane intersects the axis Oz at the point Mz.

Cartesian rectangular coordinates x , y And z points M we will call the values ​​of the directed segments accordingly OMx, OMy And OMz. The values ​​of these directed segments are calculated accordingly as x = x0 - 0 , y = y0 - 0 And z = z0 - 0 .

Cartesian coordinates x , y And z points M are called accordingly abscissa , ordinate And applicate .

Coordinate axes taken in pairs are located in coordinate planes xOy , yOz And zOx .

Problems about points in a Cartesian coordinate system

Example 1.

A(2; -3) ;

B(3; -1) ;

C(-5; 1) .

Find the coordinates of the projections of these points onto the abscissa axis.

Solution. As follows from the theoretical part of this lesson, the projection of a point onto the abscissa axis is located on the abscissa axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and an ordinate (coordinate on the axis Oy, which the x-axis intersects at point 0), which is equal to zero. So we get the following coordinates of these points on the x-axis:

Ax(2;0);

Bx(3;0);

Cx (-5; 0).

Example 2. In the Cartesian coordinate system, points are given on the plane

A(-3; 2) ;

B(-5; 1) ;

C(3; -2) .

Find the coordinates of the projections of these points onto the ordinate axis.

Solution. As follows from the theoretical part of this lesson, the projection of a point onto the ordinate axis is located on the ordinate axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and an abscissa (coordinate on the axis Ox, which the ordinate axis intersects at point 0), which is equal to zero. So we get the following coordinates of these points on the ordinate axis:

Ay(0;2);

By(0;1);

Cy(0;-2).

Example 3. In the Cartesian coordinate system, points are given on the plane

A(2; 3) ;

B(-3; 2) ;

C(-1; -1) .

Ox .

Ox Ox Ox, will have the same abscissa as the given point, and an ordinate equal in absolute value to the ordinate of the given point, and opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the axis Ox :

A"(2; -3) ;

B"(-3; -2) ;

C"(-1; 1) .

Solve problems using the Cartesian coordinate system yourself, and then look at the solutions

Example 4. Determine in which quadrants (quarters, drawing with quadrants - at the end of the paragraph “Rectangular Cartesian coordinate system on a plane”) a point can be located M(x; y) , If

1) xy > 0 ;

2) xy < 0 ;

3) x − y = 0 ;

4) x + y = 0 ;

5) x + y > 0 ;

6) x + y < 0 ;

7) x − y > 0 ;

8) x − y < 0 .

Example 5. In the Cartesian coordinate system, points are given on the plane

A(-2; 5) ;

B(3; -5) ;

C(a; b) .

Find the coordinates of points symmetrical to these points relative to the axis Oy .

Let's continue to solve problems together

Example 6. In the Cartesian coordinate system, points are given on the plane

A(-1; 2) ;

B(3; -1) ;

C(-2; -2) .

Find the coordinates of points symmetrical to these points relative to the axis Oy .

Solution. Rotate 180 degrees around the axis Oy directional segment from the axis Oy up to this point. In the figure, where the quadrants of the plane are indicated, we see that the point symmetrical to the given one relative to the axis Oy, will have the same ordinate as the given point, and an abscissa equal in absolute value to the abscissa of the given point and opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the axis Oy :

A"(1; 2) ;

B"(-3; -1) ;

C"(2; -2) .

Example 7. In the Cartesian coordinate system, points are given on the plane

A(3; 3) ;

B(2; -4) ;

C(-2; 1) .

Find the coordinates of points symmetrical to these points relative to the origin.

Solution. We rotate the directed segment going from the origin to the given point by 180 degrees around the origin. In the figure, where the quadrants of the plane are indicated, we see that a point symmetrical to the given point relative to the origin of coordinates will have an abscissa and ordinate equal in absolute value to the abscissa and ordinate of the given point, but opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the origin:

A"(-3; -3) ;

B"(-2; 4) ;

C(2; -1) .

Example 8.

A(4; 3; 5) ;

B(-3; 2; 1) ;

C(2; -3; 0) .

Find the coordinates of the projections of these points:

1) on a plane Oxy ;

2) on a plane Oxz ;

3) to the plane Oyz ;

4) on the abscissa axis;

5) on the ordinate axis;

6) on the applicate axis.

1) Projection of a point onto a plane Oxy is located on this plane itself, and therefore has an abscissa and ordinate equal to the abscissa and ordinate of a given point, and an applicate equal to zero. So we get the following coordinates of the projections of these points onto Oxy :

Axy (4; 3; 0);

Bxy (-3; 2; 0);

Cxy(2;-3;0).

2) Projection of a point onto a plane Oxz is located on this plane itself, and therefore has an abscissa and applicate equal to the abscissa and applicate of a given point, and an ordinate equal to zero. So we get the following coordinates of the projections of these points onto Oxz :

Axz (4; 0; 5);

Bxz (-3; 0; 1);

Cxz (2; 0; 0).

3) Projection of a point onto a plane Oyz is located on this plane itself, and therefore has an ordinate and applicate equal to the ordinate and applicate of a given point, and an abscissa equal to zero. So we get the following coordinates of the projections of these points onto Oyz :

Ayz(0; 3; 5);

Byz (0; 2; 1);

Cyz (0; -3; 0).

4) As follows from the theoretical part of this lesson, the projection of a point onto the abscissa axis is located on the abscissa axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and the ordinate and applicate of the projection are equal to zero (since the ordinate and applicate axes intersect the abscissa at point 0). We obtain the following coordinates of the projections of these points onto the abscissa axis:

Ax(4;0;0);

Bx (-3; 0; 0);

Cx(2;0;0).

5) The projection of a point onto the ordinate axis is located on the ordinate axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and the abscissa and applicate of the projection are equal to zero (since the abscissa and applicate axes intersect the ordinate axis at point 0). We obtain the following coordinates of the projections of these points onto the ordinate axis:

Ay(0; 3; 0);

By (0; 2; 0);

Cy(0;-3;0).

6) The projection of a point onto the applicate axis is located on the applicate axis itself, that is, the axis Oz, and therefore has an applicate equal to the applicate of the point itself, and the abscissa and ordinate of the projection are equal to zero (since the abscissa and ordinate axes intersect the applicate axis at point 0). We obtain the following coordinates of the projections of these points onto the applicate axis:

Az (0; 0; 5);

Bz (0; 0; 1);

Cz(0; 0; 0).

Example 9. In the Cartesian coordinate system, points are given in space

A(2; 3; 1) ;

B(5; -3; 2) ;

C(-3; 2; -1) .

Find the coordinates of points symmetrical to these points with respect to:

1) plane Oxy ;

2) planes Oxz ;

3) planes Oyz ;

4) abscissa axes;

5) ordinate axes;

6) applicate axes;

7) origin of coordinates.

1) “Move” the point on the other side of the axis Oxy Oxy, will have an abscissa and ordinate equal to the abscissa and ordinate of a given point, and an applicate equal in magnitude to the aplicate of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oxy :

A"(2; 3; -1) ;

B"(5; -3; -2) ;

C"(-3; 2; 1) .

2) “Move” the point on the other side of the axis Oxz to the same distance. From the figure displaying the coordinate space, we see that a point symmetrical to a given one relative to the axis Oxz, will have an abscissa and applicate equal to the abscissa and applicate of a given point, and an ordinate equal in magnitude to the ordinate of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oxz :

A"(2; -3; 1) ;

B"(5; 3; 2) ;

C"(-3; -2; -1) .

3) “Move” the point on the other side of the axis Oyz to the same distance. From the figure displaying the coordinate space, we see that a point symmetrical to a given one relative to the axis Oyz, will have an ordinate and an aplicate equal to the ordinate and an aplicate of a given point, and an abscissa equal in value to the abscissa of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oyz :

A"(-2; 3; 1) ;

B"(-5; -3; 2) ;

C"(3; 2; -1) .

By analogy with symmetrical points on a plane and points in space that are symmetrical to data relative to planes, we note that in the case of symmetry with respect to some axis of the Cartesian coordinate system in space, the coordinate on the axis with respect to which the symmetry is given will retain its sign, and the coordinates on the other two axes will be the same in absolute value as the coordinates of a given point, but opposite in sign.

4) The abscissa will retain its sign, but the ordinate and applicate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the abscissa axis:

A"(2; -3; -1) ;

B"(5; 3; -2) ;

C"(-3; -2; 1) .

5) The ordinate will retain its sign, but the abscissa and applicate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the ordinate axis:

A"(-2; 3; -1) ;

B"(-5; -3; -2) ;

C"(3; 2; 1) .

6) The applicate will retain its sign, but the abscissa and ordinate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the applicate axis:

A"(-2; -3; 1) ;

B"(-5; 3; 2) ;

C"(3; -2; -1) .

7) By analogy with symmetry in the case of points on a plane, in the case of symmetry about the origin of coordinates, all coordinates of a point symmetrical to a given one will be equal in absolute value to the coordinates of a given point, but opposite to them in sign. So, we obtain the following coordinates of points symmetrical to the data relative to the origin.

This point on the axis X'X in a rectangular coordinate system. Abscissa value of a point A equal to the length of the segment O.B.(see picture). If the point B belongs to the positive semi-axis OX, then the abscissa has a positive value. If the point B belongs to the negative semi-axis X'O, then the abscissa has a negative value. If the point A lies on the axis Y'Y, then its abscissa is zero.

In a rectangular coordinate system, a ray (straight line) X'X called the "abscissa axis". When plotting functions, the x-axis is usually used as the domain of definition of the function.

Etymology

see also

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Notes

Links

• Abscissa // Great Soviet Encyclopedia: [in 30 volumes] / ch. ed. A. M. Prokhorov. - 3rd ed. - M. : Soviet encyclopedia, 1969-1978.

Excerpt characterizing Abscissa

“However, I’m embarrassing you,” he told him quietly, “let’s go, talk about business, and I’ll leave.”
“No, not at all,” said Boris. And if you are tired, let’s go to my room and lie down and rest.
- Indeed...
They entered the small room where Boris was sleeping. Rostov, without sitting down, immediately with irritation - as if Boris was guilty of something in front of him - began to tell him Denisov’s case, asking if he wanted and could ask about Denisov through his general from the sovereign and through him deliver a letter. When they were left alone, Rostov became convinced for the first time that he was embarrassed to look Boris in the eyes. Boris, crossing his legs and stroking the thin fingers of his right hand with his left hand, listened to Rostov, as a general listens to the report of a subordinate, now looking to the side, now with the same clouded gaze, looking directly into Rostov’s eyes. Each time Rostov felt awkward and lowered his eyes.
“I have heard about this kind of thing and I know that the Emperor is very strict in these cases. I think we should not bring it to His Majesty. In my opinion, it would be better to directly ask the corps commander... But in general I think...
- So you don’t want to do anything, just say so! - Rostov almost shouted, without looking into Boris’s eyes.
Boris smiled: “On the contrary, I’ll do what I can, but I thought...
At this time, Zhilinsky’s voice was heard at the door, calling Boris.
“Well, go, go, go...” said Rostov, refusing dinner, and being left alone in a small room, he walked back and forth in it for a long time, and listened to the cheerful French conversation from the next room.

Ordinate

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Synonyms:

See what “Ordinate” is in other dictionaries:

Ordinate- When data is graphed, the ordinate corresponds to the information contained on the vertical axis, or y-axis. In experimental studies, the values ​​of the dependent variable are placed on this axis. Psychology. A I. Dictionary... ... Great psychological encyclopedia

- (from the Latin ordinatus located in order) one of the Cartesian coordinates of a point, usually the second, denoted by the letter y ... Big Encyclopedic Dictionary

ORDINATE, ordinates, female. (lat. ordinata located at equal distances) (mat.). In the coordinate system of analytical geometry, a perpendicular on a plane is lowered from a point to the abscissa axis. Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 … Ushakov's Explanatory Dictionary

Exist., number of synonyms: 1 coordinate (4) Dictionary of synonyms ASIS. V.N. Trishin. 2013… Synonym dictionary

ordinate- The difference in longitude of the beginning and end of the profile, measured at a given latitude Topics oil and gas industry EN ordinatedeparture ... Technical Translator's Guide

ordinate- In cartography, a coordinate measured in a direction perpendicular to the axial meridian... Dictionary of Geography

ORDINATE- one of two (three) numbers that determine the position of a point on a plane (in space) relative to a rectangular coordinate system... Big Polytechnic Encyclopedia

- (lat. ordinatus ordered, arranged in a certain order) eom. one of two (three) numbers that determine the position of a point on a plane (in space) relative to a rectangular coordinate system. New dictionary of foreign words. by EdwART… Dictionary of foreign words of the Russian language

Y; and. [from lat. ordinatus ordered, assigned] Mat. A quantity that determines the position of a certain point on a plane or in space along the Y axis in a rectangular coordinate system (cf. abscissa, ordinate). * * * ordinate (from Latin ordinatus ... ... encyclopedic Dictionary

ordinate- ordinatė statusas T sritis fizika atitikmenys: engl. ordinate vok. Ordinate, f rus. ordinate, f pranc. ordonnée, f … Fizikos terminų žodynas