Coordinate plane: what is it? How to mark points and construct figures on a coordinate plane? Coordinate plane
Understanding the Coordinate Plane
Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.
Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.
To construct a coordinate plane, you need to draw $2$ perpendicular straight lines, at the end of which the directions “to the right” and “up” are indicated using arrows. Divisions are applied to the lines, and the point of intersection of the lines is the zero mark for both scales.
Definition 1
The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.
Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.
Coordinate plane
Point coordinates
A point on a coordinate plane is defined by two coordinates.
To determine the coordinates of point $A$ on the coordinate plane, you need to draw straight lines through it that will be parallel to the coordinate axes (indicated by a dotted line in the figure). The intersection of the line with the x-axis gives the $x$ coordinate of point $A$, and the intersection with the y-axis gives the y-coordinate of point $A$. When writing the coordinates of a point, the $x$ coordinate is first written, and then the $y$ coordinate.
Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.
To plot a point on the coordinate plane, act in reverse order.
Constructing a point at specified coordinates
Example 1
On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$
Solution.
Construction of point $A$:
- put the number $2$ on the $x$ axis and draw a perpendicular line;
- On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.
Construction of point $B$:
- Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
- On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.
Example 2
Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.
Solution.
Construction of point $C$:
- put the number $3$ on the $x$ axis;
- coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.
Construction of point $D$:
- put the number $2$ on the $y$ axis;
- coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.
Note 1
Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.
Example 3
Determine the coordinates of points A, B, C, D.$
Solution.
Let's determine the coordinates of point $A$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. Thus, we obtain that the point $A (1; 3).$
Let's determine the coordinates of point $B$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. We find that point $B (–2; 4).$
Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.
Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$
Example 4
Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$
Solution.
Construction of point $E$:
- put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
- on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
- at the intersection of perpendicular lines we obtain the point $E (–3; –2).$
Construction of point $F$:
- coordinate $y=0$, which means the point lies on the $x$ axis;
- Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$
Construction of point $G$:
- put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
- on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
- at the intersection of perpendicular lines we obtain the point $G(3; 4).$
Construction of point $H$:
- coordinate $x=0$, which means the point lies on the $y$ axis;
- Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$
Construction of point $O$:
- both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).
Understanding the Coordinate Plane
Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.
Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.
To construct a coordinate plane, you need to draw $2$ perpendicular straight lines, at the end of which the directions “to the right” and “up” are indicated using arrows. Divisions are applied to the lines, and the point of intersection of the lines is the zero mark for both scales.
Definition 1
The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.
Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.
Coordinate plane
Point coordinates
A point on a coordinate plane is defined by two coordinates.
To determine the coordinates of point $A$ on the coordinate plane, you need to draw straight lines through it that will be parallel to the coordinate axes (indicated by a dotted line in the figure). The intersection of the line with the x-axis gives the $x$ coordinate of point $A$, and the intersection with the y-axis gives the y-coordinate of point $A$. When writing the coordinates of a point, the $x$ coordinate is first written, and then the $y$ coordinate.
Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.
To plot a point on the coordinate plane, proceed in the reverse order.
Constructing a point at specified coordinates
Example 1
On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$
Solution.
Construction of point $A$:
- put the number $2$ on the $x$ axis and draw a perpendicular line;
- On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.
Construction of point $B$:
- Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
- On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.
Example 2
Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.
Solution.
Construction of point $C$:
- put the number $3$ on the $x$ axis;
- coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.
Construction of point $D$:
- put the number $2$ on the $y$ axis;
- coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.
Note 1
Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.
Example 3
Determine the coordinates of points A, B, C, D.$
Solution.
Let's determine the coordinates of point $A$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. Thus, we obtain that the point $A (1; 3).$
Let's determine the coordinates of point $B$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. We find that point $B (–2; 4).$
Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.
Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$
Example 4
Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$
Solution.
Construction of point $E$:
- put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
- on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
- at the intersection of perpendicular lines we obtain the point $E (–3; –2).$
Construction of point $F$:
- coordinate $y=0$, which means the point lies on the $x$ axis;
- Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$
Construction of point $G$:
- put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
- on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
- at the intersection of perpendicular lines we obtain the point $G(3; 4).$
Construction of point $H$:
- coordinate $x=0$, which means the point lies on the $y$ axis;
- Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$
Construction of point $O$:
- both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).
If we construct two mutually perpendicular numerical axes on a plane: OX And OY, then they will be called coordinate axes. Horizontal axis OX called x-axis(axis x), vertical axis OY - y-axis(axis y).
Dot O, standing at the intersection of the axes, is called origin. It is the zero point for both axes. Positive numbers are depicted on the abscissa axis by points to the right, and on the ordinate axis by points upward from the zero point. Negative numbers are depicted by points to the left and down from the origin of coordinates (points O). The plane on which the coordinate axes lie is called coordinate plane.
The coordinate axes divide the plane into four parts, called in quarters or quadrants. It is customary to number these quarters with Roman numerals in the order in which they are numbered on the drawing.
Coordinates of a point on the plane
If we take an arbitrary point on the coordinate plane A and draw perpendiculars from it to the coordinate axes, then the bases of the perpendiculars will fall on two numbers. The number to which the vertical perpendicular points is called abscissa point A. The number to which the horizontal perpendicular points is - ordinate of a point A.
On the drawing, the abscissa of the point A is equal to 3, and the ordinate is 5.
The abscissa and ordinate are called the coordinates of a given point on the plane.
The coordinates of a point are written in brackets to the right of the point designation. The abscissa is written first, followed by the ordinate. So record A(3; 5) means that the abscissa of the point A is equal to three, and the ordinate is five.
The coordinates of a point are numbers that determine its position on the plane.
If a point lies on the x-axis, then its ordinate is zero (for example, a point B with coordinates -2 and 0). If a point lies on the ordinate axis, then its abscissa is equal to zero (for example, a point C with coordinates 0 and -4).
Origin - point O- has both the abscissa and ordinate equal to zero: O (0; 0).
This system coordinates is called rectangular or Cartesian.
§ 1 Coordinate system: definition and method of construction
In this lesson we will get acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes”, and learn how to construct points on a plane using coordinates.
Let us take a coordinate line x with the origin point O, a positive direction and a unit segment.
Through the origin of coordinates, point O of the coordinate line x, we draw another coordinate line y, perpendicular to x, set the positive direction upward, the unit segment is the same. Thus, we have built a coordinate system.
Let's give a definition:
Two mutually perpendicular coordinate lines intersecting at a point, which is the origin of coordinates of each of them, form a coordinate system.
§ 2 Coordinate axis and coordinate plane
The straight lines that form a coordinate system are called coordinate axes, each of which has its own name: the coordinate line x is the abscissa axis, the coordinate line y is the ordinate axis.
The plane on which the coordinate system is selected is called the coordinate plane.
The described coordinate system is called rectangular. It is often called the Cartesian coordinate system in honor of the French philosopher and mathematician René Descartes.
Each point on the coordinate plane has two coordinates, which can be determined by dropping perpendiculars from the point on the coordinate axis. The coordinates of a point on a plane are a pair of numbers, of which the first number is the abscissa, the second number is the ordinate. The abscissa is perpendicular to the x-axis, the ordinate is perpendicular to the y-axis.
Let's mark point A on the coordinate plane and draw perpendiculars from it to the axes of the coordinate system.
Along the perpendicular to the abscissa axis (x-axis), we determine the abscissa of point A, it is equal to 4, the ordinate of point A - along the perpendicular to the ordinate axis (y-axis) is 3. The coordinates of our point are 4 and 3. A (4;3). Thus, coordinates can be found for any point on the coordinate plane.
§ 3 Construction of a point on a plane
How to construct a point on a plane with given coordinates, i.e. Using the coordinates of a point on the plane, determine its position? In this case, we perform the steps in reverse order. On the coordinate axes we find points corresponding to the given coordinates, through which we draw straight lines perpendicular to the x and y axes. The point of intersection of the perpendiculars will be the desired one, i.e. a point with given coordinates.
Let's complete the task: construct point M (2;-3) on the coordinate plane.
To do this, find a point with coordinate 2 on the x-axis and draw a straight line perpendicular to the x-axis through this point. On the ordinate axis we find a point with coordinate -3, through it we draw a straight line perpendicular to the y axis. The point of intersection of perpendicular lines will be the given point M.
Now let's look at a few special cases.
Let us mark points A (0; 2), B (0; -3), C (0; 4) on the coordinate plane.
The abscissas of these points are equal to 0. The figure shows that all points are on the ordinate axis.
Consequently, points whose abscissas are equal to zero lie on the ordinate axis.
Let's swap the coordinates of these points.
The result will be A (2;0), B (-3;0) C (4; 0). In this case, all ordinates are equal to 0 and the points are on the x-axis.
This means that points whose ordinates are equal to zero lie on the abscissa axis.
Let's look at two more cases.
On the coordinate plane, mark the points M (3; 2), N (3; -1), P (3; -4).
It is easy to see that all the abscissas of the points are the same. If these points are connected, you get a straight line parallel to the ordinate axis and perpendicular to the abscissa axis.
The conclusion suggests itself: points that have the same abscissa lie on the same straight line, which is parallel to the ordinate axis and perpendicular to the abscissa axis.
If you swap the coordinates of the points M, N, P, you get M (2; 3), N (-1; 3), P (-4; 3). The ordinates of the points will be the same. In this case, if you connect these points, you get a straight line parallel to the abscissa axis and perpendicular to the ordinate axis.
Thus, points having the same ordinate lie on the same straight line parallel to the abscissa axis and perpendicular to the ordinate axis.
In this lesson you became acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes - abscissa axis and ordinate axis”. We learned how to find the coordinates of a point on a coordinate plane and learned how to construct points on the plane using its coordinates.
List of used literature:
- Mathematics. Grade 6: lesson plans for I.I.’s textbook. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. – Mnemosyne, 2009.
- Mathematics. 6th grade: textbook for students of general education institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyna, 2013.
- Mathematics. 6th grade: textbook for general education institutions/G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others/edited by G.V. Dorofeeva, I.F. Sharygina; Russian Academy of Sciences, Russian Academy of Education. - M.: “Enlightenment”, 2010
- Handbook of mathematics - http://lyudmilanik.com.ua
- Handbook for students in secondary school http://shkolo.ru
On a plane. Let one be x, the other y. And let these lines be mutually perpendicular (that is, intersect at right angles). Moreover, the point of their intersection will be the origin of coordinates for both lines, and the unit segment is the same (Fig. 1).
So we got rectangular coordinate system, and our plane has become a coordinate plane. The lines x and y are called coordinate axes. Moreover, the x-axis is the abscissa axis, and the y-axis is the ordinate axis. Such a plane is usually designated by the name of the axes and the reference point - xOy. The rectangular coordinate system is also called Cartesian coordinate system, since the French mathematician and philosopher Rene Descartes first began to actively use it.
Right angles formed by lines x and y are called coordinate angles. Each corner has its own number as shown in Fig. 2.
So, when we talked about the coordinate line, every point on this line had one coordinate. Now, when we are talking about the coordinate plane, then each point of this plane will already have two coordinates. One corresponds to straight line x (this coordinate is called abscissa), the other corresponds to straight line y (this coordinate is called ordinate). It is written this way: M(x;y), where x is the abscissa and y is the ordinate. Read as: “Point M with coordinates x, y.”
How to determine the coordinates of a point on a plane?
Now we know that every point on the plane has two coordinates. In order to find out its coordinates, we just need to draw two straight lines through this point, perpendicular to the coordinate axes. The points of intersection of these lines with the coordinate axes will be the required coordinates. So, for example, in Fig. 3 we determined that the coordinates of point M are 5 and 3.
How to construct a point on a plane using its coordinates?
It also happens that we already know the coordinates of a point on the plane. And we need to find its location. Let's say our point coordinates are (-2;5). That is, the abscissa is equal to -2, and the ordinate is equal to 5. Take a point on the x line (abscissa axis) with coordinate -2 and draw a straight line a through it, parallel to the y axis. Note that any point on this line will have an abscissa equal to -2. Now let’s find a point with coordinate 5 on the y line (ordinate axis) and draw a straight line b through it, parallel to the x axis. Note that any point on this line will have an ordinate equal to 5. At the intersection of lines a and b there will be a point with coordinates (-2;5). Let's denote it by the letter P (Fig. 4).
Let us also add that straight line a, all points of which have abscissa -2, is given by the equation
x = -2 or that x = -2 is the equation of line a. For convenience, we can say not “the straight line, which is given by the equation x = -2”, but simply “the straight line x = -2”. Indeed, for any point on the line a the equality x = -2 is true. And line b, all points of which have ordinate 5, in turn is given by the equation y = 5 or that y = 5 is the equation of line b.
