We explain what the Cartesian plane is? how it was created, quadrant in cartesian plane with elements. In addition, how functions are represented.
What is the Cartesian plane?
A Cartesian plane or Cartesian system is called an orthogonal coordinate diagram used for geometric operations in the Euclidean space (that is, the geometric space that meets the requirements formulated in ancient times by Euclid).
It is used to graphically represent mathematical functions and analytical geometry equations . It also allows to represent relationships of movement and physical position.
It is a two-dimensional system, consisting of two axes that extend from an origin to infinity (forming a cross). These axes are intercepted at a single point (denoting the point of origin of coordinates or point 0.0).
A set of length marks are drawn on each axis, which serve as a reference to locate points, draw figures or represent mathematical operations . That is, it is a geometric tool to put the latter in relation graphically.
The Cartesian plane owes its name to the French philosopher René Descartes (1596-1650), creator of the field of analytical geometry.
What is the Cartesian plane for?
The Cartesian plane is a diagram in which we can locate points, based on their respective coordinates on each axis, just as a GPS does on the globe. From there, it is also possible to graphically represent the movement (the movement from one point to another in the coordinate system).
In addition, it allows to draw two-dimensional geometric figures from straight lines and curves. These figures correspond to certain arithmetic operations , such as equations, simple operations, etc.
There are two ways to resolve these operations: mathematically and then graph it, or we can find a solution graphically, since there is a clear correspondence between what is illustrated on the Cartesian plane, and what is expressed in mathematical symbols.
In the coordinate system, to locate the points we need two values: the first corresponding to the horizontal axis X and the second to the vertical axis Y , which are denoted in parentheses and separated by a comma: (0,0) for example, is the point where both lines intersect.
These values can be positive or negative, depending on their location with respect to the lines that make up the plane.
Parts Of The Cartesian Plane
The Cartesian plane is composed of two perpendicular axes, as we know: the ordinate ( y axis ) and the abscissa ( x axis ) . Both lines extend to infinity, both in their positive and negative values. The only crossing point between the two is called origin (coordinates 0,0) .
From the origin each axis is marked with values expressed in whole numbers. The intersection point of any two points is called a point. Each point is expressed in their respective coordinates , always saying first the abscissa and then the ordinates. Joining two points you can build a line, and with several lines a figure.
So, The elements and characteristics that make up the Cartesian plane are the coordinate axes, the origin, the quadrants, and the coordinates. Next, we explain each one to you.
Coordinate axes are called the two perpendicular lines that interconnect at a point in the plane. These lines are called the abscissa and the ordinate.
- Abscissa : the abscissa axis is arranged horizontally and is identified by the letter “x”.
- Ordinate : the ordinate axis is oriented vertically and is represented by the letter “y”.
Origin or point 0
The origin is called the point at which the “x” and “y” axes intersect, the point to which the value of zero (0) is assigned. For this reason, it is also known as the zero point (point 0). Each axis represents a numerical scale that will be positive or negative according to its direction with respect to the origin.
Thus, with respect to the origin or point 0, the right segment of the “x” axis is positive, while the left is negative. Consequently, the rising segment of the “y” axis is positive, while the descending segment is negative.
Functions in a Cartesian plane
Mathematical functions can be expressed graphically on a Cartesian plane , as long as we express the relationship between a variable x and a variable and in such a way that it can be solved.
For example, if we have a function that states that the value of y will be 4 when that of x is 2, we can say that we have an expressible function like this: y = 2x. The function indicates the relationship between both axes, and allows to give value to one variable knowing the value of the other .
For example if x = 1, then y = 2. On the other hand, if x = 2, then y = 4, if x = 3, then y = 6, etc. finding all these points in the coordinate system, we will have a straight line, since the relationship between both axes is continuous and stable, predictable. If we continue the straight line towards infinity, then we will know what the value of x will be in any case of y .
The same logic will apply to other, more complex, functions that will yield curved lines, parabolas, geometric figures or dashed lines, depending on the mathematical relationship expressed in the function. However, the logic will remain the same: express the function graphically based on assigning values to the variables and solving the equation.
So, A function represented as: f (x) = y is an operation to obtain the dependent variables (against domain) from an independent variable (domain). For example: f (x) = 3x
|X function||Domain||Against domain|
|f (2) = 3x||2||6|
|f (3) = 3x||3||9|
|f (4) = 3x||4||12|
The relationship of the domain and the counter domain is one-to-one , which means that it has only two correct points.
To find the function in a Cartesian plane, we must first tabulate, that is, order the points in a table the pairs found to position them or locate them later in the Cartesian plane.
Quadrant In Cartesian Plane
As we have seen, the Cartesian plane is constituted by the crossing of two coordinate axes, that is, two infinite straight lines, identified with the letters x (horizontal) and on the other hand y (vertical). If we look at them, we will see that they form a sort of cross, thus dividing the plane into four quadrants, which are:
- Quadrant I , in the upper right region, where positive values can be represented on each coordinate axis. For example: (1,1).
- Quadrant II , in the upper left region, where positive values can be represented on the y axis but negative values on the x . For example: (-1, 1).
- Quadrant III , in the lower left region, where negative values can be represented on both axes. For example: (-1, -1).
- Quadrant IV , in the lower right region, where negative values can be represented on the y- axis but positive on the x . For example: (1, -1).
Quadrants are the four areas that are formed by the union of the two perpendicular lines. The points of the plane are described within these quadrants.
Quadrants are traditionally numbered with Roman numerals: I, II, III, and IV.
- Quadrant I : the abscissa and the ordinate are positive.
- Quadrant II : the abscissa is negative and the ordinate is positive.
- Quadrant III : both the abscissa and the ordinate are negative.
- Quadrant IV : the abscissa is positive and the ordinate negative.
The coordinates are the numbers that give us the location of the point on the plane. The coordinates are formed by assigning a certain value to the “x” axis and another value to the “y” axis. This is represented as follows:
P (x, y), where:
- P = point in the plane;
- x = axis of the abscissa (horizontal);
- y = axis of the ordinate (vertical).
If we want to know the coordinates of a point in the plane, we draw a perpendicular line from point P to the “x” axis – we will call this line a projection (orthogonal) of point P on the “x” axis.
Next, we draw another line from point P to the “y” axis – that is, a projection of point P onto the “y” axis.
In each of the crossings of the projections with both axes, a number (positive or negative) is reflected. Those numbers are the coordinates.
In this example, the coordinates of the points in each quadrant are:
- quadrant I, P (2, 3);
- quadrant II, P (-3, 1);
- quadrant III, P (-3, -1) and
- quadrant IV, P (3, -2).
If what we want is to know the location of a point from some previously assigned coordinates, then we draw a perpendicular line from the indicated number of the abscissa, and another from the number of the ordinate. The intersection or crossing of both projections gives us the spatial location of the point.
In this example, P (3,4) gives us the precise location of the point in quadrant I of the plane. The 3 belongs to the abscissa axis and the 4 (right segment) to the ordinate axis (ascending segment).
P (-3, -4) gives us the specific location of the point in quadrant III of the plane. The -3 belongs to the abscissa axis (left segment) and the -4 to the ordinate axis (descending segment).
History of the Cartesian plane
The Cartesian plane was an invention of René Descartes , as we have said, a central philosopher in the tradition of the West. His philosophical perspective was always based on the search for the point of origin of knowledge.
As part of that search, he conducted extensive studies on analytical geometry, which is considered father and founder. He managed to mathematically transfer the analytical geometry to the two – dimensional plane of the flat geometry and gave rise to the coordinate system that we still use and study today.