CONCEPTS

# What is the Cartesian Plane?

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 calledÂ 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 toÂ graphically representÂ mathematical functionsÂ and analytical geometry equationsÂ .Â It also allows to represent relationships ofÂ movementÂ and physical position.

It is aÂ two-dimensional system, consisting of two axesÂ that extend from an origin to infinity (forming a cross).Â 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 representÂ mathematicalÂ operationsÂ .Â That is, it is a geometric tool to put the latter in relation graphically.

The Cartesian plane owes its name to the French philosopher RenÃ© 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.Â From there, itÂ isÂ alsoÂ possible to graphically represent the movementÂ (the movement from one point to another in the coordinate system).

In addition, itÂ allows to drawÂ two-dimensionalÂ geometric figuresÂ from straight lines and curves.Â These figuresÂ correspond to certain arithmetic operationsÂ , 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,Â to locate the points weÂ need two values: the first corresponding to the horizontal axis X and the second to the vertical axis YÂ , which are denoted in parentheses and separated by a comma: (0,0) for example, is the point where both lines intersect.

These values â€‹â€‹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:Â the ordinate (Â yÂ axisÂ ) and the abscissa (Â xÂ axisÂ )Â .Â Both lines extend to infinity, both in their positive and negative values.Â The onlyÂ crossing point between the two is called origin (coordinates 0,0)Â .

From the origin each axis is marked with values â€‹â€‹expressed in whole numbers.Â The intersection point of any two points is called a point.Â Each point is expressed in their respective coordinatesÂ , always saying first the abscissa and then the ordinates.Â 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

Coordinate axes are called the two perpendicular lines that interconnect at a point in the plane.Â These lines are called the abscissa and the ordinate.

• AbscissaÂ : the abscissa axis is arranged horizontally and is identified by the letter “x”.
• OrdinateÂ : 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.Â For this reason, it is also known as the zero point (point 0).Â 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.Â 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 planeÂ , as long as we express the relationship between a variableÂ xÂ and a variableÂ andÂ in such a way that it can be solved.

For example, if we have a function that states that the value ofÂ yÂ will be 4 when that ofÂ xÂ is 2, we can say that we have an expressible function like this: y = 2x.Â The function indicates the relationship between both axes, and allows to give value to one variable knowing the value of the otherÂ .

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.Â 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.Â If we continue the straight line towards infinity, then we will know what the value ofÂ x will beÂ in any case ofÂ yÂ .

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.Â However, the logic will remain the same: express the function graphically based on assigning values â€‹â€‹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 isÂ one-to-oneÂ , 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.

X AND Coordinate
2 3 (2.3)
-4 2 (-4.2)
6 -1 (6, -1)

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 lettersÂ xÂ (horizontal) and on the other handÂ yÂ (vertical).Â 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 IÂ , in the upper right region, where positive values â€‹â€‹can be represented on each coordinate axis.Â For example: (1,1).
• Quadrant IIÂ , in the upper left region, where positive values â€‹â€‹can be represented on theÂ yÂ axisÂ but negativeÂ valuesÂ on theÂ xÂ .Â For example: (-1, 1).
• Quadrant IIIÂ , in the lower left region, where negative values â€‹â€‹can be represented on both axes.Â For example: (-1, -1).
• Quadrant IVÂ , in the lower right region, where negative values â€‹â€‹can be represented on theÂ y-Â axisÂ but positive on theÂ xÂ .Â For example: (1, -1).

Quadrants are the four areas that are formed by the union of the two perpendicular lines.Â The points of the plane are described within these quadrants.

Quadrants are traditionally numbered with Roman numerals: I, II, III, and IV.

• Quadrant IÂ : the abscissa and the ordinate are positive.
• Quadrant IIÂ : the abscissa is negative and the ordinate is positive.
• Quadrant IIIÂ : both the abscissa and the ordinate are negative.
• Quadrant IVÂ : the abscissa is positive and the ordinate negative.

The coordinates are the numbers that give us the location of the point on the plane.Â The coordinates are formed by assigning a certain value to the “x” axis and another value to the “y” axis.Â 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.Â Those numbers are the coordinates.

For example

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.Â The intersection or crossing of both projections gives us the spatial location of the point.

For example

In this example, P (3,4) gives us the precise location of the point in quadrant I of the plane.Â 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.Â 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 planeÂ was an invention of RenÃ© DescartesÂ , as we have said,Â aÂ centralÂ philosopherÂ in the tradition of the West.Â 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.Â He managed to mathematically transfer the analytical geometry to the twoÂ –Â dimensional planeÂ of the flat geometry and gave rise to the coordinate system that we still use and study today.