- What is the Cartesian plane?
- History of the Cartesian plane
- What is the Cartesian plane for?
- Quadrants of the Cartesian plane
- Elements of the Cartesian plane
- Functions in a Cartesian Plane
We explain what the Cartesian plane is, how it was created, its quadrants and elements. Also, how functions are represented.
What is the Cartesian plane?
A Cartesian plane or Cartesian system is called a diagram of orthogonal coordinates used for geometric operations in Euclidean space (that is, geometric space that meets the requirements formulated in ancient times by Euclid).
Used to graphically represent math functions and equations of analytical geometry. It also allows you to represent relationships of movement and physical position.
It is a two-dimensional system, made up of two axes that extend from one origin to infinity (forming a cross). These axes intersect at a single point (denoting the coordinate origin point or 0,0 point).
On each axis are drawn a set of marks of length, which serve as reference to locate points, draw figures, or represent operations math. In other words, 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 analytic geometry.
History of the Cartesian plane
The Cartesian plane was an invention of René Descartes, as we have said, philosopher central in the tradition of the West. His philosophical perspective was always based on the search for the point of origin of the knowledge.
As part of that search, he conducted extensive studies on analytical geometry, of which he considers himself the father and founder. He managed to translate analytical geometry mathematically to the two-dimensional plane of plane geometry and gave rise to the coordinate system that we still use and study today.
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 motion (the displacement from one point to another in the coordinate system).
In addition, it allows you to trace geometric figures two-dimensional from lines and curves. These figures correspond to certain arithmetic operations, such as equations, simple operations, etc.
There are two ways to solve these operations: mathematically and then graph it, or we can find a solution graphically, since there is a clear correspondence between what is illustrated in 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 X axis and the second to the vertical Y axis, which are denoted between parentheses and separated by a comma: for example, it 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.
Quadrants of the 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 contemplate 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: .
- Quadrant II. In the upper left region, where positive values can be represented on the axis Y but negative in 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 axis Y but positive in the x. For example: (1, -1).
Elements of the Cartesian plane
The Cartesian plane is made up of two perpendicular axes, as we already know: the ordinate (axis Y) and the abscissa (axis x). Both lines extend to infinity, both in their positive and negative values. The only crossing point between the two is called the origin (0,0 coordinates).
Starting from the origin, each axis is marked with values expressed in whole numbers. The point of intersection of any two points is called a point. Each point is expressed in its respective coordinates, always saying the abscissa first and then the ordinate. By joining two points you can build a line, and with several lines a figure.
Functions in a Cartesian Plane
Functions can be expressed graphically on the 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 Y in such a way that it can be resolved.
For example, if we have a function that states that the value of Y will be 4 when x Let 2 be, we can say that we have an expressible function like this: y = 2x. The function indicates the relationship between both axes, and allows giving value to a 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. By finding all those 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 in any case of Y.
The same logic It will apply to other types of functions, more complex, which will yield curved lines, parabolas, geometric figures or broken 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.