Cartesian plane

What is a Cartesian Plane?

It is known as a Cartesian plane, Cartesian coordinates or Cartesian system, to two perpendicular number lines, one horizontal and one vertical, which intersect at a point called the origin or zero point.

The purpose of the Cartesian plane is to describe the position or location of a point in the plane, which is represented by the coordinate system.

The Cartesian plane is also used to mathematically analyze geometric figures such as the parabola, hyperbole, line, circumference and ellipse, which are part of analytical geometry.

The name of the Cartesian plane is due to the French philosopher and mathematician René Descartes, who was the creator of analytical geometry and the first to use this coordinate system.

Parts of the Cartesian plane

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. 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 is assigned. For this reason, it is also known as the zero point (0 point). 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 ascending segment of the "y" axis is positive, while the descending segment is negative.

Quadrants of the Cartesian plane

Quadrants are the four areas that are formed by the union of the two perpendicular lines. Points on 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.

You may also be interested in: Analytical Geometry.

Coordinates of the Cartesian plane

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 = ordinate axis (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.

For instance,

In this example, the coordinates of the points in each quadrant are:

  • quadrant I, P;
  • 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 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.

For instance,

In this example, P 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).

Functions in a Cartesian Plane

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 dominance

f = 3x

2

6

f = 3x

3

9

f = 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.

XYCoordinate23-42(-4,2)6-1(6,-1)

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