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Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 12
Created by Boundless

The Cartesian System

The Cartesian coordinate system is used to visualize points on a graph by showing the points' distances from two axes.

Learning Objective

  • Plot points in Cartesian coordinates and explain what they mean


Key Points

    • The Cartesian coordinate system is a 2-dimensional plane with a horizontal axis, known as the xxx-axis, and a vertical axis, known as the yyy-axis.
    • A Cartesian coordinate system specifies each point uniquely in a plane with a pair of numerical coordinates, each of which is the signed distance from the point to one of the two axes.
    • The numerical coordinates of a point are represented by an ordered pair (x,y)(x,y)(x,y), where the xxx-coordinate is the point's distance from the yyy-axis, and the yyy-coordinate is the distance from the xxx-axis.
    • The Cartesian coordinate system is broken into four quadrants, labeled I, II, III, and IV, starting from the upper right hand corner and moving counterclockwise.
    • The independent variable is found on the xxx-axis and consists of the input values.  The dependent variable is found on the yyy-axis and consists of the output values.

Terms

  • ordered pair

    A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system. Notation: (x,y)(x,y)(x,y).

  • y-axis

    The axis on a graph that is usually drawn from bottom to top, with values increasing farther up.

  • quadrant

    One of the four quarters of the Cartesian plane bounded by the xxx-axis and yyy-axis.

  • x-axis

    The axis on a graph that is usually drawn from left to right, with values increasing to the right.

  • dependent variable

    An arbitrary output; on the Cartesian plane, the value of yyy.

  • independent variable

    An arbitrary input; on the Cartesian plane, the value of xxx.


Example


Full Text

Named for "the father of analytic geometry," 17th-century French mathematician René Descartes, the Cartesian coordinate system is a 2-dimensional plane with a horizontal axis and a vertical axis. Both axes extend to infinity, and arrows are used to indicate infinite length. The horizontal axis is known as the xxx-axis, and the vertical axis is known as the yyy-axis. The point where the axes intersect is known as the origin.

A Cartesian coordinate system is used to graph points. Points are specified uniquely in the Cartesian plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes. Each point can be represented by an ordered pair (x,y)(x,y) (x,y), where the xxx-coordinate is the point's distance from the yyy-axis and the yyy-coordinate is the distance from the xxx-axis. The origin where the two axes meet is therefore (0,0)(0,0)(0,0). On the xxx-axis, numbers increase toward the right and decrease toward the left; on the yyy-axis, numbers increase going upward and decrease going downward.

Cartesian coordinate system

The Cartesian coordinate system with 4 points plotted, including the origin, at (0,0)(0,0)(0,0).

Plotting Points

To plot the point (2,3)(2,3)(2,3), for example, you start at the origin (where the two axes intersect). Then, move three units to the right and two units up. 

 The point (−3,1)(-3,1)(−3,1) is found by moving three units to the left of the origin and one unit up.

The non-integer coordinates (−1.5,−2.5)(-1.5,-2.5)(−1.5,−2.5) lie between -1 and -2 on the xxx-axis and between -2 and -3 on the yyy-axis. Therefore, you move one and a half units left and two and a half units down.

Independent and Dependent Variables  

A Cartesian plane is particularly useful for plotting a series of points that show a relationship between two variables.

For example, there is a relationship between the number of cars a car wash cleans and the money the business makes (its revenue). The revenue, or output, depends upon the number of cars, or input, that they wash. Therefore, the revenue is the dependent variable (yyy), and the number of cars is the independent variable (xxx).  The revenue is plotted on the yyy-axis, and the number of cars washed is plotted on the xxx-axis.

Quadrants 

The Cartesian coordinate system is broken into four quadrants by the two axes. These quadrants are labeled I, II, III, and IV, starting from the upper right and continuing counter-clockwise, as pictured below.

Cartesian coordinates

The four quadrants of theCartesian coordinate system. The arrows on the axes indicate that they extend infinitely in their respective directions.

Some basic rules about these quadrants can be helpful for quickly plotting points:

  • Quadrant I: Points have positive xxx and yyy coordinates, (x,y)(x,y)(x,y).
  • Quadrant II: Points have negative xxx and positive yyy coordinates, (−x,y)(-x,y)(−x,y).
  • Quadrant III: Points have negative xxx and yyy coordinates, (−x,−y)(-x,-y)(−x,−y).
  • Quadrant IV: Points have positive xxx and negative yyy coordinates, (x,−y)(x,-y)(x,−y).
  • Points that have a value of 0 for either coordinate lie on the axes themselves and are not considered to be in any of the quadrants (e.g., (4,0)(4,0)(4,0), (0,−2)(0,-2)(0,−2)).

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