The Rectangular Coordinate System and Ordered Pairs

Overview of the Rectangular Coordinate System

• The rectangular coordinate system, also commonly referred to as the Cartesian coordinate system, provides a framework for graphing and geometric visualization.

• Fundamental objectives and operations using this system include:

  • Plotting ordered pairs of real numbers.

  • Determining whether given ordered pairs constitute valid solutions to equations containing $2$ variables.

  • Graphing linear equations in $2$ variables.

  • Graphing nonlinear equations in $2$ variables.

Components and Division of the Coordinate Plane

• Composition of Axes:

  • The system consists of $2$ perpendicular number lines that intersect at right angles ($90^\text{o}$).

  • The intersection occurs at the zero coordinates ($0$) of both number lines.

  • Horizontal Axis ($x$-axis): The horizontal number line used to measure horizontal distance.

  • Vertical Axis ($y$-axis): The vertical number line used to measure vertical distance.

  • Origin: The exact point of intersection of the horizontal $x$-axis and vertical $y$-axis.

  • Coordinates of the origin are $(0, 0)$.

  • Represents the zero value on the $x$-axis and the zero value on the $y$-axis simultaneously.

• Quadrants:

  • The intersecting $x$-axis and $y$-axis divide the two-dimensional plane into $4$ distinct regions.

  • These regions are called quadrants and are conventionally designated using Roman numerals:

  • Quadrant I: Top-right region.

  • Quadrant II: Top-left region.

  • Quadrant III: Bottom-left region.

  • Quadrant IV: Bottom-right region.

Principles of Ordered Pairs

• Definition and Function:

  • Points in the plane are located, graphed, or plotted using ordered pairs of numbers.

  • An ordered pair records precise distances from the coordinate axes.

• Order Convention:

  • An ordered pair consists of a pair of numbers written in a specific, mandatory order.

  • The $x$-value (or $x$-coordinate) is always written first.

  • The $y$-value (or $y$-coordinate) is always written second.

  • Standard algebraic notation: $(x, y)$.

• One-to-One Correspondence:

  • Every single ordered pair of numbers corresponds to exactly $1$ unique point in the coordinate plane.

  • Conversely, every point in the plane corresponds to exactly $1$ ordered pair.

Step-by-Step Procedure for Plotting Points

• To plot any point corresponding to an ordered pair $(x, y)$:

  • Step 1 (Starting Position): Begin at the origin, which has coordinates $(0, 0)$.

  • Step 2 (Horizontal Displacement):

  • Determine horizontal movement based on the value and sign of the $x$-coordinate.

  • If the $x$-coordinate is positive, move right along the $x$-axis by that number of units.

  • If the $x$-coordinate is negative, move left along the $x$-axis by that number of units.

  • Step 3 (Vertical Displacement):

  • Determine vertical movement from the new horizontal position, moving along a line parallel to the $y$-axis.

  • If the $y$-coordinate is positive, move upward by that number of units.

  • If the $y$-coordinate is negative, move downward by that number of units.

  • Step 4 (Marking the Point): Mark the final position in the plane, which represents the point corresponding to the ordered pair $(x, y)$.