Cartesian Coordinate System

• The Cartesian Coordinate System is a system for locating points in space using coordinates.

History

• Rene Descartes conceived the system in the 17th century.
• It uses an origin, axes, and a unit length to locate objects in space.
• It is used in algebra, analytic geometry, trigonometry, and calculus.
• It defines geometric shapes with mathematical precision.

Core Components

• X-axis: A straight horizontal line.
• Y-axis: A perpendicular vertical line that crosses the x-axis.
• Origin: The point where the x- and y-axes intersect.
• Points are located using ordered pairs (x, y), where x is the x-coordinate and y is the y-coordinate.
• The x-coordinate indicates the distance from the y-axis along the x-axis.
• The y-coordinate indicates the distance from the x-axis along the y-axis.

Quadrants

• The coordinate plane is divided into four quadrants, labeled I, II, III, and IV.
• Quadrant I: x and y are positive (e.g., (5, 1)).
• Quadrant II: x is negative, and y is positive (e.g., (-4, 7)).
• Quadrant III: x and y are negative (e.g., (-8, -4)).
• Quadrant IV: x is positive, and y is negative (e.g., (12, -5)).

Three or More Dimensions

• A z-axis can be added, perpendicular to the coordinate plane, for three-dimensional use.
• The z-axis indicates height or depth, while x- and y- axes indicate length and width.
• Mathematicians suggest that a fourth dimension may be included.

Locating Points

• The origin is at (0, 0) and lies on at least one axis, not in any quadrant.
• To locate a point (2, 3), start at the origin, move 2 units right on the x-axis, and 3 units up on the y-axis (Quadrant I).
• To locate a point (-3, 1), start at the origin, move 3 units left on the x-axis, and 1 unit up on the y-axis (Quadrant II).
• To locate a point (-1.5, -2.5), start at the origin, move 1.5 units left on the x-axis, and 2.5 units down on the y-axis (Quadrant III).

Graphing Lines

• Lines can be graphed using Cartesian coordinates.
• For y = 2x, points (-2, -4) and (2, 4) can be plotted, and a straight line drawn through them.
• For y = -3x - 2, points (-2, 4) and (0, -2) can be plotted, and a straight line drawn through them.
• For y = 1/4x - 2, points (-4, -3) and (4, -1) can be plotted, and a straight line drawn through them.