Overview
This lecture covers the Cartesian (rectangular) coordinate system: its structure, terminology, how to plot points, determine quadrants, and related concepts.
Introduction to the Cartesian Coordinate System
- The Cartesian coordinate system consists of two perpendicular number lines called axes.
- Unlike the one-dimensional number line, the Cartesian plane is two-dimensional, allowing movement in four directions.
- René Descartes introduced the Cartesian coordinate system.
Parts of the Coordinate System
- The horizontal number line is called the x-axis.
- The vertical number line is called the y-axis.
- The point where the x-axis and y-axis intersect is called the origin with coordinates (0, 0).
- The axes divide the plane into four regions called quadrants.
Understanding Coordinates and Quadrants
- An ordered pair (x, y) locates a point: x is the x-coordinate (abscissa), y is the y-coordinate (ordinate).
- Quadrant I: (x > 0, y > 0)
- Quadrant II: (x < 0, y > 0)
- Quadrant III: (x < 0, y < 0)
- Quadrant IV: (x > 0, y < 0)
- Points on the axes do not belong to any quadrant but lie on either the x-axis or y-axis.
Plotting Points
- To plot (x, y), start at the origin, move x units along the x-axis, then y units along the y-axis.
- If x = 0, the point is on the y-axis; if y = 0, the point is on the x-axis.
- Always start plotting from the origin.
Determining Quadrants and Axis Location
- Identify the signs of the coordinates to determine the quadrant.
- If a coordinate is zero, the point lies on the respective axis (not in a quadrant).
- The intersection of the axes forms four labeled quadrants in a counterclockwise direction.
Exercises and Applications
- Practice plotting points with given coordinates and identifying their quadrants or axes.
- Practice determining the ordered pair for a marked point on the plane.
- Understand that switching the order in an ordered pair changes the point’s location.
Key Terms & Definitions
- Cartesian Coordinate System — a plane defined by two perpendicular axes, used to graph ordered pairs.
- x-axis — the horizontal axis in the plane.
- y-axis — the vertical axis in the plane.
- Origin — the intersection point (0, 0) of the x and y axes.
- Quadrant — one of four regions in the coordinate plane formed by the axes.
- Ordered Pair — a pair of numbers (x, y) specifying a point.
- Abscissa — the x-coordinate in an ordered pair.
- Ordinate — the y-coordinate in an ordered pair.
Action Items / Next Steps
- Practice plotting given points and identifying their quadrants.
- Complete any assigned exercises on locating points and writing ordered pairs.
- Review quadrant signs and axis definitions before the next lesson.