Learning Objectives
- Understand the Cartesian coordinate system
- Identify axes, origin, and quadrants
- Plot points on the Cartesian plane
- Determine the coordinates of a given point
- Identify the quadrant or axis for any given point
Key Concepts
Cartesian Coordinate System
The Cartesian plane consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Their point of intersection is called the origin (O), with coordinates (0, 0).
Quadrants
The two axes divide the plane into four regions called quadrants:
- Quadrant I: x > 0, y > 0 (both positive), e.g., (3, 5)
- Quadrant II: x < 0, y > 0 (x negative, y positive), e.g., (-2, 4)
- Quadrant III: x < 0, y < 0 (both negative), e.g., (-3, -7)
- Quadrant IV: x > 0, y < 0 (x positive, y negative), e.g., (6, -1)
Coordinates of a Point
Any point P in the Cartesian plane is represented as an ordered pair (x, y) where x is the abscissa (horizontal distance from y-axis) and y is the ordinate (vertical distance from x-axis).
Points on Axes
A point on the x-axis has coordinates (x, 0) — ordinate is zero.
A point on the y-axis has coordinates (0, y) — abscissa is zero.
The origin has coordinates (0, 0).
Plotting Points
To plot point (a, b): start at origin, move |a| units along x-axis (right if positive, left if negative), then move |b| units parallel to y-axis (up if positive, down if negative).
Summary
The Cartesian coordinate system uses two perpendicular axes to uniquely locate any point in a plane using an ordered pair (x, y). The plane is divided into four quadrants with specific sign conventions. Points on the axes have one coordinate as zero.
Important Terms
- Origin: The point of intersection of the x-axis and y-axis, denoted as (0, 0)
- Abscissa: The x-coordinate; horizontal distance from the y-axis
- Ordinate: The y-coordinate; vertical distance from the x-axis
- Quadrant: One of the four regions created by the intersection of the axes
- Cartesian Plane: The plane formed by the x-axis and y-axis together
Quick Revision
- Coordinates are written as ordered pairs (x, y) — order matters
- Quadrant I: (+, +), Quadrant II: (-, +), Quadrant III: (-, -), Quadrant IV: (+, -)
- On x-axis: y = 0; on y-axis: x = 0
- The origin is the only point with coordinates (0, 0)
- Mirror image of (a, b) in x-axis is (a, -b); in y-axis is (-a, b)