Graphs and the Rectangular Coordinate System: College Algebra Study Notes

Graphs & the Rectangular Coordinate System

Introduction to the Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian Plane, is fundamental in College Algebra for graphing points and equations. It consists of two perpendicular number lines that intersect to form a two-dimensional plane.

• Horizontal axis: The x-axis

• Vertical axis: The y-axis

• Origin: The point (0, 0) where the x- and y-axes intersect

• Ordered pairs / points: Each point is represented as (x, y), where x is the horizontal position and y is the vertical position

Plotting Points

To plot a point (x, y):

• Move x units right (if x > 0) or left (if x < 0) from the origin

• Move y units up (if y > 0) or down (if y < 0) from the origin

Quadrants of the Cartesian Plane

The x- and y-axes divide the plane into four regions called quadrants:

Quadrant	x-value	y-value	Location
Quadrant I (Q1)	+	+	Top right
Quadrant II (Q2)	-	+	Top left
Quadrant III (Q3)	-	-	Bottom left
Quadrant IV (Q4)	+	-	Bottom right

Additional info: The quadrants are numbered counterclockwise starting from the top right.

Examples

• Example 1: Plot the points A (4, 3), B (–3, 2), C (–2, –3), D (5, –4), E (0, 0), F (0, –3) on the graph.

  • A (4, 3): Q1

  • B (–3, 2): Q2

  • C (–2, –3): Q3

  • D (5, –4): Q4

  • E (0, 0): Origin

  • F (0, –3): On y-axis, below origin

• Example 2: Graph the points W (1, –2), X (5, 2), Y (–3, –4), Z (–3, 4). Identify the quadrant of each point.

  • W: Q4

  • X: Q1

  • Y: Q3

  • Z: Q2

Key Terms and Definitions

• Cartesian Plane: A two-dimensional plane defined by a horizontal x-axis and a vertical y-axis.

• Origin: The point (0, 0) where the axes intersect.

• Quadrant: One of four regions into which the axes divide the plane.

• Ordered Pair: A pair of numbers (x, y) that defines a point's location.

Formulas

• Distance from origin to point (x, y): $d = \sqrt{x^2 + y^2}$