Fundamental Concepts in the Rectangular Coordinate System

Rectangular Coordinate System

Introduction to the Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is a foundational concept in College Algebra. It allows us to visually represent equations, analyze geometric relationships, and solve problems involving points, lines, and distances.

• Definition: The rectangular coordinate system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).

• Ordered Pair: Each point in the plane is identified by an ordered pair $(x, y)$, where x is the horizontal position and y is the vertical position.

• Quadrants: The plane is divided into four quadrants, numbered I to IV, starting from the upper right and moving counterclockwise.

Graphing Equations in the Rectangular Coordinate System

Graphing equations involves plotting points that satisfy a given equation and connecting them to visualize the relationship.

• Procedure:

  1. Choose values for x and solve for y to obtain ordered pairs.

  2. Plot each ordered pair on the coordinate plane.

  3. Connect the points smoothly to reveal the graph of the equation.

• Example: To graph $y = 2x + 1$, select values for x (e.g., -1, 0, 1), compute corresponding y values, and plot the points (-1, -1), (0, 1), (1, 3).

Finding the Distance Between Two Points

The distance between two points in the plane can be calculated using the Distance Formula, which is derived from the Pythagorean Theorem.

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Find the distance between (2, 3) and (5, 7): $d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Finding the Midpoint of a Line Segment

The midpoint of a line segment connecting two points is the point exactly halfway between them. The Midpoint Formula provides a method to calculate this point.

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Find the midpoint between (2, 3) and (5, 7): $M = \left( \frac{2 + 5}{2}, \frac{3 + 7}{2} \right) = (3.5, 5)$