Graphing, Distance, and Midpoint 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 two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It is used to graph equations and represent geometric figures algebraically.

• Axes: The x-axis (horizontal) and y-axis (vertical) intersect at the origin (0,0).

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

• Ordered Pairs: Each point is represented as (x, y), where x is the horizontal position and y is the vertical position.

Graphing Equations in the Rectangular Coordinate System

Plotting Points and Graphing Equations

To graph an equation, plot points that satisfy the equation and connect them to reveal the graph's shape.

• Plotting Points: Identify the x and y values and mark the corresponding location on the plane.

• Graphing Linear Equations: For equations like $y = mx + b$, plot the y-intercept (b) and use the slope (m) to find another point.

• Graphing Nonlinear Equations: Create a table of values, plot several points, and connect them smoothly.

Example: To graph $y = 2x + 1$, plot (0,1) and (1,3), then draw a straight line through these points.

Distance Between Two Points

Finding the Distance Between Two Points

The distance formula calculates the length between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane.

• Formula:

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

• Application: Useful for determining the straight-line distance between any two points.

Example: Find the distance between (2,3) and (7,11):

$ d = \sqrt{(7-2)^2 + (11-3)^2} = \sqrt{25 + 64} = \sqrt{89} $

Midpoint of a Line Segment

Finding the Midpoint

The midpoint formula finds the point exactly halfway between two points $(x_1, y_1)$ and $(x_2, y_2)$.

• Formula:

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

• Application: Used to find the center of a line segment or to bisect a segment.

Example: Find the midpoint between (2,3) and (7,11):

$ M = \left( \frac{2+7}{2}, \frac{3+11}{2} \right) = (4.5, 7) $