Rectangular Coordinate System

Introduction to the Cartesian Plane

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane divided into four regions called quadrants. Each point in the plane is identified by an ordered pair of numbers (x, y), representing its horizontal and vertical positions.

• Quadrants: The plane is divided by the x-axis (horizontal) and y-axis (vertical) into four quadrants, labeled I, II, III, and IV.

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

• Point Representation: A point P is represented as P(a, b), where 'a' is the x-coordinate and 'b' is the y-coordinate.

Example:

Point (3, -2) is located in Quadrant IV because the x-coordinate is positive and the y-coordinate is negative.

Equations of Lines and Slope

Slope of a Line

The slope of a line measures its steepness and is defined as the ratio of the vertical change to the horizontal change between two points on the line.

• Formula:

• Interpretation: 'Rise over run' – the change in y divided by the change in x.

Forms of Linear Equations

• Standard Form:

• Slope-Intercept Form:

  • Where m is the slope and b is the y-intercept.

• Vertical Line: Equation is (slope is undefined).

• Horizontal Line: Equation is (slope is zero).

Example:

The line passing through (1, 2) and (3, 6) has slope .

Distance Formula

Calculating the Distance Between Two Points

The distance formula is used to find the length of the segment connecting two points and in the plane.

• Formula:

Example:

Find the distance between (2, 3) and (5, 7):

Midpoint Formula

Finding the Midpoint of a Segment

The midpoint formula gives the coordinates of the point exactly halfway between two endpoints and .

• Formula:

Example:

Find the midpoint between (2, 3) and (6, 7):

Finding an Endpoint Given a Midpoint and Another Endpoint

Reverse Midpoint Problem

If you know the midpoint and one endpoint, you can find the other endpoint using the midpoint formula in reverse.

• Let midpoint and known endpoint . The other endpoint is found by:

Example:

If midpoint is (4, 5) and one endpoint is (2, 3): So the other endpoint is (6, 7).

Graphing Linear Equations

Plotting Lines on the Coordinate Plane

To graph a linear equation, plot points that satisfy the equation and draw a straight line through them. For standard form equations, it is often helpful to find the x- and y-intercepts.

• X-intercept: Set and solve for .

• Y-intercept: Set and solve for .

• Plot at least two points and draw the line through them.

Example:

Graph :

• Set : (y-intercept: (0, 2))

• Set : (x-intercept: (3, 0))

Plot these points and draw the line.

Homework Practice

Sample Problems from Textbook

Practice problems involve finding distances and midpoints between points, and determining endpoints given a midpoint and one endpoint.

Refer to textbook page 178, problems #11-17 odd, 31-35 odd for additional practice.

Additional info: These notes cover topics from Chapter 6 (Analytic Geometry) and Chapter 2 (Graphs and Functions) of College Algebra, including coordinate systems, equations of lines, and geometric formulas.