Coordinate Geometry Basics

The Cartesian Plane and Quadrants

The Cartesian coordinate plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). The intersection point is called the origin . The plane is divided into four quadrants:

• Quadrant I (Q1):

• Quadrant II (Q2):

• Quadrant III (Q3):

• Quadrant IV (Q4):

Each point in the plane is represented as an ordered pair .

Distance and Midpoint Formulas

Distance Between Two Points

The distance between two points and is always positive and can be found using the Pythagorean Theorem:

• Formula:

• Example: Find the distance between and :

Midpoint of a Line Segment

The midpoint of the segment from to is:

• Example: Find the midpoint from to :

Right Triangles in the Coordinate Plane

To determine if three points form a right triangle, use the Pythagorean Theorem:

• Given , , , calculate the lengths of sides and check if .

• If true, the triangle is a right triangle.

Equations and Graphs

Linear Equations

A linear equation in two variables has the form , where is the slope and is the y-intercept.

• Example:

• To graph, pick values for and solve for .

Quadratic Equations

A quadratic equation has the form .

• Example:

• Graph is a parabola; calculate for several values to plot points.

Intercepts

• x-intercepts: Points where .

• y-intercepts: Points where .

• To find intercepts algebraically, set the other variable to zero and solve.

Example: For :

• y-intercept:

• x-intercepts:

Symmetry of Graphs

Types of Symmetry

• y-axis symmetry: If is on the graph, so is .

• x-axis symmetry: If is on the graph, so is .

• Origin symmetry: If is on the graph, so is .

To test for symmetry:

• Replace with for y-axis symmetry.

• Replace with for x-axis symmetry.

• Replace both with and with for origin symmetry.

• If the equation remains unchanged, the graph has that symmetry.

Example: is symmetric about the y-axis, but not the x-axis or origin.

Slope of a Line

Definition and Calculation

The slope of a line measures its steepness and is calculated as:

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Undefined slope: Vertical line ().

Example: For points and :

Graphing and Interpreting Equations

Graphing Linear and Quadratic Equations

• To graph, create a table of values for and calculate corresponding values.

• Plot the points and connect them smoothly.

• For a complete graph, show all intercepts and relevant points.

Testing Points on a Graph

• If a point satisfies the equation, it lies on the graph.

• Substitute and into the equation to check.

Summary Table: Symmetry Tests

Key Formulas

• Distance:

• Midpoint:

• Slope: