Cartesian Coordinate System

Overview

In plane analytic geometry, the Cartesian coordinate system uniquely specifies a point by a pair of real numbers called its coordinates. These coordinates represent signed distances to the point from two fixed perpendicular oriented lines called the axes. The point where the axes meet is called the origin and have coordinates $⟨0,0⟩$.

Cartesian Equations

An equation that completely characters a figure within the Cartesian coordinate system is called a Cartesian equation.

Translations

There are two kinds of translations that we can do to a graph: shifting and scaling. A reflection is a special case of scaling.

Shifting

A vertical shift adds/subtracts a constant to every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A horizontal shift adds/subtracts a constant to every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged.

Scaling

A vertical scaling will multiply/divide every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A horizontal scaling will multiply/divide every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged.

Scaling is also known as stretching and compressing.

Bibliography

• “Cartesian Coordinate System,” in Wikipedia, October 21, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system.
• “James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, https://people.richland.edu/james/lecture/m116/functions/translations.html.
• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).