Parametric Equations and Cartesian Conversion

Introduction to Parametric Equations

Parametric equations express the coordinates of the points that make up a geometric object as functions of a variable, typically denoted as t (the parameter). These equations are especially useful for describing curves that cannot be represented as functions in the standard Cartesian form.

• Parametric Form: The coordinates x and y are given as functions of t:

• Application: Parametric equations are commonly used in physics, engineering, and computer graphics to model motion and curves.

Example: Problem #3

Given the parametric equations:

Here, both x and y are expressed in terms of the parameter t. As t varies, the pair traces a curve in the plane.

• Ordered Pair: For a given value of t, the corresponding point is .

• Graphing: To graph this curve, plot several points for different values of t and label each with its t value.

Converting Parametric to Cartesian Equation

To convert parametric equations to a Cartesian equation, eliminate the parameter t to obtain a direct relationship between x and y.

• Step 1: Express t in terms of y from the second equation:

• Step 2: Substitute t into the first equation:

• Cartesian Equation: The relationship between x and y is:

Summary Table: Parametric vs. Cartesian Forms

Example Application

Suppose t = 0:

• Point:

• Check Cartesian:

This confirms the conversion is correct for all values of t.