Parametric Equations and Cartesian Conversion

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). This approach is especially useful for describing curves that cannot be represented as functions in the standard Cartesian form.

• Definition: A parametric equation defines both x and y as functions of a third variable, usually t.

• Example: For problem 5, the parametric equations are:

• Graphing: To graph a parametric equation, plot the points for values of in the given interval. Label each point with its corresponding $t$ value for clarity.

Converting Parametric Equations to Cartesian Form

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

• Step 1: Express t in terms of x using one of the parametric equations.

• Step 2: Substitute this expression into the other equation to relate y directly to x.

• Example (Problem 6, based on Problem 5):

  • Given , so for .

  • Given , recall that .

  • Substitute for : .

  • Cartesian Equation: , with .

Key Properties and Applications

• Domain Restrictions: The domain of affects the domain of and in the Cartesian equation. For , is positive, so .

• Applications: Parametric equations are widely used in physics, engineering, and computer graphics to describe motion and curves that are not functions in the Cartesian sense.

Summary Table: Parametric vs. Cartesian Forms