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Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Up until now, we've dealt with functions of the form in the Cartesian plane. However, there are other ways to represent curves.

Choosing a different system can greatly simplify many calculations, so it's important to be familiar with these other systems.

Table of Contents​

Parametric Equations​

A parametric equation is a way to represent a curve by giving the and coordinates as functions of a third variable, usually .

Consider a ball pushed from a ledge.

You could describe the motion by having the coordinate be the horizontal distance the ball has traveled and the coordinate be the height of the ball at that time.

At some time , the ball will be at some point with an coordinate and a coordinate. Hence, we can make two functions, and , that describe the ball's position at time . These two functions are called parametric equations.

The motion is constant, so . The motion follows the equation derived previously here.

Hence, the equations for the ball are: