# Points, Vectors, and Functions

# Parametric Equations

Ashleigh, Joe, and some of their friends are playing a game of Monopoly, and the game is almost over. Joe is about to lose it all, and Joe hates to lose. Before Ashleigh is about to backrupt him, he finds a way to keep her from doing so by changing the rules. Ashleigh objects, but Joe protests, claiming that the rule was there the whole time. Ashleigh gets upset and turns over the board, flinging houses, hotels, a top hat, and an iron into Joe's drink. Sereves him right.

We are about to pull a rule change. Mathematicians are not above altering the rules to make things more convenient.

There are vector functions that take one number in and outputs multiple numbers. These sly devils are described by **parametric equations**.

### Sample Problem

The vector function *f*(*t*) = *t**y*(*t*) = *t*^{2 }*t* to the variables *x* and *y*. Here, the input *t* is called the **parameter**. Each component of the output is dependent on this parameter. *t*, *x*(*t*) and *y*(*t*) together in a single vector function *f*(*t*).**parametrization** of the function *f*(*t*). Sometimes it's more useful to link equations together using a single variable like this.*x* and *y* be the coordinates that describe the height and length of the house's path. Both of these distances are functions of time *t*. We can write the trajectory d(x,y) = *t*)>.**parametric function**, which means a function described by parametric equations.