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.
The vector function f(t) = 2 can be described by the parametric equations x(t) = t
y(t) = t2 for -∞ < t < ∞. Now we should explain the new rules.The vector function in this example actually requires three equations, or functions, to solve. The first equation is the vector itself, and the second and third are the equations that relate the variable 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. We bent the rules by linking two scalar functions of t, x(t) and y(t) together in a single vector function f(t).The particular equations used and values of the parameter included are together called a parametrization of the function f(t). Sometimes it's more useful to link equations together using a single variable like this.One good example of a parametric set of equations is the trajectory one of those houses took after Ashleigh flung the Monopoly board over. We will have 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) = , which is a vector function, instead as d(t) = y(t)>.As long as a vector function takes only one value as input, we can think of parametric equations and vector functions as the same thing, like we did with the airborne house. A parametric function, which means a function described by parametric equations.