Parametric+Equations

= **PARAMETRIC EQUATIONS** =

Parametric equations are a method of defining a function using parameters. A simple kinematical example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion. Often times parametric equations are used to express curves.

Parametric equations define x and y in terms of T (time). By plugging in values for T, one can find the coordinates (x, y) of a moving object at said time.

A curve given by:

x = x(T), y = y(T)

is called a //parametrically defined// curve and the functions.

x = x(T) and y = y(T)

are called the //parametric equations// for the curve.

To solve for the "conventional" form of the equation, solve for T and plug said value of T back into the equation. For example, if you are solving for y and y = y(T) and x = x(T), solve for T in terms of x and plug the value of T in terms of x into the y function.

A technique to identifying the curve of the parametric equations is to try to eliminate the parameter from the equations. This will result in an equation involving only //x// and //y// which we may recognize.
 * Eliminating the Parameter**

When eliminating the parameter, solve for T in one equation and plug it back into the other equation.

__Example:__ x = T² - 2T y = T + 1

It doesn't matter which equation you solve for T with, in this case we'll solve for T in terms of y. T = y - 1

Substitute this value for T into the equation for x and then simplify. x = T² - 2T x = (y - 1)² - 2(y - 1) x = y² - 2y + 1 - 2y + 2 x = y² - 4y + 3

__Example #2:__ x = T² y = sin(T)

When we solve for T, we get T = sqrt(x) Thus, we plug sqrt(x) into the y function so y = sin(sqrt(x))

Parametric equations can also be graphed, either by hand or with a calculator. Here's an example from [|Paul's Online Notes] on how to graph parametric equations by hand.
 * Graphing Parametric Equations**



[|Here] is another good source for graphing parametric equations.

Later you will learn how to derive parametric equations, and that deriving parametric equations in respect to T will give the instantaneous rate of change (velocity) at time T at said point.

Here are some good sources for extra information [|Paul's Online Notes] [|HMC Calculus Tutorial] [|Visual Calculus]
 * Extras**