10.5 Parametric Equations

Parametric Equations are a useful way to model certain realistic scenarios mathematically, especially situations involving motion-over-time.

• In fact, time is so important to the equation, it often takes over the entire role of the dependent variable, creating two separate equations where 'x' and 'y' (coordinates) are functions of the new variable 't', which is called the parameter. In some cases, 't' may not represent time.
   
• The parameter 't' is frequently restricted by the real-world conditions it is modeling. For example, if you are modeling a projectile path and ignoring air resistance, 't' will have a domain of [0, t where y=0, x ≠ 0] (from 0 to the time of landing.)

• Therefore, parametric functions are rarely continuous. In fact, they are most commonly fragments of curves, or sometimes circles when sin/cos are involved.

•  In some interesting cases, two parametric equations, once restricted, will appear to graph the exact same curve. Unless they are the same equation, this indicates their equations only change the speed at which the motion occurs, not its direction (see page 732 for example.)

Definition: A curve in the plane is said to be parameterized if the set of coordinates on the curve, (x,y), are represented as functions of a variable t. Namely, 

x = f(t),  y = g(t)   The variable t is called a parameter and the relations between x, y and t are called parametric equations. The set of ordered pairs (f(t),g(t)) on the interval 'I' is a plane curve (denoted 'C'.)

Key Idea: Because parametric equations model real physical motion, the way their graphs are sketched must be realistic. Specifically, because motion occurs along the path of the parametric curve, the curve itself is sketched in a specific direction. 

• The direction in which a parametric curve is sketched is called its orientation. This can be left to right, right to left, clockwise, or counter-clockwise.
• Orientation is determined by simply comparing coordinates that result from subsequent values of the variable 't'
  
  Eliminating the Parameter
  
  
  
  Often enough we find the need to model parametric equations in the more classical f(x) type of equation. While usually possible, restrictions on the parameter won't always transfer over to the new equation, so you must determine the domain and range of the parametric equations before eliminating the parameter.
  
  
  
  
  

 Note: While the graph above shows the new function, it fails to model the actual path including the restrictions on t, as mentioned in the first step. These should be considered to determine the domain (think about limits of x and y...)

• Trigonometric Parametric Equations are special, as solving for t and substituting is generally a poor method and will not necessarily yield a useful equation. Instead, one should recognize and attempt to apply basic trigonometric relationships, such as the always-true Pythagorean Identity, as shown below.

• By solving for cos(t) and sin(t), we can quickly use an identity to eliminate the trigonometry and the parameter in, shall we say, one fell swoop. You should probably review your other identities for more advanced equations people...