Parametric Equations

Parameter: Independent variable which both x and y depend on

An ordered pair with parameter t looks like:

$$(x(t),y(t))$$

Parametric equations are useful to define things which are not explicitly functions

(t-1, (2t)+4)  
((t^3)-3, (2t)+1)  

Deparameterizing

Parametric equations can be rewritten into single equations

$$x(t)=t^2-3,y(t)=2t+1$$

Solve one equation for t

$$t=\frac{y-1}{2}$$

Substitute into other equation

$$x={\frac{y-1}{2}}^2-3$$

Parameterize

First, it is always possible to parameterize a curve by defining x(t)=t, then replacing x with t in the equation for y(t).

$$y=2x^2-3$$ $$x(t)=t,y(t)=2t^2-3$$

If there was a limited domain, we would need to restrict the values of t

We have freedom over the second parameterization. Just make literally anything up. We only need to check that there are no restrictions of x - that the range of x(t) is all real numbers. It can be..

x(t) = 3t-2 $$ Since y=2x^2 - 3, we an substitute the above for x...

y(t) = 2(3t - 2)^2 - 3

= 2(9t^2 -12t +4) -3

= 18t^2 - 24t + 5

$$Finallygiving$$

x(t) = 3t-2, y(t) = 18t^2 - 24t + 5