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… $y(t)=2(3t-2{)}^2-3$ $=2(9t^2-12t+4)-3$ $=18t^2-24t+5$ Finally giving $x(t)=3t-2,y(t)=18t^2-24t+5$