Finding parametric equations of rectangular equation Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there some steps that might help me ? I looked at Finding Parametric Equations For A Rectangular Equation, but it did not help a lot. I am learning PreCalc so please if you use methods taught in Calc make sure to explain them well. Thank you!
 A: I think that this question requires some clarification.
First of all we can properly talk about the parametric equation of a curve, not of a function.
A function, as $f: X \to \mathbb{R} \quad y=f(x) \quad X \subset \mathbb{R} \quad$ has a graph in $\mathbb{R}^2$ that is a curve and this curve can be represented, parametrically,  as the graph of a function $p(t):I \to \mathbb{R}^2$ where $I$ is an interval in $\mathbb{R}$.
This can be done using every bijective function $g:I \to X$ so that we have a function:
$$
p:I \to X\times \mathbb{R} \qquad p(t)=(g(t),f(g(t)))
$$
As an example, if we have $f: \mathbb{R} \to \mathbb{R} \quad y=ax$, we can use $g:(-\pi/2,\pi/2)\to \mathbb{R}\quad g(t)=\tan t$ and we have:
$$
p(t):(-\pi,\pi) \to \mathbb{R}^2 \quad p(t)=(\tan t, a \tan t)
$$
note that we can use any bijective function from an interval to $X$
But this is not the only way to obtain a parametric representation of a curve. To show some example:
For the function $f:[-1,1]\to \mathbb{R} \quad y=f(x)=\sqrt{1-x^2}$, we can use a parametric representation :
$$
p:[0,\pi] \to \mathbb{R} \quad p(t)=(\cos t, \sin t)
$$
That can be found using the previous method and the trig. identity $\sin^2 x+\cos^2 x=1$.
For an hyperbola of equation 
$$
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1
$$
we have not an equation of the form $y=f(x)$ but, since it is a curve in $\mathbb{R}^2$, we can have the parametric equation:
$$
p(t)=(a \sec t, b \tan t) \quad t \in [-\pi, \pi]
$$
that is derived from the trigonometric identity $ \sec^2 \alpha - \tan^2 \alpha=1$.
Finally note that another parametric representation can also be write in the form:
$$
x=a\frac{1+t^2}{1-t^2} \quad y=b\frac{2t}{1-t^2}
$$
can you see from where this come from?
