How do you find parametric equations for other equations? I'm stuck on this problem. There is no lesson, I'm trying to teach myself.
Which of the following are NOT parametric equations for the rectangular equation?
y=x^2-2x-3
A. x=t-1; y=t^2-4t
B. x=t+2; y=t^2+2t-3
C. x=t-2; y=t^2-6t-5
D. x=t+3; y=t^2+4t
 A: We can compute, unfortunately possibly $4$ times.
For example, look at A). Substitute $t-1$ for $x$ in $x^2-2x-3$. We get 
$$x^2-2x-3=(t-1)^2-2(t-1)-3=(t^2-2t+1)-2(t-1)-3=t^2-4t.$$
This is the value for $y$ given in A), so A) is a correct parametrization. 
Now repeat for B), C), D).
The substitutions and checking may be faster if we use the factorization $x^2-2x-3=(x-3)(x+1)$. 
There are various shortcuts based on this idea. It is hard to know whether the time investment in looking for shortcuts is worthwhile. 
Remark: The bad one turns out to be C). For $(t-2)^2-2(t-2)-3=t^2-6t +5$. 
A: For this question just simply plug x and y into the equation for each set of parametric equations when you do not have equality on both sides then it cannot be a parametric equation. 
A: I think you've got the idea of the problem backwards based on your title.  The idea is not to take the equation in x and y and see if you can synthesize the other parametric versions, but rather to take the parametric versions, eliminate the parameter by substitution (solve for t in one equation, plug into the other), and see if you recover the original equation.
Let's try A.
$$x = t-1,\,\, y = t^2 -4t$$
$$t = x+1$$
$$y = (x+1)^2 -4(x+1)$$
$$y = x^2 +2x +1 -4x -4$$
$$y = x^2 -2x -3$$
That one is correct because we recover the original equation.  Let's try C.
$$x = t-2,\,\, y = t^2 -6t -5$$
$$t = x+2$$
$$y = (x+2)^2 -6(x+2)-5$$
$$y = x^2 +4x +4 -6x -12 -5$$
$$y = x^2 -2x -13$$
This one was not correct because it recovers a different equation.
B and D also work out to be correct by the same procedure.  So the answer is C.
An alternative procedure is to get the equation in terms of t only by substituting both x and y, and simplifying until the equality is either true or false.
