How Do You Convert Between Parametric and Rectangular Equations?

[LongTermStudent]

I have an upcoming exam, and I'm having trouble grasping some concepts. The things that are currently perplexing me are parametric equations and rectangular equations and converting between the two. I have a problem like this

Given the parametric equations x = e^(-t) + 1 and y = e^(-2t) - 3, find the corresponding rectangular equation and sketch the curve of orientation.

To eliminate the parameter, I set the x equation equal to t as such:
t = -ln(x-1)

and then substituted this value into the y equation to get:
y = e^[ln(x - 1)] - 3
and got y = x - 4 as my final equation, but I'm not so confident in this answer.

Then I'm told to find a set of parametric equatiions for the rectangular equation (x - 2)² + y² = 9

I set x equal to t and soved for y and ended up with
y = sqrt(t²-4t+13)

This seems like only half the answer to me, but I barely know where to start, let alone how to continue. I'm sure I'll be kicking myself when I finally figure these concepts out, but I can't for the life of me wrap my head around any of it, and I have no access to a tutor between now and my exam. Any help/guidance is appreciated.

Edit: I see now I may have posted this in the wrong forum. My bad.

 

[bob1182006]

for the first one, your substitution was wrong, you should get e^(-2t) = e^(-2(-ln(x-1))) which doesn't go down to just (x-1)

for the second one I think you get down to y^2=t^2+4t+13 right? then you square root both sides? you should get y=+ or - sqrt(...) since both +sqrt(...) and -sqrt(...) will satisfy that equation.