Drawing the lattice curves of a transformation

[ElijahRockers]

Homework Statement

Draw the lattice lines for the following transformation.

$x=se^t$
$y=se^-t$

The Attempt at a Solution

Trying to draw this transformation. So first thing I did was let t be constant. I then used substitution to show that $x=ye^{2c}$. This is the set of all lines (as C is changed) with a positive slope that go through the origin, if I am not mistaken, which is the transform of the 'S' lattice of the grid.

For the next step I let S be constant, and used substitution to get what I think should be...
$x=Ce^{-\frac{lny}{lnC}}$, and this would be the transform of the 't' lattice, if I did that right.

What am I looking at here?

I used wolframalpha to graph several instances of this, with C = 1,2 and 3, and seemed to give me wildly different graphs each time. Did I do that right? It seems like C=1,2 are hyperbolas with their asymptotes along the x and y axes, but C = 3 gives me something I've never seen before. Sort of like a hyperbola, but one of the sides has been flipped into a different quadrant.

Any push in the right direction would be helpful, thanks.

EDIT: I just realized that C can't be 1 right? because ln C would be 0, and that would make the denominator 0 in the exponent part of the equation...? Also I tried to graph it with my TI-84, but it only gave me the portion of the graph in the first quadrant. Why is that?

EDIT: Also, my graphing calculator gets funky when I graph this equation for C=1/4

 

[lippyka]

. It draws the hyperbola but then it cuts off after a certain point, where it should keep going. Why is this happening?