Please See "Clarification" for more clarity
I found a curve that I've been looking for; well sort of: The curve, shown in my fig., and Graphed Here, matches certain criteria: 1. Overall shape. 2. Height. 3. Overall Proportions (at least to a good approximation). 4. Position. etc....
Nevertheless, my curve seems arbitrary; I had to force it to do this. So, I was wondering if anyone knows of or can derive a similar curve that matches the above without requiring all kinds of constants to move it around and scale it (you CAN use the golden ratio constant, however). Basically, I want a proper function for this, not something forced. Thank you!
If it's any help, $y(t)=\frac{1-φ^{-1+t}}{1-φ}+φ^{-1}$, or something extremely similar should be able to be used as the partial solution (namely the solution for $y(t)$) in a parameterization.
Clarification:
As you can see, my equations give a very specific curve (which is virtually identical to the one I want in appearance). What I'm looking for is something that is 'naturally' almost identical to my curve in every way (really I mean every way).
A good analogy is this: Imagine if you'd never seen the equation $y=1/x$ but you had an image of it and a similar curve; you could probably modify the similar curve until it approximated the properties of $y=1/x$. The modified similar curve is to $y=1/x$ as my curve is to what I'm looking for.
