Approximating alternator output I've got this graph of the ampere output of an alternator:

or as a table read off the graph:




RPM
Ampere output




1450
42


2000
97


2500
120


3000
133


3500
143


4000
149


4500
152


5000
156


5500
159


6000
162


7000
164


8000
166




which I need to approximate using a formula, $x$ being the alternator speed in RPM (rotations per minute) and $y$ being it's output in amperes.
I've tried using the various logarithmic, square root and cubic root functions, but none of the graphs produced by those came close to the graph above.
 A: Flipped upside down, the graph strongly resembles one of an exponentially decaying function. I therefore tried fitting to an exponential function, optimizing the fit according to the minimax criterion. This results in
$$y \approx \frac{7457}{16} \left(1 - \exp \left(-\frac{63}{65536} x\right)\right) -\frac{2435}{8}$$
which has a maximum error of about $\pm3.525$. Because of similarity in shape, I also tried minimax fits to the sigmoid function and $\tanh$, but the resulting maximum error was slightly larger, a bit larger than $\pm4$.
A: Sometimes to get a good fit you need to apply a transformation to the data. I tried shifting the data horizontally to the left by 1350 amps. This is accomplished by subtracting 1350 from each of your RPM values. I was able get a pretty good logarithmic fit with a R^2 of 0.9899. The 1350 was found just by trial and error, you might find there is a different value which is more optimal.
My fit equation is,
$$Y = 30.811 \ln(X-1350) - 98.15, $$
where $Y$ is the number of amps of current and $X$ is the number of rpm's from the alternator.
