I have a series of numbers which I'd like to express as a formula.
$$22000,
25000,
25500,
25700,
26100,
26500,
26800,
27100,
27400,
27800,
28300,
28800,
29400,
30000,
30700,
31600,
32600,
34000,
35500,
37500,
40000,
43000,
46000,
49000,
53000$$
A plot of these numbers looks like this (x-axis being the term number, y-axis being the value):
I can handle the blip at the beginning separately. I've guessed that the remainder looks a bit like a quadratic curve, but my last maths class was over 30 years ago. I've been using a spreadsheet and trial-and-error to construct a quadratic equation which has come close, but isn't perfect. $$y=a(rx)^2+brx+c$$ where $r=4.5, a=1.9, b=-8.1, c=24600$
Is there a better approach I could use (bearing in mind I'm not a mathematician, so it would have to be pretty simple)?
Background: the numbers represent voltages required in a circuit ($V=3.3n/65536$) so that each successive term brings about a just-audible change in the volume of a signal passing through the circuit. Accuracy is more important near the start of the series where the term values are closer together, but some leeway in the conversion from x to y is probably acceptable.
Looking at the graph, I had the feeling that a cubic polynomial would probably be better than a quadratic. So, let $$y=a x^3+b x^2+c x+d$$ and a basic polynomial regression gives $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 4.8683 & 0.26028 & \{4.32538,5.41123\} \\ b & -128.34 & 10.2821 & \{-149.788,-106.892\} \\ c & 1421.46 & 116.246 & \{1178.97,1663.94\} \\ d & 21687.7 & 355.952 & \{20945.2,22430.2\} \\ \end{array}$$ and $R^2=0.999896$ is quite good.