I am recovering from an injury and I go for a run each morning. The distance that I run does not change but the time is getting shorter each time, i.e. I am improving. I am recording my times and would like to fit an exponential of the form $y(x)=A+Be^{Cx}$ to the data such that I can predict my future minimum time for that journey when I am at full fitness. i.e. I want to know the $A$ value. $y=Time(min)$ and $x=Days$.
Here is my data: $(Day,Time)={(0,78),(1,76),(2,76),(3,70),(4,68),(5,69),(6,66),(7,66),(8,65),(9,66),(10,64),(11,64),(12,63)}$
When I plot these data there does seem to be an exponential improvement forming. But I haven't done it long enough yet to reach my plateauing time.
My calculator has an exponential function but it assumes an $A=0$, which cannot be true. How do I estimate or calculate my $A$ value?
There are many places on this site where you can see the problem of the fit of exponential functions like $$y=a+b\, e^{cx}$$ Just have a look here.
Using that, you should find (using your data) $$y=61.7695 +17.0408 e^{-0.196822 x}$$ corresponding to $R^2=0.999742$.
The key problem is that you need "reasonable" estimates. You could have obtained them in a first step using $a\approx 60$ and then consider $$y-60=b \,e^{cx} \implies \log(y-60)=\log(b)+cx\implies z=d+ cx$$