I think I phrased my question poorly.
I want to plot $y$ at any point $x$ on a curve across any given number of points $n$ greater than 0 using growth $b$ and I want to adjust the scale and base of the curve by specifying the minimum $y_1$ and maximum $y_n$.
I believe this is how to write the formula:
$y=y_1+\frac{(y_1b^{x-1}-y_1)(y_n-y_1)}{y_1b^{n-1}-y_1}$
In Excel you would use the following formula. I'm using named ranges in the formula to make it easier to follow.
=Minimum+(((Minimum*Growth^(Position-1))-Minimum)*(Maximum-Minimum)/((Minimum*Growth^(Periods-1))-Minimum))
Here's an example result set that describes a steep curve:
$n = 10$
$b = 2$
$(x_1, y_1) = (1, 10000)$
$(x_n, y_n) = (n, 30000)$
Gives:
$(x_1, y_1) = (1, 10000)$
$(x_2, y_2) = (2, 10039)$
$(x_3, y_3) = (3, 10117)$
$(x_4, y_4) = (4, 10274)$
$(x_5, y_5) = (5, 10587)$
$(x_6, y_6) = (6, 11213)$
$(x_7, y_7) = (7, 12466)$
$(x_8, y_8) = (8, 14971)$
$(x_9, y_9) = (9, 19980)$
$(x_{10}, y_{10}) = (10, 30000)$
Here's another example result set that describes a shallow curve:
$n = 10$
$b = 1.1$
$(x_1, y_1) = (1, 10000)$
$(x_n, y_n) = (n, 30000)$
Gives:
$(x_1, y_1) = (1, 10000)$
$(x_2, y_2) = (2, 11473)$
$(x_3, y_3) = (3, 13093)$
$(x_4, y_4) = (4, 14875)$
$(x_5, y_5) = (5, 16835)$
$(x_6, y_6) = (6, 18992)$
$(x_7, y_7) = (7, 21364)$
$(x_8, y_8) = (8, 23973)$
$(x_9, y_9) = (9, 26843)$
$(x_{10}, y_{10}) = (10, 30000)$
Hopefully someone else will find this useful. Maybe someone can even find a way to simplify it.
Thanks!