How to invert a simple exponential growth formula

Question

I think this is simple but my math skills are limited. I have a basic exponential growth formula: $$y=x \cdot (1-p)^n$$ and I have $y$ and $x$ and $n$ values and I need value of $p$. Then when I solve for $p$, I have to calculate $y$ with different values of $n$. It's easy for me to do that but than I get $y$ values that decreases fast and than slow like on this graph:

But I want to decrease slower and than faster like on this graph

How to change the beginning formula to get what I said?

Answer 1

From your equation and graph I believe that your independent variable (the one on the horizontal axis) is $n$, and $0<p<1$.

One way to get the kind of graph you want is $$y=b-(1+p)^n$$ where $b$ is the $y$-intercept of the horizontal asymptote above the graph. You will probably need to shift the graph to the right, as in $$y=b-(1+p)^{n-a}$$ where $a$ is the amount of shift to the right.

Answer 2

If one starts with $f(x) = 2^x$

Step 1: Reflect across the $x$ axis: $-f(x)$

Step 3: Translate by required amount $T_{-a, b}$: $-f(-x - a) + b$