How to find function equation from a given asymptote?

I want to build a $i$ function with the following 4 parameters:

• $a$ is the left inflexion point
• $b$ is the right inflexion point
• $s$ is the maximum of the function
• $k$ is the asymptote value ($y=k$) of the function

So far, I have managed to achieve this goal with the following function:

$$f(x)=(a - x) (x - b)$$

$$g(x)=\alpha f(x)\ ;\ \alpha=\frac{4s}{(a - b)²}$$

$$h(x)=\beta exp(\ g(x)\ )\ ;\ \beta=\frac{s}{ℯ^s}$$

$$i(x)=h(x)-k$$

I'm struggling for the last parameter $k$. As I wrote $i(x)$, what I obtain is a translation of $h$.

What I want is to set $y=k$ as the asymptote of $h$ without decreasing its maximum.

How can I do that?

It looks to me as if you're missing a minus in the exponential term, but assuming for the moment that $h(x)$ ranges from $0$ to $s$, and you want it to range from $k$ to $s$ (you didn't mention this, but you need $k < s$) you can do this:
• multiply by $s-k$. Now it ranges from $0$ to $(s-k)$.
• Add $k$. Now it ranges from $k$ to $s$.
In short: define $$i(x) = k + h(x) \frac{s-k}{s}.$$