Non-symmetric bump function

I am trying to create a smooth bump function $f(x;a,b,c,k_1,k_2)$ satisfying the following properties:

• $f$ is smooth (can be relaxed)
• $f$ is supported on $[a,c]$.
• $f$ is increasing on $(a,b)$.
• $f$ is decreasing on $(b,c)$.
• $f(b) = 1$.
• $k_1$ and $k_2$ are parameters that control the decay rates on the left and right side of $b$ (I know this is a bit vague but hopefully it makes sense).

For simplicity, it can be assumed that $a=0, c=1$ and $0<b<1$. Essentially, I want it to look like a non-symmetric version of the standard bump function $\mathbb{I}_{[0,1]}\exp(\frac{1}{x^2-1})$, where the location of the peak can be controlled. If anyone has any ideas, it would be greatly appreciated.