Let's say I have problem and I want to incentivize people to give me the solution as soon as possible. If you hand me the next day you will get full 100 points. The more days it takes the lower the score. However since it's better late than never, so the score is never zero. Basically I'm looking for a function that graphs a decreasing positive line asymptoting the horizontal line. My first thought was $y = \frac{100}{x}$ or a log function, but since I acknowledge that the solution is not easy, I don't want to punish you so hard if you give me the answer at day 2, or 3, or even a week later. Those types of function have the initial interval so steep. I prefer the $Δy$ between $x-7$ and $x+7$ is not over $10$.
Is there a way to design a good function for this problem?
Working with a function of the form $$y = a+(bx+c)e^{-x/d}.$$ gives you a much more gentle slope near zero.
For example,