I'm trying to transform a set of ranks (eg. $1^{\text{st}}-1000^{\text{th}}$) into a set of scores (eg. $ 10\% - 100\% $) by way of something like a Logit function (I believe this is related to curve fitting for grades). My problem is working out how to change the shape of the curve via some parameter. Here is what I have so far:
Looking at the wikipedia page on the Logistic function and inverting the formula gives a general Logit function:
$f(x)=y_{0} + \frac{1}{k}\ln{(\frac{x}{L-x})}$
Where $y_{0}$ is the midpoint of the curve on the $y$-axis, $k$ determines the slope of the curve, and $L$ is the curve's maximum value on the x-axis.
Since I know my outputted scores will be between $10\%$ and $100\%$, $y_{0}=55$, and the maximum rank is $1000$, so $L=1000$, giving:
$f(x)=55 + \frac{1}{k}\ln{(\frac{x}{1000-x})}$
This plot for $k=5$ in Wolfram Alpha looks fairly good , but if I change the slope by changing $k$ to $20$, this plot indicates that a rank of $1-20^{\text{th}}$ all result in a score of $0\%$, which isn't ideal.
This is where I'm stuck - how can I change the shape of the curve while still having it pass through $(1^{\text{st}},10\%)$ and $(1000^{\text{th}},100\%)$?