An exponential function to calculate price depending on availability with min/max constraints? I am developing an application, the application will help users book parking spots in a mall. The price of the parking spot will be correlated with the availability of the parking spots of the mall, the prices should increase exponentially.
I was looking to write a mathematical function to calculate the price of the parking spot depending on the availability. There are 3 constants that a mall can enter. the minimum price $M$, the maximum price $M_x$ and the rate of increase $r$ (this will allow the mall to decide if they want the prices to increase dramatically at an exponential rate, or more linearly). The availability will be calculated automatically (preferably as a percentage but not necessarily) and the function should return the price. What would be the best function to use?
My Sketch of graph (showing three possible curves depending on $r$ value

 A: You are looking for the following function:
$$f_1(M,M_x,r,A) = M + (M_x-M)A^{1/r}$$
where I denoted the Availability with $A$, and $A$ ranges from $0$ to $1$, with $A=0$ meaning that the parking lot is empty, and $A=1$ meaning that the parking lot is full.
Here is what the function looks like for various values of $r$. Note: Here I simply set $M=\$10$ and $M_x=\$50$, but these can obviously be changed to different values.

This way, $r$ has to always be $>0$, but if you'd like to use negative values too, then you can try a modification of this function, with $\frac{1}{r}$ replaced with $e^{-r}$:
$$f_2(M,M_x,r,A) = M + (M_x-M)A^{e^{-r}}$$
This results in a different scaling:

Of course, in both cases you can introduce a scaling factor $c$ to make the changes less dramatic $(0 < c < 1)$, or to make the changes more dramatic $(c > 1)$:
$$f_1^{\text{scaled}}(M,M_x,r,A,c) = M + (M_x-M)A^{1/(cr)}$$
$$f_2^{\text{scaled}}(M,M_x,r,A,c) = M + (M_x-M)A^{e^{-(cr)}}$$
Feel free to experiement with more functions and more scaling methods. For example you can use a different base for $f_2^{\text{scaled}}$, or multiply $e^{-(cr)}$ with yet another scaling factor, $b$, to once again make the changes less/more dramatic: $be^{-(cr)}$.
A: You may want to try $$Price = M_{min}+c(Availability)^z.$$ where c, z are constants.
You can modify c accordingly to have Price = $M_{max}$ at maximum value of the variable ’Availability’.
 Also, a straight line occurs for exactly $z=0$, The curve below the straight line (exponential increase) occurs for z$\gt$1, and the “slow increase” graph above the straight line happens for $0<z<1$, Say $z=\frac12$.
EDIT: Based on your later comment, I would like to add that having r as negative will decrease the price as the availability goes up.
REMINDER AND DISCLAIMER: This is only a suggestion and not professional advice.
