Curve fitting problem

[JenniferMurphy]

I know this is technically not an Excel question, but there are a lot of people here with better math skills than mine, so I hope this is OK.

I would like to find a family of functions, f(x,m), that meets these conditions:

1. Maps the domain (0,10) onto the same range and is continuous over that domain.
2. f(0) = 0
3. f(10) = 10
4. f(5) = m; 0<m<10
5. f(x) is monotonically increasing. This means that it cannot be greater than 10 or less than 0.
6. If m=5, the function is y=x (straight line)

I thought I had it with quadratics (parabolas), but when m gets close to 0 or 10, the function exceeds the range. Here are the results of the parabolas:

https://www.dropbox.com/sh/hwboo6bb6i40451/AADDz82Ev1oEYycC-0QRNSXBa?dl=0

This is more or less what I want, but with the functions staying within the range.

 

[StephenCrump]

One way you might want to do this:

For m<=5:
f(x) = x^k / 10^k-1
where k = (ln(10) - ln(m)) / (ln(10) - ln(5))

For m>5:
f(x) = 10 - (10-x)^k / 10^k-1
Where k = (ln(10) - ln(10-m)) / (ln(10) - ln(5))