I am trying to find a scaling function that mimics the gas gauge in a car. I would like to map a value to a $0$ to $10$ scale, on which I have two known points. For example:
$X_1 = 2$, $Y_1 = 2.5$, $X_2 = 4$, $Y_2 = 7.5$, $X_3 =$ __
The known points, $X_1$ and $X_2$ should always correspond to $Y$ values of $2.5$ and $7.5.$ A simple linear equation is not sufficient, since its possible for the $X_3$ value to generate a $Y$ that is greater than $10$, or less than $0.$
Can you think of a suitable function that matches the known points, but will approach a limit at $0$ and $10$? Thanks for any guidance!