What function should be used to build such a graph?

Question

Mathematics

There is a graph with X and Y axes. It is
necessary to build such a curve, where with each step of X, the value of Y will increase with gradual decay (inverse exponential or something like that). If only 10 points are allowed along the X axis, then at the last point Y should also be equal to 10 or so. In short, you should get an upward curved arc that starts at Y=1 and ends at Y=10.
Here is an example chart:

Answer 1

There are many options. For the time being, we consider that there are no requirements for f(0). Let's rescale our variables:
x' = (x − 1)/9
y = 8y' + 2.
Then x' and y' will be from 0 to 1, while x and y are 1…10 and 2…10. And then the options.
1. Power function: y' = x' ^a , a = 0…1. If a \u003d ½, then the square root, ⅓ is the cube root ...
2. Quarter of the ellipse: y' = sqrt(1 - x'²)
3. Sine: y' = sin((pi/2) x')
If f(0) = 0, then scale differently:
x' = x/10
y = 10y'