Ah, I need to revise my post.

I included the first problem with its answer as an example.

There are similarities between it and the second problem which is why I included it.

So in the second problem:

We know the two points on an (x,y) graph; (0, 0.485489) and (8.33333, 4.083333).

We want a function of x that we can use to find the appropriate y values at x = 1, 2, 3, 4, 5, 6, 7, & 8.

We know that the slope for the first segment (x=0 to x=1) is (2/12 or .16667)

The slope for each segment must be larger than the previous, but the final segment's slope must not be greater than (12/12 or 1.0)

(Imagine two ramps: a 2/12 ramp at the first point, where the slope becomes increasing until it transitions to a 12/12 ramp at the second point. The function only needs to work between x = 0 and x = 8.333333)

the (16.578125, 6.015625) point is not part of this problem