Undergrad Math for Blending Numbers: Find "X

[fifthFunction]

TL;DR

blend 2 numbers by inverse square

i have not clue if this is the right place to ask
if i had 2 numbers and i wanted to blend between them but instead of a linear way it was in an inverse square way.. how would that math go?
so if i had A=1 and B=9 and wanted the number at 0.5 it would be 4.. or if i wanted the number at 0.85 it would be "X"

 

[mfb]

It depends on what exactly you need. For a parabola you need three points (or some other additional information), with just two points it is ambiguous.

As an example, f(x)=(2x+1)² satisfies f(0)=1 and f(1)=9 and it matches f(0.5)=4, but g(x)=(x+3.5)²-11.25 fits your two borders as well, it would give g(0.5)=4.75.

Anyway: If you want a parabola, find the equation for the parabola you want, then plug in different numbers.

 

[fifthFunction]

i think the most simple way to explain what i am trying to do is to map a section of a gradient of the falloff of light?
so say you have a strip of paper with a light at one end and you measure how bright one side is and how bright the other side is.. you input that into the function and it would tell you how bright the middle of the paper is

 

[mfb]

That depends on where your light source is and how its emission looks like. For a uniform emission you get an inverse square law for the intensity, modified with the incidence angle if not orthogonal. Just knowing the intensity at both sides is not sufficient.

 

[fifthFunction]

ok so it seems if i do something like A=sqrt(1),B=sqrt(9),P=0.5 then just feed that into a linear blend pow(mix(A,B,P),2) it works.. don't know if its the best way tho