Formula to scale a series that is being bent with a root / power.

I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference number and each number in the series.

Ox[n] = Abs(Sx[n] - Rx) ^ 0.5


Besides being able to control how aggressively the curve varies from linear (by changing the root), I want to be able to define what Ox[n] should be when Sx[n] is a certain % of Rx:

e.g. when Sx[n] = 0.5 Rx, then Ox[n] = 0.1 Rx


To be clear, I want to be able to keep the root index in order to control the shape of the curve, but also be able to influence the values of the output so that at a given input % of Rx, I have a pre-desired output. This would be a question of scaling.

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I figured it out:

$$S=\frac{1-Ref_{tgtPer}}{\sqrt[r]{(1-Ref_{floorPer})Ref_{val}}}$$

where

$S =$ Scale

$r =$ the index

$Ref_{tgtPer}=$ The target % of the Reference (Rx)

$Ref_{floorPer}=$ The desired value of the output at $Ref_{tgtPer}$ of Rx

$Ref_{val}=$ The reference value (Rx)

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