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> I wasn’t sure what type of math function would work for this
Usually people use "easing functions", which are convenient since they typically have a range and image both in [0.0 .. 1.0] [1] (although sometimes the range is slightly above 1 or below 0 to produce some "effect"). CSS has a cubic-bezier function [2] that can be used to replicate pretty closely all sorts of easing functions.
Functions that look like an "S" when plotted are called "sigmoids", a popular one used in graphics applications is called "Smoothstep" [3].
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1: https://easings.net
2: https://developer.mozilla.org/en-US/docs/Web/CSS/easing-func...
3: https://en.wikipedia.org/wiki/Smoothstep