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I have an alpha value with a range of 0.0 - 1.0 as well as a value N. If the alpha value is 0.5, I want to make no change to N. If the alpha value is 0, I want to double N. If the alpha value is 1.0 I want to halve N.

I want a clean formula to represent these values as well as all values in between. 0.25 should be 1.5N, and 0.75 should be 0.75N.

It seems really simple, but I'm having difficult wrapping my head around it for some reason.

Vadoff
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    Something like $2^{1-2\alpha}N$ will do what you want for the first three values of $\alpha$. I'm not sure that you will be able to get something that does what you want at all five of these values (without forcing it - e.g. a polynomial of degree 5 should work but may not do what you want outside these 5 values). – Sam Weatherhog Sep 23 '15 at 22:07
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    $2N\over4^\alpha$ is simple, but does not quite fit. ${N\over4}(|2\alpha-1| - 6\alpha + 7)$ fits, but is not quite simple. That's a dilemma to think about... – Ivan Neretin Sep 23 '15 at 22:07
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    Your problem is your Quixotic insistance on mixing arithmetic and geometric behavior. You want $0$, $1/2$, and $1$ to behave geometrically, but $1/4$, $1/2$, and $3/4$ to behave arithmetically. This can be done, but not in a simple canonical fashion so there is an obvious choice. I recommend that you give up forcing arithmetic behavior on the quarters and go with Sam Weatherhog's formula. – Paul Sinclair Sep 23 '15 at 22:23

1 Answers1

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From the data:

enter image description here

$x$-axis is $\alpha$ and $y$-axis is the multiplier of $N$.

Result: $$N\left(0.5|x-0.5|-1.5x+1.75\right).$$

f10w
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