Determine scalability between two numbers

Question

This may seem like basic math, but I'm trying to determine the relationship between the following sets of numbers to understand how they scale. I would like to write an equation to output y when given x, as follows:

when x = 1366, y = 4.67

when x = 1700, y = 5.0

when x = 2106.45, y = 5.33

I would like an equation that generates the y above when given x, in the context of the relationship between these numbers. I'm stumped.

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I explored logarithmic functions but wasn't able to come up with anything, and the only constant I was able to solve for was 1.16237188872621, which appears to be the factor by which the ratio between x and y here increments by... e.g.

1366 / 4.67 = 292.5053533191
1700 / 5.0 = 340
2106.45 / 5.33 = 395.2063789869

292.5053533191 * 1.16237188872621 = 340
340 * 1.16237188872621 = 395.2063789869

...and so on. Thanks for any help with this!

@MaximilianJanisch Can you explain how you got to these numbers (based on my numbers above)? — Zachary Beschler, Jan 02 '20 at 17:03
Sure, I used the following command in Mathematica: FindFit[{{1366,4.67},{1700,5.0},{2106.45,5.33}}, a Log[Abs[b] x],{a,b},x] For more information, have a look here: https://reference.wolfram.com/language/ref/FindFit.html — Maximilian Janisch, Jan 02 '20 at 17:08
By the way, there is also curve fitting in Python: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html — Maximilian Janisch, Jan 02 '20 at 22:38

Michael Rozenberg · Answer 1 · 2020-01-01T13:05:53.687

1

For example: $$y=\frac{(x-1700)(x-2106.45)}{(1366-1700)(1366-2106.45)}\cdot4.67+\frac{(x-1366)(x-2106.45)}{(1700-1366)(1700-2106.45)}\cdot5.0+$$ $$+\frac{(x-1366)(x-1700)}{(2106.45-1366)(2106.45-1700)}\cdot5.33.$$

edited Jan 01 '20 at 13:05

answered Jan 01 '20 at 12:51

Michael Rozenberg

194,933

Forgive me, but how can I use this in to output y when I have x? For example, if x = 3000, what would y equal? – Zachary Beschler Jan 01 '20 at 12:55
@Zachary Beschler Just substitute! I am ready to write a full expression for the function. – Michael Rozenberg Jan 01 '20 at 12:57
My math skills are pretty elementary, apologies, I'm trying to keep up, but—substitute what exactly? – Zachary Beschler Jan 01 '20 at 12:59
@Zachary Beschler I wrote a full expression. Now, substitute $x=1366.$ What did you get? – Michael Rozenberg Jan 01 '20 at 13:01
It worked! Is there a way to simplify this to only use one set of numbers?
e.g.:

when x = 1366, y = 4.67

and

when x = 1700, y = 5.0
– Zachary Beschler Jan 01 '20 at 13:16
It should be noted that, even though this function fits perfectly onto the three given points, it is not clear if the function has the desired global properties – Maximilian Janisch Jan 01 '20 at 13:19
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@Zachary Beschler It works for any $n$ numbers. It names Lagrange polynomial. – Michael Rozenberg Jan 01 '20 at 13:20

Determine scalability between two numbers

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