Find a family of curves

Question

I have a diagram with seven curves (each with a different parameter). My goal is to find a function for the family of curves (i.e. a single function not just for the seven parameters, but for all parameters in between).

The values are between 0.3 and 2.0. I manually got the values in 0.1 steps.

These are the points I got:

f50 = {{0.3, 360}, {0.4, 315}, {0.5, 280}, {0.6, 250}, {0.7, 225}, {0.8, 200}, 
{0.9, 180}, {1.0, 165}, {1.1, 150}, {1.2, 135}, {1.3, 125}, {1.4, 115}, 
{1.5, 105}, {1.6, 97}, {1.7, 93}, {1.8, 88}, {1.9, 85}, {2.0, 84}}

f100 = {{0.3, 426}, {0.4, 382}, {0.5, 347}, {0.6, 316}, {0.7, 291}, {0.8, 268},
{0.9, 249}, {1., 231}, {1.1, 216}, {1.2, 203}, {1.3,191}, {1.4, 182},
{1.5, 172}, {1.6, 165}, {1.7, 160}, {1.8, 156}, {1.9, 153}, {2., 152}}

f150 = {{0.3, 600}, {0.4, 547}, {0.5, 497}, {0.6, 454}, {0.7, 415},
{0.8, 380}, {0.9, 352}, {1., 326}, {1.1, 305}, {1.2, 287}, {1.3, 272},
{1.4, 260}, {1.5, 250}, {1.6, 242}, {1.7, 236},{1.8, 232}, {1.9, 230},
{2., 229}}

f200 = {{0.3, 750}, {0.4, 683}, {0.5, 625}, {0.6, 574}, {0.7, 530}, {0.8, 490},
{0.9, 456}, {1., 426}, {1.1, 400}, {1.2, 380}, {1.3, 358}, {1.4, 343},
{1.5, 330}, {1.6, 320}, {1.7, 310}, {1.8, 305}, {1.9, 302}, {2., 300}}

f250 = {{0.4, 900}, {0.5, 814}, {0.6, 740}, {0.7, 677}, {0.8, 622}, {0.9, 575},
{1., 535}, {1.1, 500}, {1.2, 471}, {1.3, 446}, {1.4, 425}, {1.5, 408},
{1.6, 395}, {1.7, 386}, {1.8, 380}, {1.9, 374}, {2., 372}}

f275 = {{0.7, 870}, {0.8, 785}, {0.9, 707}, {1., 642}, {1.1, 587}, {1.2, 542},
{1.3, 506}, {1.4, 476}, {1.5, 453}, {1.6, 435}, {1.7, 422}, {1.8, 413},
{1.9, 408}, {2., 405}}

f300 = {{0.9, 915}, {1., 827}, {1.1, 741}, {1.2, 671}, {1.3, 616}, {1.4, 573},
{1.5, 540}, {1.6, 510}, {1.7, 487}, {1.8, 467}, {1.9, 453}, {2., 440}}

I found several functions, that fit one particular curve quite well. What is the next step to find a function for the family of curves for values of x between 0.3 and 2.0?

I didn’t find anything online for this problem, but I admit, that I may not know the appropriate words for a search.

These are the seven curves I plotted and have the functions for. The parameters are from bottom to top:

50, 100, 150, 200, 250, 275, 300

How do I find a function with the parameter?

Answer 1

If you wish to find a formula for a set of data, you could use the FindFormulafunction:

values = {{0.3, 360}, {0.4, 315}, {0.5, 280}, {0.6, 250}, 
          {0.7, 225}, {0.8, 200}, {0.9, 180}, {1.0, 165}, 
          {1.1, 150}, {1.2, 135}, {1.3, 125}, {1.4, 115}, 
          {1.5, 105}, {1.6, 97}, {1.7, 93}, {1.8, 88}, 
          {1.9, 85}, {2.0, 84}};
FindFormula[values, x]

(* -560.276969594193` + (726.9849713264147`/(x^0.2`)) *)

Plot[-560.277 + 726.985/x^0.2, {x, -7.35603, 7.35603}]

EDIT

The question has changed and requires some adjustments.

Because of the ~abstract~ nature of the question, I figured this might be a fun time for some machine learning.

First, I'm going to borrow @Xavier's parametric fit from the comments (thanks!), while assigning the inputs to the results:

trainingSet = 
 Flatten[{#[[1]] -> {k, a, b, c} /. 
      FindFit[#[[2]], k a x^2 + k b x + k*c, {k, a, b, c}, x]}
       & /@ {{50, f50}, {100, f100}, {150, f150}, {200, f200}, 
             {250, f250}, {275, f275}, {300, f300}}]

(*{50 -> {5.22054, 20.7143, -76.5871, 87.5835}, 
   100 -> {8.18377, 13.2549, -48.8544, 64.0041}, 
   150 -> {9.83363, 16.7709, -59.5746, 76.3585}, 
   200 -> {14.5865, 12.838, -46.6688, 63.2554}, 
   250 -> {13.6602, 18.8301, -67.8033, 88.5712}, 
   275 -> {10.4912, 33.4334, -122.605, 151.243}, 
   300 -> {15.6073, 26.3044, -102.673, 129.094}} *)

Then we feed this into a predictor function:

pf = Predict[trainingSet, Method -> "NeuralNetwork", 
       PerformanceGoal -> "Quality"];

Now we can explore what these relationships look like:

Manipulate[
 TableForm@{k a x^2 + k b x + k*c, cf[{k, a, b, c}], 
   Plot[(k a x^2 + k b x + k*c ), {x, 0, 2}]}, {k, 5, 15}, {a, 11, 35}, 
    {b, -102, -40}, {c, 60, 150}]