Remove["Global`*"]

Colormap names from reference.wolfram.com/language/guide/ColorSchemes.html

Included colormaps in order:

DefaultDensityGradient
SunsetColors
WatermelonColors
CherryTones
RustTones
Rainbow
TemperatureMap

grid = Table[i,{i,.1,.9,.1}];

c=Apply[List,Map[ColorData["M10DefaultDensityGradient"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,51}],{1,x},x]
Fit[Table[r[[i]],{i,51,91}],{1,x},x]
Fit[Table[r[[i]],{i,91,101}],{1,x},x]

Fit[Table[g[[i]],{i,1,101}],{1,x},x]

Fit[Table[b[[i]],{i,1,11}],{1,x},x]
Fit[Table[b[[i]],{i,11,51}],{1,x},x]
Fit[Table[b[[i]],{i,51,91}],{1,x},x]
Fit[Table[b[[i]],{i,91,101}],{1,x},x]

c=Apply[List,Map[ColorData["SunsetColors"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,33}],{x},x]
Fit[Table[r[[i]],{i,33,51}],{1,x},x]
Fit[Table[r[[i]],{i,51,101}],{1,x},x]

Fit[Table[g[[i]],{i,1,36}],{x},x]
Fit[Table[g[[i]],{i,36,83}],{1,x},x]
Fit[Table[g[[i]],{i,83,101}],{1,x},x]

Fit[Table[b[[i]],{i,1,18}],{x},x]
Fit[Table[b[[i]],{i,18,51}],{1,x},x]
Fit[Table[b[[i]],{i,51,68}],{1,x},x]
Fit[Table[b[[i]],{i,68,83}],{1,x},x]
Fit[Table[b[[i]],{i,83,101}],{1,x},x]

c=Apply[List,Map[ColorData["WatermelonColors"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,16}],{1,x},x]
Fit[Table[r[[i]],{i,16,71}],{1,x},x]
Fit[Table[r[[i]],{i,71,86}],{1,x},x]
Fit[Table[r[[i]],{i,86,101}],{1,x},x]

Fit[Table[g[[i]],{i,1,56}],{1,x,(x^2)},x]
Fit[Table[g[[i]],{i,56,73}],{1,x},x]
Fit[Table[g[[i]],{i,73,86}],{1,x},x]
Fit[Table[g[[i]],{i,86,101}],{1,x},x]

Fit[Table[b[[i]],{i,1,31}],{1,x},x]
Fit[Table[b[[i]],{i,31,56}],{1,x},x]
Fit[Table[b[[i]],{i,56,73}],{1,x},x]
Fit[Table[b[[i]],{i,73,86}],{1,x},x]
Fit[Table[b[[i]],{i,86,101}],{1,x},x]

c=Apply[List,Map[ColorData["CherryTones"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,15}],{1,x},x]
Fit[Table[r[[i]],{i,15,31}],{1,x},x]
Fit[Table[r[[i]],{i,31,43}],{1,x},x]
Fit[Table[r[[i]],{i,43,101}],{1,x},x]

Fit[Table[g[[i]],{i,1,16}],{1,x},x]
Fit[Table[g[[i]],{i,16,31}],{1,x},x]
Fit[Table[g[[i]],{i,31,101}],{1,x,(x^2)},x]

Fit[Table[b[[i]],{i,1,16}],{1,x},x]
Fit[Table[b[[i]],{i,16,29}],{1,x},x]
Fit[Table[b[[i]],{i,29,101}],{1,x,(x^2)},x]

c=Apply[List,Map[ColorData["RustTones"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,51}],{x},x]
Fit[Table[r[[i]],{i,51,101}],{1,x},x]

Fit[Table[g[[i]],{i,1,51}],{x},x]
Fit[Table[g[[i]],{i,51,101}],{1,x},x]

Fit[Table[b[[i]],{i,1,51}],{1,x},x]
Fit[Table[b[[i]],{i,51,101}],{1,x},x]

c=Apply[List,Map[ColorData["Rainbow"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,6}],{1,x},x]
Fit[Table[r[[i]],{i,6,101}],{1,x,(x^2),(x^3)},x]

Fit[Table[g[[i]],{i,1,8}],{1,x},x]
Fit[Table[g[[i]],{i,8,101}],{1,x,(x^2),(x^4)},x]

Fit[Table[b[[i]],{i,1,41}],{1,x,(x^2)},x]
Fit[Table[b[[i]],{i,41,56}],{1,x},x]
Fit[Table[b[[i]],{i,56,63}],{1,x},x]
Fit[Table[b[[i]],{i,63,101}],{1,x},x]

c=Apply[List,Map[ColorData["TemperatureMap"],Range[0,1,.01]],{1}];

r=Table[{(i-1)/100.,c[[i]][[1]]},{i,1,101}];
g=Table[{(i-1)/100.,c[[i]][[2]]},{i,1,101}];
b=Table[{(i-1)/100.,c[[i]][[3]]},{i,1,101}];

ListPlot[r,PlotStyle->Red,GridLines->{grid,None}]
ListPlot[g,PlotStyle->Green,GridLines->{grid,None}]
ListPlot[b,PlotStyle->Blue,GridLines->{grid,None}]

Fit[Table[r[[i]],{i,1,16}],{1,x},x]
Fit[Table[r[[i]],{i,16,36}],{1,x},x]
Fit[Table[r[[i]],{i,36,68}],{1,x,(x^2)},x]
Fit[Table[r[[i]],{i,68,101}],{1,x},x]

Fit[Table[g[[i]],{i,1,41}],{1,x},x]
Fit[Table[g[[i]],{i,41,51}],{1,x},x]
Fit[Table[g[[i]],{i,51,68}],{1,x},x]
Fit[Table[g[[i]],{i,68,101}],{1,x,(x^2)},x]

Fit[Table[b[[i]],{i,1,43}],{1,x},x]
Fit[Table[b[[i]],{i,43,51}],{1,x},x]
Fit[Table[b[[i]],{i,51,59}],{1,x},x]
Fit[Table[b[[i]],{i,59,68}],{1,x},x]
Fit[Table[b[[i]],{i,68,76}],{1,x},x]
Fit[Table[b[[i]],{i,76,101}],{1,x},x]