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I would like to study the sign of this function (potential) and show the region (positive-negative)

enter image description here

These are the parameters followed by the code (left figure)

a = 1; c = 1;
b = -1; d = 1;
p = 1; q = -1;   
f1= (Exp[-2 r a] p)/(2 r c) + (Exp[-2 r b] p)/(2 r  d)
f2= ( Exp[-2 r a] q)/(2 r  c) + (Exp[-2 r b] q)/(2 r d)

Plot[{ f1,f2 } , {r, -2, 2}, PlotRange -> Automatic]

As we see for this case, we obtain different signs for f1 and f2. In four different regions I would like to show them in the plot, A legend it possible only for two regions?

For example, If we change b and d with the same sign (d=-1 and b=-1) (right figure) have only two regions. Please any suggestion and recommendation will be usefull

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  • $\begingroup$ Do you mean something like Plot[{f1, f2 } , {r, -2, 2}, ColorFunctionScaling -> False, ColorFunction -> (If[#2 > 0, Blue, Red]&)]? $\endgroup$ – MarcoB Nov 25 '20 at 6:47
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a = 1; c = 1;
b = -1; d = 1;
p = 1; q = -1;
f1 = (Exp[-2 r a] p)/(2 r c) + (Exp[-2 r b] p)/(2 r d);
f2 = (Exp[-2 r a] q)/(2 r c) + (Exp[-2 r b] q)/(2 r d);
a = 1; c = 1;
b = -1; d = 1;
p = 1; q = -1;
f1 = (Exp[-2 r a] p)/(2 r c) + (Exp[-2 r b] p)/(2 r d);
f2 = (Exp[-2 r a] q)/(2 r c) + (Exp[-2 r b] q)/(2 r d);
Plot[{f1, f2}, {r, -2, 2}, PlotRange -> Automatic, 
 MeshFunctions -> {#1 &,#1*#2 &}, 
 MeshShading -> {{Red, Green}, {Blue, Yellow}}, Mesh -> {{0}}]

enter image description here

Since the sign of $x$, $x y$ can distinguish the four quadrant, so this method can also be used to ParametricPlot who cross the four quadrant.

ParametricPlot[{t/4 Cos[t], t/4 Sin[t]}, {t, 0, 20}, 
 MeshFunctions -> {#1 &, #1*#2 &}, 
 MeshShading -> {{Red, Green}, {Blue, Yellow}}, Mesh -> {{0}}]

enter image description here

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Clear["Global`*"]

a = 1; c = 1;
b = -1; d = 1;
p = 1; q = -1;
f1 = (Exp[-2 r a] p)/(2 r c) + (Exp[-2 r b] p)/(2 r d);
f2 = (Exp[-2 r a] q)/(2 r c) + (Exp[-2 r b] q)/(2 r d);

Plot[Evaluate@Outer[
   Tooltip@ConditionalExpression[#2, #1] &,
   {r > 0, r < 0}, {f1, f2}], {r, -2, 2},
 PlotLegends ->
  {"f1;\[ThinSpace]r\[ThinSpace]>\[ThinSpace]0", 
   "f2;\[ThinSpace]r\[ThinSpace]>\[ThinSpace]0", 
   "f1;\[ThinSpace]r\[ThinSpace]<\[ThinSpace]0", 
   "f2;\[ThinSpace]r\[ThinSpace]<\[ThinSpace]0"}]

enter image description here

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