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I am trying to evaluate the properties of a multivariate function $f(a,b,x,y)$ with respect to $x$ and $y$ given assumptions for $a$ and $b$.

An example without $a$ and $b$.

RegionPlot[{x*y > 0, x*y < 0}, {x, -2, 2}, {y, -2, 2}]

How can I include that for example $a > 0$ and $b < 0$ for $a\,b\,x\,y$? The ultimate goal is to evaluate the sign for $f(a,b,x,y)$ given a set of parameters $a$ and $b$.

I have tried Plot, RegionPlot, Plot[Sign[...]], etc., without success.

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    $\begingroup$ I don't understand what you want to plot for the condition $a>0$. Do you want $f(.1, b, x, y)$, $f(.2, b, x, y)$, and so on for an infinite number of graphs? Also, please eliminate extraneous code that is irrelevant to your question (viz. PlotLegends). $\endgroup$ – David G. Stork Sep 18 '15 at 18:19
  • $\begingroup$ I want a two-dimensional graph for (x,y) whose effects are most important but (a,b) are parameters with any positive or negative values that can be kept constant for evaluation. I could set specific values for (a,b) as examples but I would prefer a general discussion of (x,y)-effects. $\endgroup$ – Tom G Sep 18 '15 at 18:23
  • $\begingroup$ What can you possibly mean--rigorously, mathematically--by "most important"? How can you possibly get a "general discussion..."? Show us one concrete example (done by hand, if necessary). Also, please edit your question to eliminate irrelevant code, as I suggested before. $\endgroup$ – David G. Stork Sep 18 '15 at 18:30
  • $\begingroup$ A simple example would be an output function with multiple inputs. All inputs have an effect on output but I am mostly interested in the effects of, say, capital or labor. However, whether the effects are positive or negative may depend on the specific values of other inputs - why I want to keep their exact values unspecified. $\endgroup$ – Tom G Sep 18 '15 at 18:41
  • $\begingroup$ Your "example" is far too vague. Please write an equation involving $a,b,x,f$ and what you seek. I've given a possible answer using Manipulate but you are too vague about what you're seeking. Don't talk about capital or labor. Give equations! $\endgroup$ – David G. Stork Sep 18 '15 at 18:44
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This is what you asked for:

Manipulate[
  RegionPlot[a x y - b > 0, {x, 0, 1}, {y, 0, 1}],
  {{a, .5}, 0, 1}, {{b, .25}, 0, .5}]
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  • $\begingroup$ Thank you. Is there a possibility to specify that 0<b<a -- e.g., a>0, b>0 and b<a? $\endgroup$ – Tom G Sep 22 '15 at 21:21
  • $\begingroup$ Certainly: Manipulate[RegionPlot[a x y - b > 0, {x, 0, 1}, {y, 0, 1}], {{a, .5}, 0, 1}, {{b, .25}, 0, a}]. I urge you to read the documentation on Manipulate, which describes all these concepts clearly. $\endgroup$ – David G. Stork Sep 22 '15 at 21:24
  • $\begingroup$ My apologies for the misunderstanding. The question was regarding RegionPlot without using Manipulate and allowing a, b to be greater than 1. $\endgroup$ – Tom G Sep 22 '15 at 21:34
  • $\begingroup$ Now I have no idea what you want, as I stated in my first comment (above). $\endgroup$ – David G. Stork Sep 22 '15 at 22:00
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I am not sure what the aim is here. I post this to illustrate shading approaches in case it may be helpful. Shading by sign of function:

fun[f_] := 
 Row[{ContourPlot[ Sign[f], {x, -3, 3}, {y, -3, 3}, 
    Contours -> {-1, 1}, ContourShading -> {Red, Blue}, 
    ImageSize -> 300],
   Plot3D[f, {x, -3, 3}, {y, -3, 3}, MeshFunctions -> (Sign@#3 &), 
    MeshShading -> {Blue, Red}, ImageSize -> 300]}]

Examples:

Manipulate[
 fun[test], {test, {Sin[x y], Sin[x + y], Sin[x - y], Cos[x y], x y, 
   x + y, x - y}}]

enter image description here

Obviously, you can use manipulate for parameter tuning a and b.

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