Quadric Surfaces

Hyperboloid of Two Sheets  (x^2) - (y^2) -(z^2) = 1

Define Function

f[x_,y_]:=Sqrt[x^2-y^2-1];

3D Surface Plot - Standard Style

P1 = Plot3D[f[x,y],{x,-3,3},{y,-3,3}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) - (y^2) - (z^2) = 1",BaseStyle->Directive[FontSize->16, FontFamily->"Times"], PlotPoints->30, BoxRatios->Automatic];

P2 = Plot3D[-f[x,y],{x,-3,3},{y,-3,3}, AxesLabel->{"x","y","z"}, PlotLabel->"Plot of Surface (x^2) - (y^2) - (z^2) = 1", PlotPoints->30, BoxRatios->Automatic,BaseStyle->Directive[FontSize->16, FontFamily->"Times"]];

Show[{P1,P2}, PlotRange->All]

Contour Plot of z = f(x,y) Level Curves

ContourPlot[f[x,y],{x,-3,3},{y,-3,3}, FrameLabel->{"X","Y"}, PlotLabel->"Level Curves For (x^2) - (y^2) - (z^2) = 1", ColorFunction->Hue, Contours -> 40,BaseStyle->Directive[FontSize->16, FontFamily->"Times"]]

3D Surface Plot - Hue Color Function Option

P1 = Plot3D[f[x,y],{x,-3,3},{y,-3,3}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) - (y^2) - (z^2) = 1", ColorFunction->Function[{x,y,z},Hue[z]], PlotPoints->30, BoxRatios->Automatic,BaseStyle->Directive[FontSize->16, FontFamily->"Times"]];

P2 = Plot3D[-f[x,y],{x,-3,3},{y,-3,3}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) - (y^2) - (z^2) = 1", ColorFunction->Function[{x,y,z},Hue[z]], BoxRatios->Automatic, PlotPoints->30,BaseStyle->Directive[FontSize->16, FontFamily->"Times"]];

Show[{P1,P2}, PlotRange->All]