Quadric Surfaces

Hyperbolic Paraboloid z = (x^2) - (y^2) (a.k.a., a “saddle surface”)

Define Function

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

3D Surface Plot - Standard Style

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

3D Surface Plot - Hue Color Function Option

Plot3D[f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"x","y","z"}, PlotLabel->"Plot of Surface z = f[x,y] = (x^2) - (y^2)", ColorFunction->Function[{x,y,z},Hue[z]],BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic]

Contour Plot of z=f(x,y)

ContourPlot[f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"x","y","z"}, PlotLabel->"Contour Plot for Surface z = (x^2) - (y^2)", ColorFunction->Hue, Contours -> 40,BaseStyle->Directive[FontSize->16, FontFamily->"Times"], FrameLabel->{"X","Y"}]

3D Surface Plot and Contour Plot using RegionFunction restriction to circular domain

Plot3D[f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"x","y","z"}, PlotLabel->"Plot of Surface z = f[x,y] = (x^2) - (y^2)", ColorFunction->Function[{x,y,z},Hue[z]], RegionFunction->Function[{x,y,z},x^2+ y^2≤ 4],BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic]

ContourPlot[f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"x","y","z"}, PlotLabel->"Plot of Surface z = f[x,y] = (x^2) - (y^2)", ColorFunction->Hue, RegionFunction->Function[{x,y,z},x^2+y^2≤ 4], Contours->40,BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic]