Contours

f[x_, y_] := (x^2) + (y^2) + 5

p1=ParametricPlot3D[{x, y, 6},{x, -2.1, 2.1},{y, -2.1, 2.1},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p2=ParametricPlot3D[{x, y, 8},{x, -2.1, 2.1},{y, -2.1, 2.1},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p3 = 
        ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], (R^2) + 5}, {ϕ, 
            0, 2  π}, {R, 0, 2.1}, 
          DisplayFunction -> Identity];
Show[{p1, p2, p3}, 
        DisplayFunction -> $DisplayFunction];

p1=ContourPlot[f[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},DisplayFunction -> Identity];
p2=ContourPlot[f[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourLines -> False,DisplayFunction -> Identity];
p3=ContourPlot[f[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourShading -> False,DisplayFunction -> Identity];
Show[
        GraphicsArray[{p1, p2, p3}]];

f[x_, y_] := Exp[-(x^2) - (y^2) + 2  x + 4  y - 3]
p1 = Plot3D[f[x, y],{x, -4, 4},{y, -3, 3},PlotRange -> All,DisplayFunction -> Identity];
p2=ContourPlot[f[x, y],{x, -1, 3},{y, 0, 4},DisplayFunction -> Identity];
p3=ContourPlot[f[x, y],{x, -1, 3},{y, 0, 4},ContourLines -> False,DisplayFunction -> Identity];
p4=ContourPlot[f[x, y],{x, -1, 3},{y, 0, 4},ContourShading -> False,DisplayFunction -> Identity];
Show[p1, 
        DisplayFunction -> $DisplayFunction];
Show[GraphicsArray[{p2, p3, p4}]];

Even and Odd Symmetry

f1[x_, y_] := -(x^2) - (y^2)
f2[x_, y_] := (x^2) + (y^1)
f3[x_, y_] := (x^1) - (y^2)
f4[x_, y_] := (x^3) + (y^3)
p11 = Plot3D[f1[x, y],{x, -4, 4},{y, -4, 4},AxesLabel -> {"x", "y", "z"},PlotRange -> All,DisplayFunction -> Identity];
p12=ContourPlot[f1[x, y],{x, -4, 4},{y, -4, 4},DisplayFunction -> Identity];
p13=ContourPlot[f1[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourLines -> False,DisplayFunction -> Identity];
p14=ContourPlot[f1[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourShading -> False,DisplayFunction -> Identity];
p21 = Plot3D[f2[x, y],{x, -4, 4},{y, -4, 4},AxesLabel -> {"x", "y", "z"},PlotRange -> All,DisplayFunction -> Identity];
p22=ContourPlot[f2[x, y],{x, -4, 4},{y, -4, 4},DisplayFunction -> Identity];
p23=ContourPlot[f2[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourLines -> False,DisplayFunction -> Identity];
p24=ContourPlot[f2[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourShading -> False,DisplayFunction -> Identity];
p31 = Plot3D[f3[x, y],{x, -4, 4},{y, -4, 4},AxesLabel -> {"x", "y", "z"},PlotRange -> All,DisplayFunction -> Identity];
p32=ContourPlot[f3[x, y],{x, -4, 4},{y, -4, 4},DisplayFunction -> Identity];
p33=ContourPlot[f3[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourLines -> False,DisplayFunction -> Identity];
p34=ContourPlot[f3[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourShading -> False,DisplayFunction -> Identity];
p41 = Plot3D[f4[x, y],{x, -4, 4},{y, -4, 4},AxesLabel -> {"x", "y", "z"},PlotRange -> All,DisplayFunction -> Identity];
p42=ContourPlot[f4[x, y],{x, -4, 4},{y, -4, 4},DisplayFunction -> Identity];
p43=ContourPlot[f4[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourLines -> False,DisplayFunction -> Identity];
p44=ContourPlot[f4[x, y],{x, -2.1, 2.1},{y, -2.1, 2.1},ContourShading -> False,DisplayFunction -> Identity];
Show[
            GraphicsArray[{{p11, p12, p13, p14}, {p21, p22, p23, p24}, {p31, 
                  p32, p33, p34}, {p41, p42, p43, p44}}]];

ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], 
          (R^3) (Cos[ϕ]^3 +  Sin[ϕ]^3)}, {ϕ, 0, 
          2  π}, {R, 0, 1.2}];

Let's look at a saddle function

f[x_, y_] := (x^2) - (y^2)
p1 = Plot[f[3, y],{y, -2, 2},AxesLabel -> {"y", "z"},DisplayFunction -> Identity];
p2 = Plot[f[x, (Sqrt[2])],{x, -2, 2},AxesLabel -> {"y", "z"},DisplayFunction -> Identity];
Show[
        GraphicsArray[{p1, p2}]];
Null

p1 = Plot3D[f[x, y],{x, -4, 4},{y, -4, 4},AxesLabel -> {"x", "y", "z"},PlotRange -> All,DisplayFunction -> Identity];
p2=ContourPlot[f[x, y],{x, -4, 4},{y, -4, 4},DisplayFunction -> Identity];
p3=ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], 
                (R^2) (Cos[2 ϕ])},{ϕ, 0, 2  π},{R, 0, 1.2},DisplayFunction -> Identity];

Show[
        GraphicsArray[{p1, p2, p3}]];