Curves in (ℝ^2)

Explicit Function y=f(x)

f[x_]=4Sqrt[1-x^2/4]
Plot[f[x],{x,-5,5},PlotStyle->{Red,Thick},BaseStyle->16,AxesLabel->{"x","y"},PlotRange->{{-5,5},{-5,5}},AspectRatio->1]

Implicit Function F(x,y)=0

F[x_,y_]=x^2/4+y^2/16 ==1
ContourPlot[Evaluate[F[x,y]],{x,-5,5},{y,-5,5},ContourStyle->{Red,Thick},BaseStyle->16,FrameLabel->{"x","y"}]

Parametric Function x(t),y(t), a≤t≤b

x[t_]=2Cos[t]
y[t_]=4Sin[t]
ParametricPlot[{x[t],y[t]},{t,-5,5},PlotStyle->{Red,Thick},BaseStyle->16,AxesLabel->{"x","y"},PlotRange->{{-5,5},{-5,5}}]

Space Curves in (ℝ^3)

Parametric Space Curve x(t),y(t),z(t) a≤t≤b

x[t_]= Cos[2Pi*t]
y[t_]=Sin[2Pi*t]
z[t_]=t
ParametricPlot3D[{x[t],y[t],z[t]},{t,-2,2},PlotStyle->{Red,Thick},BaseStyle->16,AxesLabel->{"x","y","z"},AspectRatio->1]

Surfaces in (ℝ^3)

Surface z=f(x,y)

f[x_,y_]=Sqrt[x^2-y^2+4]
Plot3D[f[x,y],{x,-5,5},{y,-5,5},PlotStyle->{Red,Thick},BaseStyle->16,AxesLabel->{"x","y","z"},AspectRatio->1]

Contour plot of z=f(x,y)

f[x_,y_]=Sqrt[x^2-y^2+4]
ContourPlot[f[x,y],{x,-5,5},{y,-5,5},ContourStyle->{Thick},BaseStyle->16,FrameLabel->{"x","y"},AspectRatio->1]

Parametric Surface x=f(t,s), y=g(t,s), z = h(t,s), a ≤t≤b, c≤s≤d

Note this example is a mobius strip.

x[t_,s_]=(1+s/2Cos[t/2])Cos[t]
y[t_,s_]=(1+s/2Cos[t/2])Sin[t]
z[t_,s_]=s/2Sin[t/2]
ParametricPlot3D[{x[t,s],y[t,s],z[t,s]},{t,0,2Pi},{s,-1,1},PlotStyle->{Red,Thick},BaseStyle->16,AxesLabel->{"x","y","z"},AspectRatio->1]

For Your Information: Surfaces in (ℝ^4): w=f(x,y,z)

Contour plot of w=f(x,y,z)

We cannot plot a four dimensional function w=f(x,y,z), but we can plot three dimensional contours by setting w=k, where k is a real number. These plots take a long time to be created, so be patient. By default, Mathematica will create multiple “contour surfaces” of its own choosing, which can be confusing to view, so I have set it to only produce the three contour surfaces:
 w=-2=f(x,y,z) (the red surface),  
 w=19=f(x,y,z) (the orange surface). and 
 w=35=f(x,y,z) (the yellow surface).

f[x_,y_,z_]=x^2-y^2+z^2
ContourPlot3D[f[x,y,z],{x,-6,6},{y,-6,6},{z,-6,6},BaseStyle->16,AspectRatio->1,Contours->{-2,19,35},Mesh->None,ContourStyle->{Red,Orange,Yellow},Lighting->"Neutral"]