There are three demonstrations in this notebook.  First we trace a parametric plot as t goes from 0 to 2 π. Then we see a family of parametric curves as we vary a parameter a and finally we see tangents to a parametic curve.

Plot the parametic curve x=f(t); y=g(t) in the (a,b)x(c,d) window for t in [start,end] Note if you change the two sets of functions to f1,g1 and then to f2,g2 in the manipulate command you will see other demos.

Clear[f,g,a,b,c,d,start,end]
start:=0
end:=12π
a:=-4
b:=4
c:=-2
d:=2
f[t_]:=Cos[t]
g[t_]:=Sin[t]
f1[t_] := Sin[t] - Sin[2.3t]
g1[t_]:=Cos[t]
f2[t_]:=Sin[t]+Sin[5t]/2+Cos[2.3t]/4
g2[t_]:=Cos[t]+Cos[5t]/2+Sin[2.3t]/4

Manipulate[ Show[{ListPlot[{{f[s],g[s]}},PlotStyle->PointSize[0.02],AxesOrigin->{0,0},AspectRatio->Automatic,PlotRange->{{a,b},{c,d}},PlotStyle->{Purple}, PlotLabel-> "Parameter t = π"<>ToString[s/π]],ParametricPlot[{f[t],g[t]},{t,start,end}]}],{s,start,end}]

In This next plot we see the effect of the parameter a in the family of parametric curves x= a+ cos(t) y = a tan(t) + sin (t).

Manipulate[   ParametricPlot[{{a + Cos[t],a Tan[t]+Sin[t]}},{t,-π,π},PlotLabel-> "Parameter a = "<>ToString[a],AxesOrigin->{0,0},AspectRatio->Automatic,PlotRange->{{-4,4},{-5,5}}]
           ,{a,-2,2}]

Now Let’s look at a parametric plot and it’s tangents:

Manipulate[ParametricPlot[{{(t^2),(t^3)-3 t},{t+2,1.5 (s-1/s)(t+2-(s^2))+(s^3)-3s}},{t,-2,2}, PlotRange->{{-1,4},{-3,3}}],{s,-2,2}]