f[x_] := x^2 + 2 x - 3
    p[a_] := {{a, f[a]}}
    Animate[Plot[f[x], {x, -2, 4}, Epilog -> Point[p[a]]], {a, -2, 4}]

Hi, Marcel, I posted this new code for the tangent line as a suggestion for you to post other pieces of code that you are using in the course. They may be useful for others in the Wolfram Community and the Community may suggest new solutions and useful pieces of code. Sure, it's your decision...

f[x_] := 2 x^3 + 2 x - 3
p[a_] := {{a, f[a]}}
t[x_, a_] := f[a] + (x - a) f'[a]
Animate[Plot[f[x], {x, -2, 4}, PlotRange -> {{-2, 4}, {-24, 128}}, 
  Epilog -> {PointSize[Medium], Point[p[a]], Red, 
    InfiniteLine[{p[a][[1]], {0, t[0, a]}}]}], {a, -2, 4}, 
 AnimationDirection -> ForwardBackward]