Pxy[u_, ax_, bx_, cx_, dx_, ay_, by_, cy_, 
        dy_] :=  {ax*u^3 + bx*u^2 + cx*u  + dx,  
        ay*u^3 + by*u^2 + cy*u + dy}

P0 = {2, 2};
P1  =  {3, 2.5};
P2  =  {4, 1};
P3  =  {5, 2};

v  =  
        Solve[
{Pxy[0, ax, bx,  cx, dx, ay, by, cy, 
                dy]  ==  
              P0, 
Pxy[1/3, ax, bx,  cx, dx, ay, by, cy, 
                dy] == P1, 
 
            Pxy[2/3, ax, bx,  cx, dx, ay, by, cy, dy] == 
              P2, 
Pxy[1, ax, bx,  cx, dx, ay, by, cy, 
                dy] == P3}, 
{ax, bx,  cx, dx, ay, 
            by, cy, dy}];

For[u = 0; w = {}, u ≤ 1.0, u += 0.04, 
      w =  Append[w, 
          Pxy[u, ax, bx,  cx, dx, ay, by, cy, dy]  /.  
            v[[1]]]]

g1  =  Graphics[Line[w]]

g2  =  
      Graphics[{PointSize[0.025], Point[P0], Point[P1], Point[P2], 
          Point[P3]}]

Show[{g1, g2}, Axes -> Automatic]

v[[1]]

Qxy[u_] :=  {3*u  + 2, 20.25*u^3 - 29.25`*u^2 + 9*u + 2}

Q1  =  Qxy[0] + 3*Qxy'[0]

Q2  =  Qxy[1] - 3*Qxy'[1]

g3  =  Graphics[{PointSize[0.025], Point[Q1], Point[Q2]}]

Show[{g1, g2, g3}, 
      PlotRange -> {{-5, 12}, {-35, 30}}]