\(as = u; 
bs = \(-2\)\ u\ v\  - \ 4\ v^3; 
cs = \ 3\ v^4\  + \ u\ v^2; 
    \)

swallowtail = 
      ParametricPlot3D[{as, bs, cs -  .01}, {v, -1, 1}, {u, -3, 3}]

p = Graphics3D[Point[{as, bs, cs}]]

v = 0; u = -1;

\(au = 1; av = 0; 
bu = \(-2\)\ v; \ bv = \ \(-2\)\ u\  - 12\ v^2; 

    cu\  = v^2; cv = 12\ v^3\  + \ 2\ u\ v; \)

\(a = 1; b = 2; c = 0; 
a2 = \(-2\); b2 = 1; c2 = 0; \)

plane = 
      ParametricPlot3D[{a s  +  a2 t,  b s  +  b2 t,  
          c s  +  c2 t}, {s, -1.7, 1.7}, {t, -1.7, 1.7}]

Show[{plane, swallowtail, p}]

s = -1; t = -1.2; 
    pt3 = Graphics3D[{PointSize[ .03], RGBColor[1, 0, 0], 
          Point[{a s  +  a2 t,  b s  +  b2 t,  
              c s  +  c2 t}]}];

Show[pt3, plane]

pic3 = 
      GraphicsArray[{
          Show[{swallowtail, plane, pt3}, Boxed -> False, Axes -> None], 
          Plot[x^4  + (a s  +  a2 t) x^2  +   
              (b s  +  b2 t) x  +  c s  +  c2 t, {x, -5, 
              5}, AxesLabel -> {"x", "p(x)"}]}]

Show[pic1]

Display["tanpt3", pic3, "GIF", ImageSize -> {600, 371}]

pic2 = 
      GraphicsArray[{
          Show[{swallowtail, plane, pt2}, Boxed -> False, Axes -> None], 
          Plot[x^4  + (a s  +  a2 t) x^2  +   
              (b s  +  b2 t) x  +  c s  +  c2 t, {x, -5, 
              5}, AxesLabel -> {"x", "p(x)"}]}]

pic1 = 
      GraphicsArray[{
          Show[{swallowtail, plane, pt1}, Boxed -> False, Axes -> None], 
          Plot[x^4  +  as x^2  + bs x  +  cs, {x, -5, 5}, 
            AxesLabel -> {"x", "p(x)"}]}]