Create a slider for a constant used in a function

Question

I want to create a slider for the $k$ constant in the function $h(u)$ on the second line. The code outputs a graph that seems to be using too few points. The two pictures below use the same $k$-value, one with the "Manipulate"-function and the other without.

Manipulate[
     h[u_] = k*u^2;
     fun[u_, v_] = -v^2 + u^5 + u^4 + 4 u^3 + 4 u^2 + 3 u + 3 - h[u]*v ;

     P1 = {-1, y /. (Solve[fun[-1, y] == 0, y])[[1]]};
     P1spec = { P1[[1]], -P1[[2]] - h[P1[[1]]]};
     P2 = {0, y /. (Solve[fun[0, y] == 0, y])[[1]]};
     P2spec = { P2[[1]], -P2[[2]] - h[P2[[1]]]};
     P3 = {1, y /. (Solve[fun[1, y] == 0, y])[[1]]};
     P3spec = { P3[[1]], -P3[[2]] - h[P3[[1]]]};


     ContourPlot[fun[x, y] == 0, {x, -2, 2}, {y, -15, 15}, 
      Epilog -> {PointSize[0.03],
        Red, Tooltip[#, #[[1]]] &@Point[P1], 
        Tooltip[#, #[[1]]] &@Point[P1spec],
        Blue, Tooltip[#, #[[1]]] &@Point[P2], 
        Tooltip[#, #[[1]]] &@Point[P2spec],
        Orange, Tooltip[#, #[[1]]] &@Point[P3], 
        Tooltip[#, #[[1]]] &@Point[P3spec]}],
     {k, -2, 7}
]

Answer 1

Try to add the option PlotPoints -> 50 to your plot statement as follows:

Manipulate[h[u_] = k*u^2;
 fun[u_, v_] = -v^2 + u^5 + u^4 + 4 u^3 + 4 u^2 + 3 u + 3 - h[u]*v;
 P1 = {-1, y /. (Solve[fun[-1, y] == 0, y])[[1]]};
 P1spec = {P1[[1]], -P1[[2]] - h[P1[[1]]]};
 P2 = {0, y /. (Solve[fun[0, y] == 0, y])[[1]]};
 P2spec = {P2[[1]], -P2[[2]] - h[P2[[1]]]};
 P3 = {1, y /. (Solve[fun[1, y] == 0, y])[[1]]};
 P3spec = {P3[[1]], -P3[[2]] - h[P3[[1]]]};
 ContourPlot[fun[x, y] == 0, {x, -2, 2}, {y, -15, 15}, 

  PlotPoints -> 50, 

  Epilog -> {PointSize[0.03], Red, Tooltip[#, #[[1]]] &@Point[P1], 
    Tooltip[#, #[[1]]] &@Point[P1spec], Blue, 
    Tooltip[#, #[[1]]] &@Point[P2], 
    Tooltip[#, #[[1]]] &@Point[P2spec], Orange, 
    Tooltip[#, #[[1]]] &@Point[P3], 
    Tooltip[#, #[[1]]] &@Point[P3spec]}], {k, -2, 7}]

I separated it out by empty lines just to make it visible in the code. It may improve the quality of the plot. Play with the number of points. The result should look as follows:

Have fun!