Comparing and classifying real and complex-valued graphs of cubic polynomials

Question

There seem to be six essentially different types of cubic polynomials with real coefficients, giving rise to 1, 2 or 3 real roots in different ways.

Consider $f(z) = z^3 + a_2z^2 + a_1z + a_0$ and let $(a_2,a_1,a_0)$ be $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, $(2,0,0)$, $(2,0,1)$, $(2,0,-1)$ which yield the following graphs:

Is this classification sound? How can it be made precise?

To each of these types there correspond types of curves in the complex plane whose intersections give the complex roots (red for $\operatorname{Re}f(z) = 0$, blue for $\operatorname{Im}f(z) = 0$, for more details see here):

But interestingly there are cases where some cubic polynomials look essentially the same as real-valued functions (and are very close in "coefficient space"), while their complex counter parts look quite different as curves. For example for $(1,3\frac{1}{9},1), (1,3\frac{2}{9},1) (1,3\frac{3}{9},1)$:

What does this mean? "There's more structure/complexity in the complex numbers than is visible in the real numbers"?

Another question: Which formula yields the critical value $a_1 = 3\frac{2}{9}$ as a function of the other two $(a_2 = a_0 = 1)$? What's the significance of this number (which happens to be rational)? For arbitrary rational $a_2, a_0$, or only when $a_2 = a_0$? (It turns out that for $a_2 = a_0 = 2$ the critical value is $a_1 = 3\frac{8}{9}$ and for $a_2 = a_0 = 0$ it's $a_1 = 1$ - see above.)

If you want to play around with the coefficients, you can do it here.

Answer 1

For the real cubic polynomials, there are only three shapes of graphs, corresponding to whether the derivative has zero, one (double), or two roots. The other graphs correspond to vertical translations of these. The translations determine the number and nature of the roots of the original cubic but do not change the shape of the graph.

Answer 2

In your last three graphs, the red lines in the upper half of the plane are level curves near a "saddle point" in the graph of $\operatorname{Re} f(z).$ Depending on whether you get the level curves from a plane above, below, or exactly at the saddle point, you'll get level curves on the "high" part of the saddle, the "low" part of the saddle, or all meeting together at the saddle point.

The same thing is happening in the lower half of the plane.

Notice that none of the curves move very much between one figure and the next. Even the apparently dramatic changes between the orientations of the red curves is just a result of small deflections of four sections of the curves near a single point where they either all meet exactly at that point or only meet in pairs.

And yes, there is a lot more to see in the behavior over complex numbers than you can see in the real numbers alone.