How to correctly plot x^(1/3)

edit

Another way is to use the method x.nth_root(3) for real numbers x.

Using this

sage: plot(lambda x: RR(x).nth_root(3), (-1, 1))

gives the desired plot.

After many, many years, we finally have real_nth_root as well. From the documentation:

sage: plot(real_nth_root(x, 3), (x, -1, 1))

See Trac 12074 for some of the story.

[Edited after the question was updated, clarifying it is not "why does the graph not start at zero", but "why is the negative part missing".]

Together with the unsatisfactory plot, the command

sage: plot(x^(1/3))

gives some warnings:

verbose 0 (3797: plot.py, generate_plot_points)
WARNING: When plotting, failed to evaluate function at 100 points.
verbose 0 (3797: plot.py, generate_plot_points)
Last error message: 'can't convert complex to float'

They give a clue as to what goes wrong.

By default, plot plots on the interval $[-1, 1]$ using 200 points regularly spaced along that interval to evaluate the function being plotted.

The warning that it "failed to evaluate function at 100 points", and the plot it produces, reveal that it failed to get a point to plot for all the negative values.

Here is how Sage computes powers one-third of negative numbers, depending on whether they are rational or floating-point:

sage: a = (-1)^(1/3)
sage: a
(-1)^(1/3)
sage: b = (-1.)^(1/3)
sage: b
0.500000000000000 + 0.866025403784439*I

And here is what happens when trying to convert them to floats (plot does that):

sage: float(a)
---------------------------------------------------------------------------
Traceback (most recent call last)
...
TypeError: can't convert complex to float
During handling of the above exception, another exception occurred:
TypeError: unable to simplify to float approximation
sage: float(b)
Traceback (most recent call last)
...
TypeError: unable to convert 0.500000000000000 + 0.866025403784439*I
to float; use abs() or real_part() as desired

To plot the real-cube-root function, i.e. the composition inverse of the $(x \mapsto x^3)$ bijection, there are various options.

One is indicated in @rburing's answer, and uses the sign and the absolute value to work around this limitation in Sage. It is actually part of the SageMath FAQ, see:

• SageMath FAQ: How do I plot the cube root (or other odd roots) for negative input?

One other option is to plot $x = y^3$ instead of $y = x^(1/3)$.

This can be done using a parametric plot:

sage: parametric_plot((lambda t: t^3, lambda t: t), (-1, 1))

This can also be done using an implicit plot since the desired graph is the locus where $x - y^3$ is zero:

sage: implicit_plot(lambda x, y: x - y^3, (-1, 1), (-1, 1))

By default implicit plot comes with a frame instead of axes.

One can add axes:

sage: implicit_plot(lambda x, y: x - y^3, (-1, 1), (-1, 1), axes=True)

One can additionally remove the frame:

sage: implicit_plot(lambda x, y: x - y^3, (-1, 1), (-1, 1), axes=True, frame=False)

Another option is to specify the xmin, xmax, ymin, ymax or just one of them:

sage: plot(x**(1/3), ymin=0)