# How to correctly plot x^(1/3)

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

This command produces a graph that looks like this:

I was expecting a graph like this:

How to correctly plot x^(1/3)

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

This command produces a graph that looks like this:

I was expecting a graph like this:

3

SageMath sometimes chooses complex cube roots, which explains this behavior.

There's not much you can do about this internal choice. But you can do this:

```
plot(sgn(x)*abs(x)^(1/3),(x,-13,13))
```

4

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.

3

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.

2

[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:

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)
```

Suppose we want to visualise the set of points $(x, y)$ where $x^2 + y^2 = 1$. This cannot be put in the form $y = f(x)$. But one can think of it as the set of points where $x^2 + y^2 - 1$ is zero, and so plot it as

```
sage: implicit_plot(lambda x, y: x^2 + y^2 - 1, (-1, 1), (-1, 1))
```

It is called an implicit plot because instead of the plot of a function, it is the plot of some relation between $x$ and $y$. Some pieces of the plot could be put in the form $y = f(x)$ or $x = g(y)$, but these functions remain implicit here, as we only use some relation between $x$ and $y$ to get the plot.

0

Another option is to specify the `xmin`

, `xmax`

, `ymin`

, `ymax`

or just one of them:

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

So why does a simple graphing calculator do the correct graph but Sage can't?

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Asked: ** 2020-04-25 13:53:08 +0100 **

Seen: **946 times**

Last updated: **Sep 30 '21**

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What do you expect to get instead?

I was expecting this