To test for symmetry about the x-axis,

   Substitute -y in place of y and see if it is possible to 
   simplify the equation back to its original form. If this
   is possible it is symmetrical with respect to the x-axis.


To test for symmetry about the y-axis,

   Substitute -x in place of x and see if it is possible to 
   simplify the equation back to its original form. If this
   is possible it is symmetrical with respect to the y-axis.

To test for symmetry about the origin,

   Substitute -x in place of x and -y in place of y and see 
   if it is possible to simplify the equation back to its 
   original form. If this is possible it is symmetrical with 
   respect to the x-axis.

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Let's test for symmetry about the x-axis

     x² = y³ + 4

Substitute -y for y

     x² = (-y)³ + 4

     x² = -y³ + 4

That is not like the original and has one term with the opposite sign
as the corresponding term in the original, so it cannot
be made into the same equation as the original. 

Therefore it is not symmetrical about the x axis.

-------------------------

Let's test for symmetry about the y-axis

     x² = y³ + 4

Substitute -x for x

  (-x)² = y³ + 4

     x² = y³ + 4

That is the same as the original so it is
symmetrical with respect to the y-axis.

--------------------------

Let's test for symmetry about the origin

     x² = y³ + 4

Substitute -s for x and -y for y

   (-x)² = (-y)³ + 4

     x² = -y³ + 4

That is not like the original and has one term with the opposite sign
as the corresponding term in the original, so it cannot
be made into the same equation as the original. 

Therefore it is not symmetrical about the origin.

The graph is below, but I don't think you had to do that,
just test for symmetry. 

Yes you can take the cube root (or any odd root) of any number.

 y = ∛x²-4  

We can see it is symmetrical with respect
to the y-axis, but not the x-axis or origin.



Edwin