
Sphere/hemisphere
Sketch $\displaystyle \sqrt{13  x^2  y^2}$
Attempt:
$\displaystyle z^2 = 13  x^2  y^2$
$\displaystyle z^2 = x^2  y^2 + 13$
$\displaystyle x^2 + y^2 + z^2 = 13$
Now I dont understand why this is half of a sphere (which opens up downwards) and not a whole sphere?

If this was $\displaystyle \displaystyle z = \sqrt{13  x^2  y^2}$, this is really $\displaystyle \displaystyle z = +\sqrt{13  x^2 y^2}$, which is the upper half of the sphere.
To be a full sphere it would need to be $\displaystyle \displaystyle z = \pm \sqrt{13  x^2  y^2}$.