Describe the level surfaces and a section of the graph of each function:
$a)$ $ f: \mathbb {R} ^ 3 \rightarrow \mathbb {R}, \: (x, \: y, \: z) \mapsto -x ^ 2-y ^ 2-z ^ 2 $
$b)$ $ f: \mathbb {R} ^ 3 \rightarrow \mathbb {R}, \: (x, \: y, \: z) \mapsto x ^ 2-y ^ 2-z ^ 2 $
For $a)$, Level surfaces have the equation $ c = -x ^ 2-y ^ 2-z ^ 2 $. If $ c> 0 $ there is no level surface. If $ c = 0 $, the level surface is the origin. If $ c <0 $, we have
$$-x ^ 2-y ^ 2-z ^ 2 = c$$
$$x ^ 2 + y ^ 2 + z ^ 2 = -c$$
$$\sqrt{x ^ 2 + y ^ 2 + z ^ 2} = \sqrt{-c}$$
For $ c <0 $, the level surface is the sphere of radius $ \sqrt{-c} $ centered at $ (0, \: 0, \: 0) $. Now, we want to find a section of the graph of the function $ f $. A section of the graph determined by $ z = a $ is given by $ t = -x ^ 2-y ^ 2-a ^ 2 $. We know that it is a paraboloid of revolution opening in the space $ xyt $.
For $b)$, well...
two-sheeted hyperboloid if c>0
one-sheeted hyperboloid if c<0
infinite cone if c=0