Question

In Exercises \(19-26,\) use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ z=4-\sqrt{|x y|} $$

Short answer

The graph of the given surface equation results in a surface that opens downward with a peak at \(z=4\). For \(x y\) >= 0, the surface is determined by \(z=4-\sqrt{x y}\) and for \(x y\) < 0, the surface is represented by \(z=4-\sqrt{-x y}\).

Step-by-step solution

Consider both positive and negative values for \(x y\)

This surface equation has an absolute sign on \(x y\), which means it will hold the same value whether \(x y\) is positive or negative. So two separate cases must be considered: one for \(x y\) >= 0 and one for \(x y\) < 0.

Solve for \(z\) in terms of \(x\) and \(y\)

In the case where \(x y\) >= 0, the equation becomes \(z=4-\sqrt{x y}\). And for the case where \(x y\) < 0, the equation becomes \(z = 4 - \sqrt{-x y}\).

Use a Computer Algebra System to graph the surface

Using a Computer Algebra System, enter these two equations separately to get two graphs. One represents the surface for \(x y\) >= 0, and the other is for \(x y\) < 0. Combine both graphs for the complete representation of the surface for all possible values of \(z\), \(x\), and \(y\).