This modeling exploration follows the exploration seen here.
In 2D, consider the equation . It's graph is a circle with center (0,0) and radius = 4.
To see why, check out this resource here.
Yet in this equation, if we replace y with z, this equation becomes .
If we solve explicitly for z, we get (upper semicircle) and (lower semicircle).
In 3D, think of z as the new DEPENDENT VARIABLE.
If we graph these 2 surfaces in 3D, the value of y doesn't matter. Thus, these semicircles become infinitely long half cylinders. When they're put together, we get an infinitely long cylinder.
What happens when we add the terms (containing y) to the right side of each equation? Why does this occur?
What other kinds of surfaces can we create?