Notice that the equation of a circle is `x^2+y^2=r^2` . So your first equation describes a set of circles where the radius = `sqrt(z-1)` .
Your graph is empty until `z>=1`...then you get a "stack" of circles whose radius is increasing.
If the radius were increasing at a linear rate, you'd have a cone. But the radius is increasing ` `like a square root graph, so you get a nice vase shape, or bullet shape. Imagine the graph of `y=sqrt(z-1)` rotated around the z-axis.
OK, now x - y = 0 means x = y. So imagine the regular graph of y = x, but extend it out in the z direction to make a plane. The intersection of the plane and the "bullet vase" is the curve we're after.
It's just a parabola!
If you projected it onto the x-z plane it would look like this:
Same thing for the y-z plane. So let's just let
x(t) = t
y(t) = t
z(t) = t^2 + 1
Double-check the condition that z increases when x and y (in other words, t) are positive. Looks good.
I know it's hard to visualize this stuff from words alone, so try to work it through with a series of graphs, and talk with your instructor!