Sketching z = x² + y²

Calculus 3 · Axiom Academy

EXAMPLE Sketching z = x^2 + y^2 Reading the traces and level curves of a surface to sketch it as a circular paraboloid Sketch the surface z = x^2 + y^2 . Identify its traces in the coordinate planes and its level curves, then describe its shape, vertex, and the direction it opens. A bowl that opens upward: every horizontal slice is a circle of radius √k, and every vertical slice through the z-axis is an upward parabola. Nice work. You sketched a 3D surface by reading its traces and level curves — the same routine that handles any quadric surface. Intercepts: setting two variables to zero showed z = x^2 + y^2 touches every axis only at the origin. Vertical traces: in the xz - and yz -planes the surface reduces to z = x^2 and z = y^2 — upward parabolas. Level curves: z = k gives x^2 + y^2 = k , a circle of radius for (and nothing for k < 0 ). The shape: circular slices under parabolic walls make a circular paraboloid — vertex (0,0,0) , axis the +z -axis, opening up. Traces + level curves is the go-to strategy for any quadric surface — try it next on ellipsoids and hyperboloids.

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