surface example

3D Saddle Surface Graph

A saddle surface curves up in one direction and down in the perpendicular direction.

z = x^2 - y^2

Teacher prompt

What type of critical point sits at the origin?

The origin is a saddle point because one slice curves up while the perpendicular slice curves down.

min z 0.00max z 0.0056 samples

What this graph represents

It is the classic hyperbolic paraboloid used to explain why a critical point can be neither a maximum nor a minimum.

Where it appears in calculus

Students should rotate the graph and compare cross-sections along x and y.

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Related graphs

Open another surface page and compare shape, slices, and contour behavior.

Gaussian Surface

z = exp(-(x^2 + y^2))

A smooth bell-shaped surface centered at the origin.

Elliptic Paraboloid

z = x^2 + y^2

A bowl-shaped surface that opens upward.

Inverted Paraboloid

z = 12 - x^2 - y^2

A dome-shaped surface with a highest point at the center.

Monkey Saddle

z = x^3 - 3*x*y^2

A three-way saddle with three valleys and three ridges.