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

Quick facts

  • Formula: z = x^2 - y^2
  • Category: surface
  • Tags: surface, calculus, z=f(x,y)

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.

How to use this graph

  1. Rotate the surface by dragging with your mouse or finger
  2. Use the Embed button to copy an <iframe> for your LMS or lesson page
  3. Click PNG to export a classroom-ready image
  4. Click Formula or LaTeX to copy the equation for your notes

Embed this graph

Use the Embed button in the calculator to copy a ready iframe for blogs, LMS pages, and lesson notes.

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Teacher prompt

Question: What type of critical point sits at the origin?

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

Related graphs

Explore more surface examples and compare shapes, 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.

Sine Wave Surface

z = sin(x) + sin(y)

Two perpendicular sine waves combined into one surface.

Ripple Surface

z = sin(sqrt(x^2 + y^2))

Circular waves radiate outward from the center.

Explore more tools

Use these free browser tools to extend your graphing workflow.

3D Surface Plotter

Plot z=f(x,y), rotate, and export

Embed Builder

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Worksheet Generator

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How to Graph z=f(x,y)

Step-by-step guide for beginners