surface example

3D Monkey Saddle Graph

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

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

Quick facts

  • Formula: z = x^3 - 3*x*y^2
  • Category: surface
  • Tags: surface, calculus, z=f(x,y)

What this graph represents

It shows how higher-order terms can create shapes that are not visible from second derivatives alone.

Where it appears in calculus

Use this after students learn the second derivative test.

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.

Open embed page

Teacher prompt

Question: Why can this surface confuse the second derivative test?

Answer: The usual quadratic test degenerates at the origin, so higher-order behavior matters.

Related graphs

Explore more surface examples and compare shapes, slices, and contour behavior.

Saddle Surface

z = x^2 - y^2

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

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.

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

Create ready-to-use iframe embeds

Worksheet Generator

Print graph-based classroom worksheets

How to Graph z=f(x,y)

Step-by-step guide for beginners