surface example

3D Inverted Paraboloid Graph

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

z = 12 - x^2 - y^2

Quick facts

  • Formula: z = 12 - x^2 - y^2
  • Category: surface
  • Tags: surface, calculus, z=f(x,y)

What this graph represents

The negative quadratic terms make z fall as x or y moves away from 0.

Where it appears in calculus

This is the standard local maximum example.

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: Where is the highest point?

Answer: The highest point is at the origin, where z equals 12.

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.

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

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