What Size Fence 3D

What Size Fence 3D

Q: What size fence should I use when graphing 3D functions?

A: Many functions can be analyzed best over the 2x2x2 "unit" box: -1 < x,y,z < 1. However, certain functions may be easier to analyze over a larger fence. Take, for example, "Twin Peaks," f(x,y) = -x⁴ + 2x² - y². When analyzed over the unit box it appears to be a saddle function quite similar to f(x,y) = x² - y²:

However, upon further analysis of the function's slice curves we may conclude that there is crucial information left out of the above graph.

Slice curves y = 1, -1 (above left) and y = 0 (above right) contain critical points at x = 1, -1. Here the graph changes direction and proceeds downwards. We find that the graph hits the x axis when x = 2^1/2, -2^1/2, so perhaps we should extend our fence: -2^1/2 < x,y < 2^1/2:

As you can see, using a larger box (or rather a smaller scale) allows us to see the true shape of the function, as well as to understand its name. Remember that this is always an option when graphing.




	What Size Fence 3D Q: What size fence should I use when graphing 3D functions? A: Many functions can be analyzed best over the 2x2x2 "unit" box: -1 < x,y,z < 1. However, certain functions may be easier to analyze over a larger fence. Take, for example, "Twin Peaks," f(x,y) = -x⁴ + 2x² - y². When analyzed over the unit box it appears to be a saddle function quite similar to f(x,y) = x² - y²: However, upon further analysis of the function's slice curves we may conclude that there is crucial information left out of the above graph. Slice curves y = 1, -1 (above left) and y = 0 (above right) contain critical points at x = 1, -1. Here the graph changes direction and proceeds downwards. We find that the graph hits the x axis when x = 2^1/2, -2^1/2, so perhaps we should extend our fence: -2^1/2 < x,y < 2^1/2: As you can see, using a larger box (or rather a smaller scale) allows us to see the true shape of the function, as well as to understand its name. Remember that this is always an option when graphing.