Graphing Multivariable Polynomials
Date: 02/20/2005 at 12:30:55 From: Eric Subject: Multivariable Polynomials I have a few questions about multivariable polynomials. Say we have the polynomial x + y + z, can this be graphed? I also really don't know anything about the z axis or how to graph on it. I guess I'm really just looking for more information about multivariable polynomials. Thanks.
Date: 02/20/2005 at 14:27:12 From: Doctor Tom Subject: Re: Multivariable Polynomials Hi Eric, We live in a three-dimensional world, so it's tough to "graph" polynomials with more than two variables. Let's go back to single-variable polynomials for a second. When you graph f(x) = x^2 - 3x + 2, you usually graph the input, x, on one axis and the output, f(x), on the other. You need two dimensional paper to do this. Now consider something like this: f(x, y) = x^2 - xy + y - 6. For each pair of values you choose to be x and y, there is a single output. This can be "graphed" as follows: Draw an x-axis and a y-axis on a sheet of paper and place that paper on a table. Now, each point on the paper corresponds to an x and y pair, so calculate f(x, y) for that pair. The output is a number, and (somehow) put a dot in space that distance above the paper. If the output is negative, put the dot below the paper. If you do this (conceptually) for every (x, y) pair, you'll get a surface whose height above or below the paper represents the output value. Let's look at a simpler example: f(x, y) = x + y - 5. The "surface" in this case will be a plane that slices through your paper along the line x + y - 5 = 0. Every point along this line is a solution--there are, as you noticed, an infinite number of them. With curved surfaces, the intersection with the paper may be curved lines or combinations of curved lines. Unfortuantely, it's tough to go to the next stage. For something like your example of f(x, y, z) = x + y + z, you need a three-dimensional volume just to represent the inputs. Since we only live in a three-dimensional world, there's no fourth direction you can go to graph the output. There are tricks you can do that might help visualize what's going on. Imagine that the output happened to be a number between 0 and 1 in all cases. You could say to yourself: "I'll represent 0 by red and 1 by violet, and the numbers in between by the colors of the rainbow going from red to violet. Then you could have a 3-D grid where the output of each triple is represented by the appropriate color as a series of dots in the grid. You can't fill in all the dots or you can't see inside, but you could space them, and imagine looking at the whole mess on a computer screen where you could change your viewpoint. The problem of "graphing" high-dimensional systems is very important, and there's a lot of work in this area of visualization. Edward Tufte has a wonderful series of books on visualization of these sorts of things and many others. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
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