Expert: Scotto - 1/5/2011

Question
QUESTION: Greetings Sir,
Is it possible to have two different cubic curves that have the same shape in a particular section but dont have that same shape adjacent to that particular section.If not what sort of different degree or same degree pair of curves have these properties.

ANSWER: Without special definition for each section, I don't think so.

If the two curves f(x) = x + 2 and g(x) = (x²-4)/(x-2) are looked at,
the only have a disagreement at one point.  That is, g(x) is undefined at 2
where f(2) = 4, but they only disagree at one points.

Now then again, if f(x) = x + 2 and g(x) = (x^1.5 + 2x^0.5)/x^0.5,
the g(x) is not even defined for negative numbers the wats its written.
For 0 and all positive numbers, they both agree.

If g(x) was defined this way for x > 0 and as 2 for x<=0, the functions would be different.
Here, howeever, we've given one function a piecewise definition.

Using piecewise definitions, almost anything could be introduced.


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QUESTION: Thanks so much for replying.The examples you gave were perfect apart from the fact that the point(s) of difference were undefined.I was wondering if the point(s) of difference could be at finite values and if the curves could be restricted to polynomials?Better yet if you know of graphing software that allows one to draw the curve(s) and then get the equations of the curve that would fit the shape of the curve(s) exactly ,at least in the sections that are being considered,that would be greatly appreciated.Thanks

Answer
I have used several varieties of software for plotting curves, but that was back in the 90's.
The computers today, however, are for more advanced.  A good place to look for graphics software is http://en.wikipedia.org/wiki/List_of_information_graphics_software

After reviewing this list, I'm sure that you can see which software is best.