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April 23rd, 2012, 04:19 AM   #1
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equation of a split parabola

what is the equation that gives a curve (similar x = y) but with half curve in the first and half in the third quadrant?

If that is not clear : I mean a curve, a function, similar (mutatis mutandis) to x = y
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April 23rd, 2012, 08:41 AM   #2
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Re: equation of a split parabola

You could use:

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April 24th, 2012, 04:36 AM   #3
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Re: equation of a split parabola

Quote:
Originally Posted by MarkFL
You could use:
consider the equation (x|x|) - 5x = -6
This equation has 3 real roots, right? how do we find them?
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April 24th, 2012, 04:52 AM   #4
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Re: equation of a split parabola

According to the computer, that equation has no real roots.
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April 24th, 2012, 09:08 PM   #5
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Re: equation of a split parabola

Quote:
Originally Posted by MarkFL
According to the computer, that equation has no real roots.
What about 2 ?: 4-10 =-6

consider also: x|x| - 5x =-4,
can you find the 3 roots?
For all y beween the extrema (+,-6.25) you get 3 real roots. How do you find them?
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April 24th, 2012, 09:45 PM   #6
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Re: equation of a split parabola

Quote:
What about 2 ?: 4-10 =-6


Quote:
consider also: x|x| - 5x =-4,
can you find the 3 roots?
For all y between the extrema (+,-6.25) you get 3 real roots. How do you find them?
where

Case 1:



From this you find two real roots if k is non-positive (repeated if k is on the boundary). If k is negative you get 1 real root.

Case 2:



If k is non-positive, you get 1 real negative root, and you discard the positive root.

If k is positive, you get 2 real negative roots.
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April 25th, 2012, 03:13 AM   #7
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Re: equation of a split parabola

Quote:
Originally Posted by MarkFL

sorry, I squared it, but I meant x|x|, of course
Quote:
can you find the 3 roots?
For all y between the extrema (+,-6.25) you get 3 real roots. How do you find them?
Case 2:
If k is non-positive, you get 1 real negative root, and you discard the positive root.
If k is positive, you get 2 real negative roots.
That's fine , but how do you calculate their value, what is the formula?
consider : x|x| - 5x =+,-(makes no difference the value doew not change) 3,
how do you find the 3 roots?
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April 25th, 2012, 08:12 AM   #8
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Re: equation of a split parabola

There is no formula, you essentially have a piecewise defined function, and you have to consider each piece as a separate case.
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April 25th, 2012, 08:19 AM   #9
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Re: equation of a split parabola

Quote:
Originally Posted by MarkFL
There is no formula, you essentially have a piecewise defined function, and you have to consider each piece as a separate case.
why piecewise, isn't it like x- 5x = y? can't we apply something like Cardano?
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April 25th, 2012, 11:26 PM   #10
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Re: equation of a split parabola

Quote:
Originally Posted by MarkFL
There is no formula, you essentially have a piecewise defined function, and you have to consider each piece as a separate case.
Pardon my ignorance, Mark, aren't we talking of a parabola? why piecewise defined?
Try it a wolfram and you'll find the 3 real solutions.
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