My Math Forum equation of a split parabola

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 April 23rd, 2012, 04:19 AM #1 Member   Joined: Nov 2011 Posts: 45 Thanks: 0 equation of a split parabola what is the equation that gives a curve (similar x² = 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 x³ = y Thanks
 April 23rd, 2012, 08:41 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 465 Math Focus: Calculus/ODEs Re: equation of a split parabola You could use: $y=x|x|$
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:$y=x|x|$
consider the equation (x|x|)² - 5x = -6
This equation has 3 real roots, right? how do we find them?

 April 24th, 2012, 04:52 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 465 Math Focus: Calculus/ODEs Re: equation of a split parabola According to the computer, that equation has no real roots.
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
$$$2|2|$$^2-5(2)\ne-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?
$y=x|x|-5x+k$ where $-6.25\le k\le6.25 \in\mathbb R$

Case 1: $0\le x$

$x^2-5x+k=0$

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: $x<0$

$x^2+5x-k=0$

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.

April 25th, 2012, 03:13 AM   #7
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Re: equation of a split parabola

Quote:
Originally Posted by MarkFL
$$$2|2|$$^2-5(2)\ne-6$
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: $x<0$
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?

 April 25th, 2012, 08:12 AM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 465 Math Focus: Calculus/ODEs 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.
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 x³- 5x = y? can't we apply something like Cardano?

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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