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You are not logged in. #1 20130923 04:43:48
Relation between hyperbolas and rational expressionsHello. and a rational function of the form such that both are symmetric in the same fasion. What I know: The function f(x) is a hyperbolic that has been rotated 45 degrees and shifted up and over in some manner. What I want to know: How do the terms a, p, q, and r relate and in what way does one convert from either form to the other and vice versa? #4 20130923 05:17:49
Re: Relation between hyperbolas and rational expressionsWell, well well. You've taught me something new. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #5 20130923 05:25:47
Re: Relation between hyperbolas and rational expressionsAccording to your signature I have taught you nothing but have only helped you find it in yourself. Last edited by Reuel (20130923 05:27:24) #6 20130923 05:27:53
Re: Relation between hyperbolas and rational expressionsThey do not give the same graph. How do you expect to get one from the other??? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #7 20130923 05:31:40
Re: Relation between hyperbolas and rational expressionsYou should be able to work via the asymptotes. Find those for the rf and then construct a hyperbola with those.
I think the idea is to transform one into the other. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #8 20130923 05:33:28
Re: Relation between hyperbolas and rational expressionsanonimnystefy: They are the same kind of graph only, as I have said, different by 45 degrees in a rotational sense. Suppose you want to get a rational function that is some hyperbola without knowing p, q, or r or anything about it such as its center, foci, vertices, etc. and all you have is a hyperbola in standard form in terms of a and perhaps b. How do you get the hyperbolic "standard" into the form of the rational function? That is the problem I am seeking help with. #9 20130923 05:36:32
Re: Relation between hyperbolas and rational expressionsbob bundy  I have already successfully converted from a given rational function to a standard hyperbolic by finding a and b (which turn out to be equal because the hyperbola is rectangular) but as I have said in my reply to another user I want to also know how to move in the other direction which is more difficult. #10 20130923 05:37:45
Re: Relation between hyperbolas and rational expressions
If by a rational function you mean a quotient of two polynomials, then what you are asking is impossible. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #11 20130923 05:42:21
Re: Relation between hyperbolas and rational expressionsIn general, you are correct. But it seems that a rf, in that format only, is a hyperbola, at least according to the page I linked earlier. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #12 20130925 00:55:50
Re: Relation between hyperbolas and rational expressionsA "proof" is here given. I put proof in quotations as this is rather inductive and I still do not know how to go from one to the other. It could be very difficult given the thing is full of transcendentals but maybe there is an easier way than the following I have yet to see. As it is, graphing can show the following is apparently accurate. is a rectangular hyperbola of the standard form ... The value of "a" has been analytically found to be that of and the center (h,k) is given by both of which are values crucial to the problem. Please note that "a" in the hyperbola is not the same "a" in f. I went ahead and used the same symbol as we use the hyperbolic a very little. The difference between them is obvious based on the context of my work. We can solve the equation for y, but as it is unclear what to do with that we can switch to a parametric form of the hyperbola The transformation matrix for the rotation of the parametric system by some angle is that of such that its application for a 7π/4 rotation upon our vector curve <x,y> yields the vectorvalued functions Next, the curve, while the correct shape, is in the wrong location on the plane and simply needs shifted over and up by the original center of the hyperbola which yields the solution I apologize for any errors made in my work. Example: If then f is a rectangular hyperbola. In standard form that hyperbola has the equation . Omitting all the steps already given in the general form above, the solution is If f(x) is plotted so as to go from x = 0 to x = 5/3 the parametric equivalent is approximately t=0.3398369094 to t=0.3398369094 or, if you prefer, the exact value is . The reason for choosing these bounds is completely by preference; it is where the domain is equal to the range such that f(5/3) = 5/3. Last edited by Reuel (20130925 01:06:43) #13 20130925 18:13:43
Re: Relation between hyperbolas and rational expressionshi Reuel, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #16 20130926 07:11:10
Re: Relation between hyperbolas and rational expressionsOK, that looks good to me. And I seem (as you pointed out earlier) to have lived up to my signature. That's good. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #17 20130926 23:37:28
Re: Relation between hyperbolas and rational expressionsAh, I remember. If the rational expression is a hyperbola that means it is a conic section; if a conic section then it is defined as the intersection of a plane with a double cone. Last edited by Reuel (20130926 23:38:16) #18 20130927 01:45:52
Re: Relation between hyperbolas and rational expressionsI have a maths book that has the standard equations. It's in the loft so I'll have to go mountaineering. Gardening over for today, I've been laying a turfs for part of a new lawn. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #19 20131002 20:39:36
Re: Relation between hyperbolas and rational expressionshi Reuel, is a hyperbola with its lines of symmetry at 45 degrees to the axes, going through (0,0) and with a horizontal, and a vertical, asymptote. The plane you're after; is it as follows ?: (i) Find the equation of a cone (ii) Find the equation of a plane. (iii) Where this plane intersects the cone, that's the above hyperbola. If so, I'll see if I can do (i) and (ii). It may be necessary to move the hyp. across so the cone, itself, has an axis as its vertical axis. Bob You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #21 20131011 17:32:44
Re: Relation between hyperbolas and rational expressionshi Reuel, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei 