Rotation of f(x) = k/x
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From: Yimin Rong on 14 Jul 2010 15:18 Given f(x) = k/x where k and x > 0 Rotating the function by pi/4 (i.e. 45 degrees) can be done by multiplying (x, f(x)) with a transformation matrix to give (x*, f*(x)). After the transformation, y* is still > 0, but x* can be any value. Is there a way to convert this to a "regular" function? E.g. suppose g() is f() rotated by pi/4, then: g(0) = k^(1/2) g(a) = g(-a) g(k - 1/2) = k + 1/2 What is g()? Thanks for reading. Regards, Yimin
From: Ross on 14 Jul 2010 15:56 On Jul 14, 12:18 pm, Yimin Rong <yiminr...(a)yahoo.ca> wrote: > Given f(x) = k/x where k and x > 0 > > Rotating the function by pi/4 (i.e. 45 degrees) can be done by > multiplying (x, f(x)) with a transformation matrix to give (x*, > f*(x)). After the transformation, y* is still > 0, but x* can be any > value. Is there a way to convert this to a "regular" function? > > E.g. suppose g() is f() rotated by pi/4, then: > > g(0) = k^(1/2) > g(a) = g(-a) > g(k - 1/2) = k + 1/2 > > What is g()? > > Thanks for reading. > > Regards, > > Yimin Your rotation matrix corresponds to y*=(x+y)/sqrt(2), x*=(x-y)/ sqrt(2). If you solve these for x and y in terms of x* and y* you can substitute into your original equation y=k/x, getting y^2=x^2+2k or y=sqrt(x^2+2k)
From: Henry on 14 Jul 2010 16:07 On 14 July, 20:56, Ross <rmill...(a)pacbell.net> wrote: > On Jul 14, 12:18 pm, Yimin Rong <yiminr...(a)yahoo.ca> wrote: > > > > > Given f(x) = k/x where k and x > 0 > > > Rotating the function by pi/4 (i.e. 45 degrees) can be done by > > multiplying (x, f(x)) with a transformation matrix to give (x*, > > f*(x)). After the transformation, y* is still > 0, but x* can be any > > value. Is there a way to convert this to a "regular" function? > > > E.g. suppose g() is f() rotated by pi/4, then: > > > g(0) = k^(1/2) > > g(a) = g(-a) > > g(k - 1/2) = k + 1/2 > > > What is g()? > > > Thanks for reading. > > > Regards, > > > Yimin > > Your rotation matrix corresponds to y*=(x+y)/sqrt(2), x*=(x-y)/ > sqrt(2). > If you solve these for x and y in terms of x* and y* you can > substitute > into your original equation y=k/x, getting y^2=x^2+2k or y=sqrt(x^2+2k) Indeed. and so g(0) = (2k)^(1/2) [the other two are correct]
From: cwldoc on 14 Jul 2010 14:10 > Given f(x) = k/x where k and x > 0 > > Rotating the function by pi/4 (i.e. 45 degrees) can > be done by > multiplying (x, f(x)) with a transformation matrix to > give (x*, > f*(x)). After the transformation, y* is still > 0, > but x* can be any > value. Is there a way to convert this to a "regular" > function? > > E.g. suppose g() is f() rotated by pi/4, then: > > g(0) = k^(1/2) > g(a) = g(-a) > g(k - 1/2) = k + 1/2 > > What is g()? > > Thanks for reading. > > Regards, > > Yimin The coordinates (X*, Y*) are rotated clockwise by pi/4 (counterclockwise by -pi/4) from (X, Y), so a point with coordinates (x*, y*) in the (X*, Y*) system has coordinates x = cos(-pi/4) x* - sin(-pi/4) y* = (x* + y*)/sqrt(2) y = sin(-pi/4) x* + cos(-pi/4) y* = (-x* + y*)/sqrt(2) in the (X, Y) system. Substitution into xy = k gives (x* + y*)/(-x* + y*) = 2k simplifying, y* = sqrt[(x*)^2 + 2k] where we are justified in choosing y* > 0, because we were given that x and k are > 0 (and thus y > 0), so y* = (x + y)/sqrt(2) > 0.
From: cwldoc on 14 Jul 2010 14:13
> > Given f(x) = k/x where k and x > 0 > > > > Rotating the function by pi/4 (i.e. 45 degrees) > can > > be done by > > multiplying (x, f(x)) with a transformation matrix > to > > give (x*, > > f*(x)). After the transformation, y* is still > 0, > > but x* can be any > > value. Is there a way to convert this to a > "regular" > > function? > > > > E.g. suppose g() is f() rotated by pi/4, then: > > > > g(0) = k^(1/2) > > g(a) = g(-a) > > g(k - 1/2) = k + 1/2 > > > > What is g()? > > > > Thanks for reading. > > > > Regards, > > > > Yimin > > The coordinates (X*, Y*) are rotated clockwise by > pi/4 (counterclockwise by -pi/4) from (X, Y), so a > point with coordinates (x*, y*) in the (X*, Y*) > system has coordinates > x = cos(-pi/4) x* - sin(-pi/4) y* = (x* + > y*)/sqrt(2) > y = sin(-pi/4) x* + cos(-pi/4) y* = (-x* + > y*)/sqrt(2) > in the (X, Y) system. > > Substitution into xy = k gives > (x* + y*)/(-x* + y*) = 2k CORRECTION: (x* + y*)(-x* + y*) = 2k > > simplifying, > y* = sqrt[(x*)^2 + 2k] > > where we are justified in choosing y* > 0, because we > were given that x and k are > 0 (and thus y > 0), so > y* = (x + y)/sqrt(2) > 0. |