document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=715758>Question 715758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> equation of a tilted parabola </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #439576 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=439576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, notice that the equation of the parabola y = x^2 can be parametrized by x = t, y = t^2, as t goes from -infinity to infinity; or, as a column vector,<BR>\n" );
document.write( " [x] = [t]<BR>\n" );
document.write( " [y] = [t^2].<BR>\n" );
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document.write( "To rotate the graph of the parabola about the origin, we rotate each point individually. Rotation clockwise by 45 degrees is a linear transformation; the transformation sends the point (1, 0) to (1/sqrt(2), -1/sqrt(2)), and it sends the point (0, 1) to (1/sqrt(2), sqrt(2)). So the standard matrix for the linear transformation is<BR>\n" );
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document.write( "[1/sqrt(2) 1/sqrt(2)]<BR>\n" );
document.write( " [-1/sqrt(2) 1/sqrt(2)]<BR>\n" );
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document.write( "Thus, if we apply this linear transformation to a point (t, t^2) on the graph of the parabola, we get<BR>\n" );
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document.write( "[1/sqrt(2) 1/sqrt(2)][t]<BR>\n" );
document.write( " [-1/sqrt(2) 1/sqrt(2)][t^2]<BR>\n" );
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document.write( "Which gives<BR>\n" );
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document.write( "[(t^2 + t)/sqrt(2)]<BR>\n" );
document.write( " [(t^2 - t)/sqrt(2)].<BR>\n" );
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document.write( "So, as t goes from -infinity to infinity, this is a parametrization of the graph of the rotated parabola. We must now convert back to x and y.<BR>\n" );
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document.write( "We have<BR>\n" );
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document.write( "[x] = [(t^2 + t)/sqrt(2)]<BR>\n" );
document.write( " [y] = [(t^2 - t)/sqrt(2)]<BR>\n" );
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document.write( "By adding and subtracting x and y, we find<BR>\n" );
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document.write( "x - y = t * sqrt(2)<BR>\n" );
document.write( " x + y = t^2 * sqrt(2)<BR>\n" );
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document.write( "Solving the first equation for t, we get t = (x - y) / sqrt(2). Plug this into the second equation:<BR>\n" );
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document.write( "x + y = ((x - y) / sqrt(2))^2 * sqrt(2)<BR>\n" );
document.write( " x + y = ((x^2 - 2xy + y^2) / 2) * sqrt(2)<BR>\n" );
document.write( " x + y = (x^2 - 2xy + y^2) / sqrt(2)<BR>\n" );
document.write( " x * sqrt(2) + y * sqrt(2) = x^2 - 2xy + y^2<BR>\n" );
document.write( " x^2 - 2xy + y^2 - x * sqrt(2) - y * sqrt(2) = 0.<BR>\n" );
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document.write( "So an equation for the parabola y = x^2 rotated clockwise by 45 degrees is<BR>\n" );
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document.write( "x^2 - 2xy + y^2 - x * sqrt(2) - y * sqrt(2) = 0.<BR>\n" );
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document.write( "Copied from somewhere, I lost the link to the site. </SPAN>\n" );
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