document.write( "Question 329720: Please sketch this hyperbola and show me how you did it: x^2-y^2=1. \n" ); document.write( "
Algebra.Com's Answer #236342 by Edwin McCravy(20056)\"\" \"About 
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Please sketch this hyperbola and show me how you did it: x^2-y^2=1.
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document.write( "The equation\r\n" );
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document.write( "\"x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1\" is the equation of a hyperbola that opens\r\n" );
document.write( "right and left with these properties:\r\n" );
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document.write( "1.  The center is at the origin (0,0).\r\n" );
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document.write( "2.  The vertices are the ends of the transverse axis (-a,0) and (a,0).\r\n" );
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document.write( "3.  The endpoints of the conjugate axis are (0,b) and (0,-b).\r\n" );
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document.write( "4.  The foci are the points (-c,0) and (c,0) where \"c=sqrt%28a%5E2%2Bb%5E2%29\"\r\n" );
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document.write( "5.  The defining rectangle has the 4 corners \r\n" );
document.write( "    (-a,b), (-a,-b), (a,-b), and (a,b)\r\n" );
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document.write( "6.  The asymptotes are the extended diagonals of the defining rectangle\r\n" );
document.write( "    and have equations \"y\"\"%22%22=%22%22\"\"%22%22+%2B-+%28b%2Fa%29\"\"x\"\r\n" );
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document.write( "Your equation is \r\n" );
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document.write( "\"x%5E2-y%5E2=1\"\r\n" );
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document.write( "Write it as \r\n" );
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document.write( "\"x%5E2%2F1-y%5E2%2F1=1\"\r\n" );
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document.write( "and we see that \"a%5E2=1\" and \"b%5E2=1\", so \"a=1\" and \"b=1\",\r\n" );
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document.write( "so:\r\n" );
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document.write( "1.  The center is at the origin (0,0).\r\n" );
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document.write( "2.  The vertices are the ends of the transverse axis (-1,0) and (1,0).\r\n" );
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document.write( "3.  The endpoints of the conjugate axis are (0,1) and (0,-1).\r\n" );
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document.write( "4.  The foci are the points (-c,0) and (c,0) where \"c=sqrt%28a%5E2%2Bb%5E2%29\"\r\n" );
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document.write( "    So we calculate \"c=sqrt%28a%5E2%2Bb%5E2%29=sqrt%281%5E2%2B1%5E2%29=sqrt%281%2B1%29=sqrt%282%29\"\r\n" );
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document.write( "    and the foci are (\"-sqrt%282%29\",0) and (\"sqrt%282%29\",0) \r\n" );
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document.write( "5.  The defining rectangle has the 4 corners \r\n" );
document.write( "    (-1,1), (-1,-1), (1,-1), and (1,1)\r\n" );
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document.write( "6.  The asymptotes are the extended diagonals of the defining rectangle\r\n" );
document.write( "    and have equations \"y\"\"%22%22=%22%22\"\"%22%22+%2B-+%281%2F1%29\"\"x\", or\r\n" );
document.write( "    \"y\"\"%22%22=%22%22\"\"%22%22+%2B-+x\"\r\n" );
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document.write( "So we draw the defining rectangle and its extended diagonals:\r\n" );
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document.write( "    \r\n" );
document.write( "Now we sketch in the hyperbola:\r\n" );
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document.write( "Edwin

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