document.write( "Question 480568: Graph each hyperbola. Include labels indicating the coordinates of the foci and vertices and the equations of the asymptotes.\r
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\n" ); document.write( "thanks for ur time \n" ); document.write( "
Algebra.Com's Answer #329112 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!

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document.write( "First get it in standard form, which is either\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\" if the hyperbola opens right and left, \r\n" );
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document.write( "and the slopes of the asymptotes are \"%22%22%2B-b%2Fa\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1\" if the hyperbola opens upward and downward.\r\n" );
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document.write( "and the slopes of the asymptotes are \"%22%22%2B-a%2Fb\"\r\n" );
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document.write( "\"4x%5E2-40x-7y%5E2-14y%2B65=0\"\r\n" );
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document.write( "Get the 65 off the left side:\r\n" );
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document.write( "\"4x%5E2-40x-7y%5E2-14y=-65\"\r\n" );
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document.write( "Factor out the coefficient of \"x%5E2\" out of the \r\n" );
document.write( "first two terms on the left. \r\n" );
document.write( "Factor out the coefficient of \"y%5E2\" out of the \r\n" );
document.write( "last two terms on the left. \r\n" );
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document.write( "\"4%28x%5E2-10x%29-7%28y%5E2%2B2y%29=-65\"\r\n" );
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document.write( "Complete the square on \"%28x%5E2-10x%29\" by multiplying\r\n" );
document.write( "the coefficient of x, which is -10, by \"1%2F2\" getting -5,\r\n" );
document.write( "and then squaring -5, getting 25.  And we add that inside the\r\n" );
document.write( "first parentheses.  However since there is a 4 in front of the\r\n" );
document.write( "first parentheses, adding 25 inside the parentheses amounts\r\n" );
document.write( "to adding 4*25 or 100 to the left side, so we must add 100 \r\n" );
document.write( "to the right side:\r\n" );
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document.write( "\"4%28x%5E2-10x%2Bred%2825%29%29-7%28y%5E2%2B2y%29=-65%2Bred%28100%29\"\r\n" );
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document.write( "Complete the square on \"%28y%5E2%2B2y%29\" by multiplying\r\n" );
document.write( "the coefficient of y, which is 2, by \"1%2F2\" getting 1,\r\n" );
document.write( "and then squaring 1, getting 1.  And we add that inside the\r\n" );
document.write( "second parentheses.  However since there is a -7 in front of the\r\n" );
document.write( "second parentheses, adding 1 inside the parentheses amounts\r\n" );
document.write( "to adding \"-7%2A1\" or -7 to the left side, so we must add -7 \r\n" );
document.write( "to the right side:\r\n" );
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document.write( "Factor the parentheses as squares of binomials, and combine\r\n" );
document.write( "the numbers on the right:\r\n" );
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document.write( "\"4%28x-5%29%5E2-7%28y%2B1%29%5E2=28\"\r\n" );
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document.write( "Get a 1 on the right by dividing through by 4\r\n" );
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document.write( "\"4%28x-5%29%5E2%2F28-7%28y%2B1%29%5E2%2F28=28%2F28\"\r\n" );
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document.write( "\"%28x-5%29%5E2%2F7-%28y%2B1%29%5E2%2F4=1\"\r\n" );
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document.write( "Since the variable x comes first in the standard form, the\r\n" );
document.write( "hyperbola opens right and left.\r\n" );
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document.write( "So we compare that to:\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "The center is at (5,-1).  So we plot the center:\r\n" );
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document.write( "\"a%5E2=7\" so \"a=sqrt%287%29\", the semi-transverse axis is\r\n" );
document.write( "\"sqrt%287%29\" units long, and that's about 2.6. So we draw the \r\n" );
document.write( "complete transverse axis right and left about 2.6 unit from the \r\n" );
document.write( "center.  The tranverse axis is the horizontal green line below:\r\n" );
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document.write( "\"Trans\" means \"across\" and \"VERse\" means \"VERtices\", so the \r\n" );
document.write( "transverse axis goes across from vertex to vertex, and so the \r\n" );
document.write( "ends of the transverse axis are the vertices and they are found \r\n" );
document.write( "by adding and subtracting \"a=sqrt%287%29\" from the x-coordinate \r\n" );
document.write( "of the center, so \r\n" );
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document.write( "The VERTICES are (5+\"sqrt%287%29\",-1) and (5-\"sqrt%287%29\",-1)\r\n" );
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document.write( "\"b%5E2=4\" so \"b=2\", the semi-conjugate axis is 2 units\r\n" );
document.write( "long, so we draw the complete conjugate axis up and down\r\n" );
document.write( "2 units from the center, that is, the conjugate axis is the \r\n" );
document.write( "vertical green line below:\r\n" );
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document.write( "Now we draw in the defining rectangle\r\n" );
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document.write( "Now we can draw the asymptotes which are the extended diagonals\r\n" );
document.write( "of the defining rectangle:\r\n" );
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document.write( "and we can sketch in the hyperbola:\r\n" );
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document.write( "The slopes of the asymptotes are \"%22%22%2B-b%2Fa+=+%22%22%2B-+2%2Fsqrt%287%29=%22%22%2B-+2sqrt%287%29%2F7\"\r\n" );
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document.write( "They pass through the center (5,-1) so we use the point-slope\r\n" );
document.write( "formula:\r\n" );
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document.write( "\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"\r\n" );
document.write( "\"y-%28-1%29=expr%282sqrt%287%29%2F7%29%28x-5%29\"\r\n" );
document.write( "\"y%2B1=expr%282sqrt%287%29%2F7%29%28x-5%29\"\r\n" );
document.write( "\"7y%2B7=14sqrt%287%29%28x-5%29\"\r\n" );
document.write( "\"7y%2B7=14sqrt%287%29x-90sqrt%287%29\"\r\n" );
document.write( "\"7y-14sqrt%287%29x=-90sqrt%287%29-7\"\r\n" );
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document.write( "That's the equation of one of the asymptotes. The other one:\r\n" );
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document.write( "\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"\r\n" );
document.write( "\"y-%28-1%29=expr%28-2sqrt%287%29%2F7%29%28x-5%29\"\r\n" );
document.write( "\"y%2B1=expr%28-2sqrt%287%29%2F7%29%28x-5%29\"\r\n" );
document.write( "\"7y%2B7=-14sqrt%287%29%28x-5%29\"\r\n" );
document.write( "\"7y%2B7=-14sqrt%287%29x%2B90sqrt%287%29\"\r\n" );
document.write( "\"7y%2B14sqrt%287%29x=90sqrt%287%29-7\"\r\n" );
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document.write( "The foci are the two points \"c\" units right and left of the\r\n" );
document.write( "vertex, where \r\n" );
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document.write( "c² = a² + b²\r\n" );
document.write( "c² = (sqrt(7))² + 2²\r\n" );
document.write( "c² = 7 + 4\r\n" );
document.write( "c² = 11\r\n" );
document.write( "c = ±\"sqrt%2811%29\"\r\n" );
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document.write( "So the foci are  (5+\"sqrt%2811%29\",-1) and (5-\"sqrt%2811%29\",-1),\r\n" );
document.write( "Here they are plotted.  They are inside the branches of they\r\n" );
document.write( "hyperbola:\r\n" );
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document.write( "I don't have time to do the other one.  It's a hyperbola\r\n" );
document.write( "that opens upward and downward.  But I think you can do it.\r\n" );
document.write( "Just remember that the transverse axis is vertical and the\r\n" );
document.write( "conjugate axis is horizontal.\r\n" );
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document.write( "Edwin
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