\r\n" );
document.write( " y²-4y-6x+13 = 0\r\n" );
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document.write( "Since y is squared the standard (vertex) form is\r\n" );
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document.write( " (y-k)² = 4p(x-h) (Some books use \"a\" instead of \"p\")\r\n" );
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document.write( "The vertex is the point (h,k)\r\n" );
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document.write( "The parabola will open right if the sign of p turns\r\n" );
document.write( "out to be positive and it will open left if it turns \r\n" );
document.write( "out to be negative.\r\n" );
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document.write( "y²-4y-6x+13 = 0\r\n" );
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document.write( "Isolate the terms in y on the left side of the equation:\r\n" );
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document.write( " y²-4y = 6x-13\r\n" );
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document.write( "Complete the square on the left:\r\n" );
document.write( "1. Multiply the coefficient of y by 
. 
\r\n" );
document.write( "2. Square the result of step 1. 
\r\n" );
document.write( "3. Add the result of step 2 to both sides of the equation.\r\n" );
document.write( "\r\n" );
document.write( " y²-4y+4 = 6x-13+4\r\n" );
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document.write( "1. Factor the trinomial on the left as a perfect square. \r\n" );
document.write( " (y-2)(y-2) = (y-2)²\r\n" );
document.write( "2. Combine the numbers on the right side of the equation:\r\n" );
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document.write( " (y-2)² = 6x-9\r\n" );
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document.write( "Factor the coefficient of x out on the right side. This will \r\n" );
document.write( "involve a fraction since we have to factor 6 out of -9. Had \r\n" );
document.write( "it been -12, it would have been easy but we have no choice \r\n" );
document.write( "but to factor 6 out of -9 and get a fraction 
which is \r\n" );
document.write( "what we get when we divide -9 by 6. Then reduces to 
.\r\n" );
document.write( "So the standard form is:\r\n" );
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document.write( " (y-2)² = 6(x-
)\r\n" );
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document.write( "We compare that to:\r\n" );
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document.write( " (y-k)² = 4p(x-h)\r\n" );
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document.write( "and see that k=2 and h= and 4p=6 or p=
=\r\n" );
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document.write( "Therefore the vertex is (h,k) = (,2) and since\r\n" );
document.write( "p is a positive number the parabola will open to the right.\r\n" );
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document.write( "So we plot the vertex:\r\n" );
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document.write( "
\r\n" );
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document.write( "Since p= or 
and the parabola opens to the right\r\n" );
document.write( "the focus is units to the right of the vertex. A parabola\r\n" );
document.write( "always curves around its focus, and since the parabola opens right,\r\n" );
document.write( "that's how we know that the focus is to the right of the vertex. The\r\n" );
document.write( "focus will have the same y-coordinate as the vertex, but its x-coordinate\r\n" );
document.write( "will be or units to the right of the vertex\r\n" );
document.write( "(,2), and since 
= 3, the focus will be\r\n" );
document.write( "the point (3,2). We plot the focus:\r\n" );
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document.write( "
\r\n" );
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document.write( "The directrix will be a vertical line p= units left of the\r\n" );
document.write( "vertex, which will mean that the directrix will coincide with the\r\n" );
document.write( "y-axis. So the y-axis IS the directrix and its equation is x=0.\r\n" );
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document.write( "To sketch the parabola we draw the focal chord (the so-called \"latus rectum\")\r\n" );
document.write( "which has length 4p = 6 units with the focus at its midpoint:\r\n" );
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document.write( "
\r\n" );
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document.write( "Now we can sketch in the parabola:\r\n" );
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document.write( "
\r\n" );
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document.write( "Summary. The standard form equation is (y-2)² = 6(x-)\r\n" );
document.write( " The parabola opens right\r\n" );
document.write( " The vertex is (,2) \r\n" );
document.write( " The focus is (3,2)\r\n" );
document.write( " The directrix is the line whose equation is x=0 which\r\n" );
document.write( " happens to be the y-axis.\r\n" );
document.write( " The focal chord, focal width, or latus rectum is 6 units\r\n" );
document.write( " and extends from the point (3,-1) to (3,5)\r\n" );
document.write( " The axis of symmetry is the line through the vertex and focus\r\n" );
document.write( " which bisects the parabola. It is the dotted line and its \r\n" );
document.write( " equation is y=2. \r\n" );
document.write( " \r\n" );
document.write( "That's more information that what you asked for, but you may need\r\n" );
document.write( "to give that on other parabola problems.\r\n" );
document.write( " \r\n" );
document.write( "Edwin
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
