document.write( "Question 1041223: Find the vertex,focus,directrix,and the axis of symmetry in the given equation:\r
\n" ); document.write( "\n" ); document.write( "(x+6)^2 = -24y+51 \n" ); document.write( "
Algebra.Com's Answer #656190 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Question 1041196
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document.write( "\"%28x%2B6%29%5E2+=+-24y%2B51\"\r\n" );
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document.write( "We have to get it in the standard form:\r\n" );
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document.write( "\"x-h%29%5E2=4p%28y-k%29\"\r\n" );
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document.write( "where (h,k) is the vertex, and |p| is the distance\r\n" );
document.write( "from the vertex to the focus and also the distance\r\n" );
document.write( "from the vertex to the directrix.  If p is positive\r\n" );
document.write( "the parabola opens upward and if p is negative the\r\n" );
document.write( "parabola opens downward.\r\n" );
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document.write( "The left side of\r\n" );
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document.write( "\"%28x%2B6%29%5E2+=+-24y%2B51\"\r\n" );
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document.write( "is already in that form.  We factor out -24\r\n" );
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document.write( "\"%28x%2B6%29%5E2=-24%28y-51%2F24%29\"\r\n" );
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document.write( "Reduce the fraction:\r\n" );
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document.write( "\"%28x%2B6%29%5E2=-24%28y-17%2F8%29\"\r\n" );
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document.write( "Now we can compare it to\r\n" );
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document.write( "\"x-h%29%5E2=4p%28y-k%29\"\r\n" );
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document.write( "and see that \r\n" );
document.write( "-h=+6 so h=-6\r\n" );
document.write( "-k = -17/8 so k = 17/8\r\n" );
document.write( "4p = -24, so p = -6\r\n" );
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document.write( "The vertex is (h,k) or (-6,17/8).\r\n" );
document.write( "We sketch the parabola. It goes through the \r\n" );
document.write( "vertex (-6,17/8) as opens downward since p \r\n" );
document.write( "is a negative number.\r\n" );
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document.write( "The focus (which is inside the parabola is |p| or 6\r\n" );
document.write( "units below the vertex.  It has the same x-coordinate -6\r\n" );
document.write( "So to find its y-coordinate we subtract 6 from the\r\n" );
document.write( "y-coordinate of the vertex 17/8:\r\n" );
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document.write( "\"17%2F8-6=17%2F8-48%2F8=-31%2F8\"  \r\n" );
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document.write( "So the focus is (-6,-31/8)\r\n" );
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document.write( "Finally we will draw the directrix, which is a horizontal\r\n" );
document.write( "line (in green below) which is |p| = 6 units outside (above)\r\n" );
document.write( "the vertex.  We determine how far above the x-axis that is\r\n" );
document.write( "by adding 6 to the y-coordinate of the vertex 17/8:\r\n" );
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document.write( "\"17%2F8%2B6=17%2F8%2B48%2F8=65%2F8\"\r\n" );
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document.write( "So the directrix is a horizontal line y = 65/8\r\n" );
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document.write( "  \r\n" );
document.write( "The axis of symmetry is the vertical line (in blue below) \r\n" );
document.write( "through both the vertex and the focus which cuts the\r\n" );
document.write( "parabola in two.  It is the line x=(the x-coordinate of\r\n" );
document.write( "the vertex and focus).  In this case the equation of the\r\n" );
document.write( "axis of symmetry is x=-6:\r\n" );
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document.write( "Edwin
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