document.write( "Question 1041196: Find the vertex,focus,directrix and axis of symmetry in the given equation:
\n" ); document.write( "(x+6)^2=8(-3y+17) \n" ); document.write( "
Algebra.Com's Answer #656138 by AnlytcPhil(1806) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We have to get it in the standard form:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
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" );
document.write( "\r\n" );
document.write( "The left side of\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "is already in that form.  We multiply out the right\r\n" );
document.write( "side\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we factor out -24\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Reduce the fraction:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we can compare it to\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and see that \r\n" );
document.write( "-h=+6 so h=-6\r\n" );
document.write( "-k = -17/3 so k = 17/3\r\n" );
document.write( "4p = -24, so p = -6\r\n" );
document.write( "\r\n" );
document.write( "The vertex is (h,k) or (-6,17/3)\r\n" );
document.write( "The parabola opens downward, since\r\n" );
document.write( "p is a negative number.\r\n" );
document.write( "\r\n" );
document.write( "We sketch the parabola by getting a few points\r\n" );
document.write( "It goes through the vertex (-6,17/3) as well\r\n" );
document.write( "as the points (-14,3), (-10,5), (-2,5), (2,3),\r\n" );
document.write( "(10,-5)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "The focus (which is inside the parabola is |p| or 17/6\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 17/3:\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "So the focus is (-6,-1/3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
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 17/3:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So the directrix is a horizontal line y = 35/3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
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" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );