Algebra.Com's Answer #557520 by AnlytcPhil(1807)![\"\"](\"/images/stars/stars-12.gif\")  
You can put this solution on YOUR website!
How can I find the equation of a parabola when i'm given the focus and directrix
\n" );
document.write( "of it?
\n" );
document.write( "In this case, the focus is (0,8) and the directrix is y=0
\n" );
document.write( "\r\n" );
document.write( "We first draw the the focus and the directrix (which is y=0 which is the x-axis,\r\n" );
document.write( "which I have made bluish-purple below):\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Next we know that since every point on a parabola is the same distance\r\n" );
document.write( "from the vertex as it is directly to the directrix, and since the vertex\r\n" );
document.write( "is on the parabola, it must also be exactly half way between the the\r\n" );
document.write( "focus (0,8) and the directrix (the x-axis). So the vertex is (0,4),\r\n" );
document.write( "since it's halfway between the focus (0,8) and the directrix (the bluish\r\n" );
document.write( "purple x-axis). So we draw the vertex (0,4):\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "There are two other points we can plot that are exactly the same distance \r\n" );
document.write( "from the focus (0,4) and they are to the directrix (the x-axis). They\r\n" );
document.write( "are the two points (8,8) and (-8,8). (Sometimes they are called \"the ends\r\n" );
document.write( "of the 'latus rectum', or 'focal chord', the line connecting them which I\r\n" );
document.write( "won't bother to draw ). We plot those two points also:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Now we can sketch in the parabola since it must go through the vertex\r\n" );
document.write( "(0,4) and those two points (8,8) and (-8,8):\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "The standard equation of a parabola opening upward or downward is:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "where the vertex is (h,k). Some books use \"a\" instead of \"p\". I will \r\n" );
document.write( "use p. \"p\" (or \"a\") is the distance from the vertex to the focus which\r\n" );
document.write( "is also the distance from the vertex to the directrix. p is taken to \r\n" );
document.write( "be positive if the parabola opens upward and negative if it opens \r\n" );
document.write( "downward. This one opens upward so we take p as positive. By counting \r\n" );
document.write( "the units we see that the vertex (0,4) is 4 units from the focus (0,8) \r\n" );
document.write( "and also 4 units from the directrix (the x-axis). So p=+4, and \r\n" );
document.write( "(h,k) = (0,4). So the standard equation:\r\n" );
document.write( "\r\n" );
document.write( " becomes upon substituting for h,k and p:\r\n" );
document.write( "\r\n" );
document.write( " or\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin \n" );
document.write( " |