document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=587098>Question 587098</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the parabola with vertex at (3,-2), focus at (3,4). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #373992 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20059)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=373992\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the parabola with vertex at (3,-2), focus at (3,4).\r<BR>\n" );
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document.write( "Let's plot the vertex and focus\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%2C%0D%0Acircle%283%2C-2%2C.15%29%2C+circle%283%2C4%2C.15%29%2C+locate%283%2C-2%2Cvertex%29%2C+locate%283%2C4%2Cfocus%29%2C%0D%0Acircle%283%2C-2%2C.1%29%2C+circle%283%2C4%2C.1%29%2C%0D%0Acircle%283%2C-2%2C.05%29%2C+circle%283%2C4%2C.05%29%2C%0D%0A%0D%0Agraph%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The equation of a parabola with focus above or below the vertex is\r<BR>\n" );
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document.write( "   (x - h)² = 4p(y - k)\r<BR>\n" );
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document.write( "where the vertex is (h,k) and p = distance from vertex to to focus,<BR>\n" );
document.write( "taken as a positive number if the focus is above the vertex and<BR>\n" );
document.write( "negative if the focus is below the vertex.\r<BR>\n" );
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document.write( "In this case the vertex = (h,k) = (3,-2)\r<BR>\n" );
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document.write( "By counting the units from the vertex up to the focus we see<BR>\n" );
document.write( "that p = 6, and is taken positive since the focus is above the<BR>\n" );
document.write( "vertex, which also means that the parabola opens upward, \r<BR>\n" );
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document.write( "So the equation is:\r<BR>\n" );
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document.write( "   (x - h)² = 4p(y - k)\r<BR>\n" );
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document.write( "   (x - 3)² = 4(6)(y - (-2))\r<BR>\n" );
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document.write( "   (x - 3)² = 24(y + 2)\r<BR>\n" );
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document.write( "That's all you were asked to find.  But you might have<BR>\n" );
document.write( "been asked to graph it and find the directrix.\r<BR>\n" );
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document.write( "We draw the focal chord (also called \"latus rectum\")<BR>\n" );
document.write( "which has length 4p = 4(6) = 24 and whose midpoint is <BR>\n" );
document.write( "the focus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%2C%0D%0Acircle%283%2C-2%2C.15%29%2C+circle%283%2C4%2C.15%29%2C+locate%283%2C-2%2Cvertex%29%2C+locate%283%2C4%2Cfocus%29%2C%0D%0Acircle%283%2C-2%2C.1%29%2C+circle%283%2C4%2C.1%29%2C%0D%0Acircle%283%2C-2%2C.05%29%2C+circle%283%2C4%2C.05%29%2C%0D%0Agreen%28line%28-12%2C4%2C15%2C4%29%29%2C%0D%0Agraph%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And we can sketch in the parabola:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%2C%0D%0Acircle%283%2C-2%2C.15%29%2C+circle%283%2C4%2C.15%29%2C+locate%283%2C-2%2Cvertex%29%2C+locate%283%2C4%2Cfocus%29%2C%0D%0Acircle%283%2C-2%2C.1%29%2C+circle%283%2C4%2C.1%29%2C%0D%0Acircle%283%2C-2%2C.05%29%2C+circle%283%2C4%2C.05%29%2C%0D%0Agreen%28line%28-9%2C4%2C15%2C4%29%29%2C%0D%0Agraph%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%2C-2%2B%28x-3%29%5E2%2F24%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can also find the directrix which is a horizontal line p = 6<BR>\n" );
document.write( "units below the vertex, the dotted line below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%2C%0D%0Acircle%283%2C-2%2C.15%29%2C+circle%283%2C4%2C.15%29%2C+locate%283%2C-2%2Cvertex%29%2C+locate%283%2C4%2Cfocus%29%2C%0D%0Acircle%283%2C-2%2C.1%29%2C+circle%283%2C4%2C.1%29%2C%0D%0Acircle%283%2C-2%2C.05%29%2C+circle%283%2C4%2C.05%29%2C%0D%0Agreen%28line%28-9%2C4%2C15%2C4%29%29%2C%0D%0Agraph%28800%2C16000%2F33%2C-15%2C18%2C-10%2C10%2C-2%2B%28x-3%29%5E2%2F24%2C20%2C%0D%0A%0D%0A-8sqrt%28sin%2810x%29%29%0D%0A%2F%0D%0Asqrt%28sin%2810x%29%29%29+%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The directrix has equation y = -8\r<BR>\n" );
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document.write( "Edwin</pre> </SPAN>\n" );
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