document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012164>Question 1012164</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the vertex directrix and focus of the equation (x+5)^2=4y </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #628018 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=628018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%5E2=4y+ ALIGN=MIDDLE  ALT=\"%28x%2B5%29%5E2=4y+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4%29%28x%2B5%29%5E2+ ALIGN=MIDDLE  ALT=\"y=%281%2F4%29%28x%2B5%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....compare to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-5 ALIGN=MIDDLE  ALT=\"h=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "so, vertex is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "or, we can use  standard form and calculate coordinates of the vertex like this:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4%29%28x%5E2%2B10x%2B25%29+ ALIGN=MIDDLE  ALT=\"y=%281%2F4%29%28x%5E2%2B10x%2B25%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4%29x%5E2%2B%281%2F4%2910x%2B%281%2F4%2925+ ALIGN=MIDDLE  ALT=\"y=%281%2F4%29x%5E2%2B%281%2F4%2910x%2B%281%2F4%2925+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4%29x%5E2%2B%285%2F2%29x%2B%2825%2F4%29+ ALIGN=MIDDLE  ALT=\"y=%281%2F4%29x%5E2%2B%285%2F2%29x%2B%2825%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "the x-coordinate of the focus is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b+%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+-b+%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%285%2F2%29%2F+%282%281%2F4%29%29 ALIGN=MIDDLE  ALT=\"x+=+-%285%2F2%29%2F+%282%281%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%285%2F2%29%2F+%281%2F2%29 ALIGN=MIDDLE  ALT=\"x+=+-%285%2F2%29%2F+%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%2810%2F2%29 ALIGN=MIDDLE  ALT=\"x+=+-%2810%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-5 ALIGN=MIDDLE  ALT=\"x+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "then, y-coordinate is\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4%29%28-5%29%5E2%2B%285%2F2%29%28-5%29%2B%2825%2F4%29+ ALIGN=MIDDLE  ALT=\"y=%281%2F4%29%28-5%29%5E2%2B%285%2F2%29%28-5%29%2B%2825%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4%2925-%2825%2F2%29%2B%2825%2F4%29+ ALIGN=MIDDLE  ALT=\"y=%281%2F4%2925-%2825%2F2%29%2B%2825%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=25%2F4-50%2F4%2B%2825%2F4%29+ ALIGN=MIDDLE  ALT=\"y=25%2F4-50%2F4%2B%2825%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2F4-50%2F4+ ALIGN=MIDDLE  ALT=\"y=50%2F4-50%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0+ ALIGN=MIDDLE  ALT=\"y=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "so, we got same result for vertex\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the focus is:\r<BR>\n" );
document.write( "\n" );
document.write( "The focus lies on the axis of symmetry of the parabola, and has the y-coordinate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2B1%2F%284a%29 ALIGN=MIDDLE  ALT=\"k%2B1%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Because we just found the vertex to be (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we know the axis of symmetry to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the focus lies on that line.\r<BR>\n" );
document.write( "\n" );
document.write( "y-coordinate is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%284a%29+=1%2F%284%281%2F4%29%29=1%2F%284%2F4%29=1 ALIGN=MIDDLE  ALT=\"1%2F%284a%29+=1%2F%284%281%2F4%29%29=1%2F%284%2F4%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "and the coordinates of the focus are:(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once you know the y=coordinate of the vertex, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+k+%96+p ALIGN=MIDDLE  ALT=\"y+=+k+%96+p\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+1%2F%284a%29 ALIGN=MIDDLE  ALT=\"p+=+1%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Thus, as we calculated for the focus, above:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+1%2F%284a%29+=+1%2F%284%2A%281%2F4%29%29+=+1 ALIGN=MIDDLE  ALT=\"p+=+1%2F%284a%29+=+1%2F%284%2A%281%2F4%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "and the directrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0+%96+1+=-1 ALIGN=MIDDLE  ALT=\"y+=+0+%96+1+=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Acircle%28-5%2C0%2C.12%29%2Ccircle%28-5%2C1%2C.12%29%2C%0D%0Alocate%28-5%2C-0.5%2CV%28-5%2C0%29%29%2Clocate%28-5%2C1%2CF%28-5%2C0%29%29%2C%0D%0A+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C-1%2C-1%2C+%281%2F4%29%28x%5E2%2B10x%2B25%29%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );