document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=949683>Question 949683</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me find the vertex and the directrix of the following parabola : 9(x-1)^2=36(y+2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #579819 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=579819\">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=36%28y%2B2%29=9%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"36%28y%2B2%29=9%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B2=%289%2F36%29%2A%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"y%2B2=%289%2F36%29%2A%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the equation of a parabola<BR>\n" );
document.write( "with axis of symmetry <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >>---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "That is easy to see even if you know nothing about parabolas.<BR>\n" );
document.write( "You do not need to know anything about parabolas to realize that the equation<BR>\n" );
document.write( "gives the same value to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B2 ALIGN=MIDDLE  ALT=\"y%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0.1 ALIGN=MIDDLE  ALT=\"x-1=0.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=-0.1 ALIGN=MIDDLE  ALT=\"x-1=-0.1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Generalizing, we would say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B2 ALIGN=MIDDLE  ALT=\"y%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the same value<BR>\n" );
document.write( "when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%3E0 ALIGN=MIDDLE  ALT=\"d%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the right of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=d ALIGN=MIDDLE  ALT=\"x-1=d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),<BR>\n" );
document.write( "as when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%3E0 ALIGN=MIDDLE  ALT=\"d%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the left of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-x=d ALIGN=MIDDLE  ALT=\"1-x=d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=-d ALIGN=MIDDLE  ALT=\"x-1=-d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( " <BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%3E=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2B2%29=%289%2F36%29%2A%28x-1%29%5E2=%289%2F36%29%2A0=0 ALIGN=MIDDLE  ALT=\"%28y%2B2%29=%289%2F36%29%2A%28x-1%29%5E2=%289%2F36%29%2A0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B2=0 ALIGN=MIDDLE  ALT=\"y%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E1 ALIGN=MIDDLE  ALT=\"x%3C%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1%3C%3E0 ALIGN=MIDDLE  ALT=\"x-1%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%3E0 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2B2%29=%289%2F36%29%2A%28x-1%29%5E2%3E%289%2F36%29%2A0=0 ALIGN=MIDDLE  ALT=\"%28y%2B2%29=%289%2F36%29%2A%28x-1%29%5E2%3E%289%2F36%29%2A0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E1 ALIGN=MIDDLE  ALT=\"x%3C%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B2%3E0 ALIGN=MIDDLE  ALT=\"y%2B2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E-2 ALIGN=MIDDLE  ALT=\"y%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "To either side of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > the value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > is greater than for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) is the minimum of the graph,<BR>\n" );
document.write( "and that the point (1,-2) is the vertex of the parabola.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Finding the directrix requires knowing a bit about parabolas.<BR>\n" );
document.write( "Maybe you remember (or finding in your textbook) a definition that says that a parabola is the set of points on a plane, such that<BR>\n" );
document.write( "the distance from the vertex to the focus and<BR>\n" );
document.write( "the distance from the vertex to a line called the directrix are the same.<BR>\n" );
document.write( "From that definition, with a little algebra knowledge, and a lot of thinking,<BR>\n" );
document.write( "you could get all you need to know about parabolas<BR>\n" );
document.write( "(unless they ask you what is the \"latus rectum\" ).<BR>\n" );
document.write( "Most people just read on until they find a formula that they can apply to the problem, whether they understand why it is so or not.<BR>\n" );
document.write( "Many try to memorize that formula and remember it,<BR>\n" );
document.write( "at least until they take the test on parabolas.<BR>\n" );
document.write( "However, just from the definition of parabola,<BR>\n" );
document.write( "you could get to the conclusion that<BR>\n" );
document.write( "1) the equation for any parabola with a horizontal directrix,<BR>\n" );
document.write( "and a vertex at the origin can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2 ALIGN=MIDDLE  ALT=\"y=ax%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ;<BR>\n" );
document.write( "2) the focus is at (0,1/4a) ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4a ALIGN=MIDDLE  ALT=\"1%2F4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > above the vertex), and<BR>\n" );
document.write( "3) the directrix is the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1%2F4a ALIGN=MIDDLE  ALT=\"y=-1%2F4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4a ALIGN=MIDDLE  ALT=\"1%2F4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > below the vertex.<BR>\n" );
document.write( "If the vertex were instead the point (h,k),<BR>\n" );
document.write( "the formula would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-k=a%28x-h%29%5E2 ALIGN=MIDDLE  ALT=\"y-k=a%28x-h%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but the rest would be the same.<BR>\n" );
document.write( "Your equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B2=%289%2F36%29%2A%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"y%2B2=%289%2F36%29%2A%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-k=a%28x-h%29%5E2 ALIGN=MIDDLE  ALT=\"y-k=a%28x-h%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28h=1%2Ck=-2%2Ca=9%2F36%29 ALIGN=MIDDLE  ALT=\"system%28h=1%2Ck=-2%2Ca=9%2F36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the vertex, (h,k), is (1,-2), <BR>\n" );
document.write( "and the directrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4a=1%2F%284%2A%289%2F36%29%29=1 ALIGN=MIDDLE  ALT=\"1%2F4a=1%2F%284%2A%289%2F36%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > below the vertex.<BR>\n" );
document.write( "So the directrix is the horizontal line with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2-1 ALIGN=MIDDLE  ALT=\"y=-2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-3%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Here is what the parabola and directrix look like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C6%2C-4%2C6%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-4%2C6%2C-4%2C6%2C%289%2F36%29%2A%28x-1%29%5E2-2%2C-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );