document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204077>Question 1204077</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you graph the parabola of -12(x+3)=(y-3)^2 that includes the directrix and focus on the graph?\r<BR>\n" );
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document.write( "Focus = (-6,3)<BR>\n" );
document.write( "Directrix = x=0<BR>\n" );
document.write( "Vertex = (-3,3)\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840094 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=840094\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-12%28x%2B3%29=%28y-3%29%5E2 ALIGN=MIDDLE  ALT=\"+-12%28x%2B3%29=%28y-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Focus = (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6 ALIGN=MIDDLE  ALT=\"+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3 ALIGN=MIDDLE  ALT=\"+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "Directrix = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=0 ALIGN=MIDDLE  ALT=\"+x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Vertex = (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-3 ALIGN=MIDDLE  ALT=\"+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3 ALIGN=MIDDLE  ALT=\"+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > )\r<BR>\n" );
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document.write( "Two points determine any line. However, since a parabola is curved, we should find more than two points. In this text, we will determine at least five points as a means to produce an acceptable sketch. \r<BR>\n" );
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document.write( "To begin, we graph our first parabola by plotting points. Given a quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++-12%28x%2B3%29=%28y-3%29%5E2 ALIGN=MIDDLE  ALT=\"++-12%28x%2B3%29=%28y-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<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 the independent variable and <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 the dependent variable (so, you choose it). Choose some values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x ALIGN=MIDDLE  ALT=\"+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and then determine the corresponding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y ALIGN=MIDDLE  ALT=\"+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > -values. Then plot the points and sketch the graph.\r<BR>\n" );
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document.write( "since you are already given parabola opening sideways and focus is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6 ALIGN=MIDDLE  ALT=\"+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3 ALIGN=MIDDLE  ALT=\"+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ), means the parabola is opening to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+left ALIGN=MIDDLE  ALT=\"+left\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and graph will be in II and III quadrant<BR>\n" );
document.write( "you are also given two points, use them and use formula to find three more\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-12%28x%2B3%29=%28y-3%29%5E2 ALIGN=MIDDLE  ALT=\"+-12%28x%2B3%29=%28y-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=-4 ALIGN=MIDDLE  ALT=\"+x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-12%28-4%2B3%29=%28y-3%29%5E2 ALIGN=MIDDLE  ALT=\"+-12%28-4%2B3%29=%28y-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=6.46+ ALIGN=MIDDLE  ALT=\"+y=6.46+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.46 ALIGN=MIDDLE  ALT=\"y=-0.46\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-12%28x%2B3%29=%28y-3%29%5E2 ALIGN=MIDDLE  ALT=\"+-12%28x%2B3%29=%28y-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=-10 ALIGN=MIDDLE  ALT=\"+x=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-12%28-10%2B3%29=%28y-3%29%5E2 ALIGN=MIDDLE  ALT=\"+-12%28-10%2B3%29=%28y-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=12.16 ALIGN=MIDDLE  ALT=\"+y=12.16\"   DISABLED_style= \"max-width:90%; height:auto;\" >   or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6.16 ALIGN=MIDDLE  ALT=\"-6.16\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+ ALIGN=MIDDLE  ALT=\"x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.46 ALIGN=MIDDLE  ALT=\"6.46\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.46 ALIGN=MIDDLE  ALT=\"-0.46\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12.16 ALIGN=MIDDLE  ALT=\"12.16\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6.16 ALIGN=MIDDLE  ALT=\"-6.16\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "plot these points and sketch the graph\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+600%2C+600%2C+-20%2C+10%2C+-20%2C20%2C%0D%0Alocate%28-6%2C3%2CF%28-6%2C3%29%29%2Ccircle%28-6%2C3%2C.15%29%2Ccircle%28-3%2C3%2C.15%29%2C%0D%0Agreen%28line%28-0.2%2C-20%2C-0.2%2C20%29%29%2Clocate%28-3%2C3%2CV%28-6%2C3%29%29%2C%0D%0Acircle%28-4%2C6.46%2C.15%29%2Ccircle%28-4%2C-0.46%2C.15%29%2C%0D%0Acircle%28-10%2C12.16%2C.15%29%2Ccircle%28-10%2C-6.16%2C.15%29%2C%0D%0Agraph%28+600%2C+600%2C+-20%2C+10%2C+-20%2C+20%2C+3+-2sqrt%283%29%2Asqrt%28-x-3%29%2C2sqrt%283%29%2Asqrt%28-x-3%29%2B3%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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