document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=855915>Question 855915</A>: <I><SPAN STYLE=\"word-break: break-all;\"> write the equation of the parabola with vertex at the origin which satisfies the given condition:<BR>\n" );
document.write( "a.axis on the y-axis and passes through (6,-3)<BR>\n" );
document.write( "b.focus (0,4/3) and the equation of the directrix is y+4/3=0<BR>\n" );
document.write( "c.directrix is x-4=0<BR>\n" );
document.write( "d.focus at(0,2)<BR>\n" );
document.write( "e. latus rectum is 6 units and the parabola opens to the left<BR>\n" );
document.write( "f.focus on the x-axis and passes through (4,3)\r<BR>\n" );
document.write( "\n" );
document.write( "ive tried answering but i dont know if i got the ryt answers . can anybody help me ? thank u so much in advance god bless </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #515655 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=515655\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Equation of the parabola with vertex at the origin which satisfies the given condition:<BR>\n" );
document.write( " <BR>\n" );
document.write( "a. axis on the y-axis and passes through (6,-3)<BR>\n" );
document.write( "Since the axis is on the y-axis, the equation is of the form <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( "We just have to find the constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since it passes through the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3 ALIGN=MIDDLE  ALT=\"y=-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=a%2A6%5E2 ALIGN=MIDDLE  ALT=\"-3=a%2A6%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3=36a ALIGN=MIDDLE  ALT=\"-3=36a\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F36=a ALIGN=MIDDLE  ALT=\"-3%2F36=a\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1%2F12 ALIGN=MIDDLE  ALT=\"a=-1%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-%281%2F12%29x%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-%281%2F12%29x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-x%5E2%2F12%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-x%5E2%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "b. focus (0,4/3) and the equation of the directrix is y+4/3=0<BR>\n" );
document.write( "That is twice as much information as needed.<BR>\n" );
document.write( "We do not need both directrix and focus when we already have the vertex.<BR>\n" );
document.write( "The focus and directrix must be at the same distance from the vertex,<BR>\n" );
document.write( "so knowing that the vertex is at (0,0),<BR>\n" );
document.write( "if the focus is at (0,4/3) the directrix equation had to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4%2F3 ALIGN=MIDDLE  ALT=\"y=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B4%2F3=0 ALIGN=MIDDLE  ALT=\"y%2B4%2F3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In a parabola with a focal distance of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the absolute value of the coefficient of the squared term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4p ALIGN=MIDDLE  ALT=\"1%2F4p\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since the focus has the same x-coordinate as the vertex, which is the origin,<BR>\n" );
document.write( "and the focus has a positive y-coordinate,<BR>\n" );
document.write( "the rest of the parabola is also above the x-axis.<BR>\n" );
document.write( "So the coefficient is positive, and the equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F%284%2A%284%2F3%29%29%29x%5E2 ALIGN=MIDDLE  ALT=\"y=%281%2F%284%2A%284%2F3%29%29%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%283%2F16%29x%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%283%2F16%29x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "c. directrix is x-4=0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4=0 ALIGN=MIDDLE  ALT=\"x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><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;\" > , meaning that the focal distance is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=4 ALIGN=MIDDLE  ALT=\"p=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that the directrix is to the right of the vertex (the origin).<BR>\n" );
document.write( "That means that the focus and the parabola (except for the vertex) are to the left of the y-axis.<BR>\n" );
document.write( "Then the equation must be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%281%2F4p%29y%5E2 ALIGN=MIDDLE  ALT=\"x=-%281%2F4p%29y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%281%2F%284%2A4%29%29y%5E2 ALIGN=MIDDLE  ALT=\"x=-%281%2F%284%2A4%29%29y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-%281%2F16%29y%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-%281%2F16%29y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "d. focus at(0,2)<BR>\n" );
document.write( "That means the focal distance is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=2 ALIGN=MIDDLE  ALT=\"p=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the focus is above the (0,0) vertex.<BR>\n" );
document.write( "The equation of the parabola is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F%284%2A2%29x%5E2 ALIGN=MIDDLE  ALT=\"y=1%2F%284%2A2%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%281%2F8%29x%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%281%2F8%29x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "e. latus rectum is 6 units and the parabola opens to the left.<BR>\n" );
document.write( "With the vertex at the origin and the parabola opening to the left, the equation  is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=ay%5E2 ALIGN=MIDDLE  ALT=\"x=ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a negative coefficient, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "with the focal distance represented by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the latus rectum length is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4p ALIGN=MIDDLE  ALT=\"4p\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the absolute value of the coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4p ALIGN=MIDDLE  ALT=\"1%2F4p\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the reciprocal of the latus rectum length.<BR>\n" );
document.write( "So the equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-%281%2F6%29y%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-%281%2F6%29y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );