document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1032247>Question 1032247</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Use the focus-directrix definition of a parabola to answer the following questions.\r<BR>\n" );
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document.write( "1) How would the shape of the parabola change if the focus were moved up, away from the directrix? How would we describe p?<BR>\n" );
document.write( "2) How would the shape of the parabola change if the focus were moved down, toward the directrix? How would we describe p?<BR>\n" );
document.write( "3) How would the shape of the parabola change if the focus were moved down, below the directrix?\r<BR>\n" );
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document.write( "2. Given:\r<BR>\n" );
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document.write( "y = 3x^2\r<BR>\n" );
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document.write( "1) Write the equation of a parabola that contains the point (2, &#8210;12) that is congruent to the parabola that is given. Describe the series of transformations that would move the given parabola to your parabola.\r<BR>\n" );
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document.write( "2) Write the equation of a parabola that contains the point (0, 8) that is congruent to the parabola that is given. Describe the series of transformations that would move the given parabola to your parabola.\r<BR>\n" );
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document.write( "3) Write the equation of a parabola that is similar (not congruent) to the given parabola that does NOT contain the point (0, 0), but does contain the point (2, 2).\r<BR>\n" );
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document.write( "Note: Please show all work so that I can understand how to do the problem. Thank you. <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 #647068 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=647068\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> PROBLEM !:<BR>\n" );
document.write( "1. The focus-directrix definition of a parabola with axis parallel to the y-axis and vertex at the origin leads to the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4p%29x%5E2 ALIGN=MIDDLE  ALT=\"y=%281%2F4p%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for a parabola with directrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-p ALIGN=MIDDLE  ALT=\"y=-p\"   DISABLED_style= \"max-width:90%; height:auto;\" > and focus at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+0+%2C+p+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+0+%2C+p+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is valid (and useful) to think of the problem based on such a parabola, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E0 ALIGN=MIDDLE  ALT=\"p%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > parabola, because all other parabolas are variations, flipped and/or translated, and conclusions that works for one works for all (with the adaptations dictated by the flipping and translations used.\r<BR>\n" );
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document.write( "1) Moving the focus \"up, away from the directrix\" means that<BR>\n" );
document.write( "away from the directrix was up, meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E0 ALIGN=MIDDLE  ALT=\"p%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to begin with, and that the parabola opens up.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.5%2C0.5%2C-0.25%2C0.75%2C%0D%0Arectangle%280%2C-0.1%2C0.2%2C0.1%29%2C%0D%0Agraph%28300%2C300%2C-0.5%2C0.5%2C-0.25%2C0.75%2C-1%2C2.5x%5E2%29%2C%0D%0Agreen%28circle%280%2C0.1%2C0.01%29%29%2Cgreen%28line%28-.5%2C-.1%2C.5%2C-.1%29%29%2C%0D%0Alocate%28-.4%2C-.1%2Cgreen%28y=-p%29%29%2C%0D%0Agreen%28circle%280.2%2C0.1%2C0.01%29%29%2C%0D%0Alocate%280.01%2C0.16%2Cgreen%28F%280%2Cp%29%29%29%2C%0D%0Alocate%280.22%2C0.13%2Cgreen%28P%282p%2Cp%29%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The vertex of the parabola is midway between focus F and directrix, at a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > form both. The focus is at a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p ALIGN=MIDDLE  ALT=\"2p\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the directrix. A horizontal line through F intersects the parabola at P, which is at a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p ALIGN=MIDDLE  ALT=\"2p\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the directrix and Focus F. Focus F and point P are vertices of a square, with sides on the parabola's axis, and its directrix.<BR>\n" );
document.write( "As we move F up, the square expands, and the parabola opes wider:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.5%2C0.5%2C-0.25%2C0.75%2C%0D%0Arectangle%280%2C-0.1%2C0.2%2C0.1%29%2Crectangle%280%2C-0.2%2C0.4%2C0.2%29%2C%0D%0Agraph%28300%2C300%2C-0.5%2C0.5%2C-0.25%2C0.75%2C-1%2C2.5x%5E2%2C1.25x%5E2%29%2C%0D%0Agreen%28circle%280%2C0.1%2C0.01%29%29%2Cgreen%28line%28-.5%2C-.1%2C.5%2C-.1%29%29%2C%0D%0Alocate%28-.4%2C-.1%2Cgreen%28y=-p%29%29%2C%0D%0Agreen%28circle%280.2%2C0.1%2C0.01%29%29%2Cblue%28circle%280.4%2C0.2%2C0.01%29%29%2C%0D%0Ablue%28circle%280%2C0.2%2C0.01%29%29%2Cblue%28line%28-.5%2C-.2%2C.5%2C-.2%29%29%2C%0D%0Alocate%280.01%2C0.26%2Cblue%28F%29%29%2Clocate%280.42%2C0.22%2Cblue%28P%29%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > You could say the parabola gets compressed vertically, gets flatter, or that it gets stretched horizontally.<BR>\n" );
document.write( "How would we describe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > ? It is the distance between the focus and the vertex of the parabola. In the problem, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E0 ALIGN=MIDDLE  ALT=\"p%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and as we move the focus \"up and away\" from the directrix, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases.<BR>\n" );
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document.write( "2) How would the shape of the parabola change if the focus were moved down, toward the directrix? How would we describe p?<BR>\n" );
document.write( "We reverse the motion, from the blue parabola to the green one.<BR>\n" );
document.write( "The parabola gets \"taller, stretched vertically, hugging its axis tighter, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > decreases.\r<BR>\n" );
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document.write( "3) How would the shape of the parabola change if the focus were moved down, below the directrix?<BR>\n" );
document.write( "You have to jump over the directrix, of course. The focus cannot be on the directrix, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=0 ALIGN=MIDDLE  ALT=\"p=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > . As you get the focus down, closer and closer to the directrix, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > gets smaller, and smaller, closer to zero. At some point, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > is incredibly small, the focus is a hair above the directrix, and the parabola has hugged the positive y-axis so tight that you do not see the separation. At that point, you blink and then see that the focus is a hair below the directrix. By then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a negative but almost zero value, the parabola is tightly hugging the negative y-axis. As the focus keeps moving down, away from the directrix, the parabola opens up hugging the negative y-axis less tightly.<BR>\n" );
document.write( "You have flipped the parabola upside down, so it would be a frown rather than a smile. It now open downs, with a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3C0 ALIGN=MIDDLE  ALT=\"p%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the more you move the focus away from the directrix the parabola opens wider.<BR>\n" );
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document.write( "PROBLEM 2:<BR>\n" );
document.write( "A generic parabola, with vertex at the origin, axis on the y-axis, and opening up, with focus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%280%2Cp%29 ALIGN=MIDDLE  ALT=\"F%280%2Cp%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E0 ALIGN=MIDDLE  ALT=\"p%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  has the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4p%29x%5E2 ALIGN=MIDDLE  ALT=\"y=%281%2F4p%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Changing the sign 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;\" > , to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3C0 ALIGN=MIDDLE  ALT=\"p%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > would flip the parabola upside down. We could call the flipped parabola congruent, because we can superimposing it by rotating the graph around the vertex of the parabola (the origin).<BR>\n" );
document.write( "Horizontal and/or vertical translations of a parabola, with vertex at the origin, and axis on the y-axis would yield other parabolas that we would call \"congruent\" to each other, because we can superimpose them by using those translations.<BR>\n" );
document.write( "Those parabolas have an equation of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F4p%29%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=%281%2F4p%29%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the horizontal rightwards translation,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the vertical upwards translation.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2 ALIGN=MIDDLE  ALT=\"y=3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the equation of a parabola with vertex at the origin, axis on the y-axis, and opening up, with focus above the origin, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E0 ALIGN=MIDDLE  ALT=\"p%3E0\"   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=y=3x%5E2 ALIGN=MIDDLE  ALT=\"y=3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4p=3 ALIGN=MIDDLE  ALT=\"1%2F4p=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "1) We can get \"a parabola that contains the point (2, &#8210;12) that is congruent to the parabola that is given\" by translating the vertex to the point (2, &#8210;12) .<BR>\n" );
document.write( "That means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4p=3 ALIGN=MIDDLE  ALT=\"1%2F4p=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-12 ALIGN=MIDDLE  ALT=\"k=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which would give us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%28x-2%29%5E2%2B-12 ALIGN=MIDDLE  ALT=\"y=3%28x-2%29%5E2%2B-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That parabola could be obtained easily by<BR>\n" );
document.write( "translating <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2 ALIGN=MIDDLE  ALT=\"y=3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > 2 units to the left and 12 units down.<BR>\n" );
document.write( "We can find an unlimited number of different examples of \"a parabola that contains the point (2, &#8210;12) that is congruent to the parabola that is given,\" but that would require more computation.<BR>\n" );
document.write( "For parabolas opening up, they would need to<BR>\n" );
document.write( "have an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=3%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be such that they satisfy<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12=3%282-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"-12=3%282-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "2) Similarly, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2%2B8 ALIGN=MIDDLE  ALT=\"y=3x%5E2%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is \"a parabola that contains the point (0, 8) that is congruent to the parabola that is given\". translating up by 8 units \"would move the given parabola to\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2%2B8 ALIGN=MIDDLE  ALT=\"y=3x%5E2%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "3) All parabolas are similar, meaning that if you change the scale factor<BR>\n" );
document.write( "(making the units on both axes larger or smaller by the same factor),<BR>\n" );
document.write( "you can superimpose them.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-2%29%5E2%2B2 ALIGN=MIDDLE  ALT=\"y=%28x-2%29%5E2%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is \"a parabola that is similar (not congruent) to\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x3x%5E2 ALIGN=MIDDLE  ALT=\"y=x3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-2%29%5E2%2B2%3E=2 ALIGN=MIDDLE  ALT=\"y=%28x-2%29%5E2%2B2%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the graph is above the x-axis, and \"does NOT contain the point (0, 0), but does contain the point (2, 2)\".<BR>\n" );
document.write( "Using the same scale, you see that they are clearly not congruent:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C7%2C-1%2C9%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-3%2C7%2C-1%2C9%2C3x%5E2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C7%2C-1%2C9%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-3%2C7%2C-1%2C9%2C-2%2C%28x-2%29%5E2%2B2%29%2C%0D%0Aline%28-4%2C0%2C11%2C0%29%2Cline%280%2C-2%2C0%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C7%2C-1%2C9%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-3%2C7%2C-1%2C9%2C3x%5E2%2C%28x-2%29%5E2%2B2%29%2C%0D%0Aline%28-4%2C0%2C11%2C0%29%2Cline%280%2C-2%2C0%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, when we scale down appropriately, we see that they would superimpose:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-4%2F3%2C6%2F3%2C-1%2C7%2F3%2C3x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-4%2F3%2C6%2F3%2C-1%2C7%2F3%2C3x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C7%2C-1%2C9%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-3%2C7%2C-1%2C9%2C-2%2C%28x-2%29%5E2%2B2%29%2C%0D%0Aline%28-4%2C0%2C11%2C0%29%2Cline%280%2C-2%2C0%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C7%2C-1%2C9%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-3%2C7%2C-1%2C9%2C-2%2C%28x-2%29%5E2%2B2%29%2C%0D%0Agraph%28300%2C300%2C-4%2F3%2C6%2F3%2C-1%2C7%2F3%2C3x%5E2%29%2C%0D%0Aline%28-4%2C0%2C11%2C0%29%2Cline%280%2C-2%2C0%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3.1%2C6.9%2C-1%2C9%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-3.1%2C6.9%2C-1%2C9%2C-2%2C%28x-2%29%5E2%2B2%29%2C%0D%0Agraph%28300%2C300%2C-5%2F3%2C5%2F3%2C-1%2C7%2F3%2C3x%5E2%29%2C%0D%0Aline%28-4%2C0%2C11%2C0%29%2Cline%280%2C-2%2C0%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );