document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=973815>Question 973815</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello! I really need help with this! My teacher never gave us a lesson on this, so I am struggling through! It is all pretty much how to identify a conic, but I do not really get what that means or how to do it.  Here are a few of the problems, but if you want, you can only use one of them to show me how to do the rest.\r<BR>\n" );
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document.write( "identify the conic represented by the equation x^2-6x = y+3\r<BR>\n" );
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document.write( "identify the conic with the equation 3x^2 - y^2 - 7x + 2 = 0\r<BR>\n" );
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document.write( "identify the conic section of the equation [(x-2)^2]/9   -  [(y+2)^2]/4  = 1\r<BR>\n" );
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document.write( "please help me!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595955 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=595955\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Conics with axes of symmetry parallel to the x-axis and/or y-axis are easy to identify and graph.<BR>\n" );
document.write( " <BR>\n" );
document.write( "PARABOLAS have only one variable squared, and you can transform them into something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=a%28y-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"x=a%28y-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > with some constants <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <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;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6x=y%2B3 ALIGN=MIDDLE  ALT=\"x%5E2-6x=y%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6x%2B9=y%2B3%2B9 ALIGN=MIDDLE  ALT=\"x%5E2-6x%2B9=y%2B3%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2=y%2B12 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2=y%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-3%29%5E2-12 ALIGN=MIDDLE  ALT=\"y=%28x-3%29%5E2-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C-14%2C6%2Cgrid%281%29%2C%0D%0Agraph%28300%2C300%2C-2%2C8%2C-14%2C6%2C%28x-3%29%5E2-12%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "ELLIPSES and CIRCLES have equations that can be transformed into something like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b=R ALIGN=MIDDLE  ALT=\"a=b=R\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get a circle: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2FR%5E2%2B%28y-k%29%5E2%2FR%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2FR%5E2%2B%28y-k%29%5E2%2FR%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2B%28y-k%29%5E2=R%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2B%28y-k%29%5E2=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3Eb ALIGN=MIDDLE  ALT=\"a%3C%3Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" > the circle gets stretched to make it longer in the direction of the variable atop that larger of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you see an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > that would both have a positive coefficient if they were on the same side of the equal sign, suspect an ellipse<BR>\n" );
document.write( "(you don't have one of those in your examples).<BR>\n" );
document.write( " <BR>\n" );
document.write( "HYPERBOLAS have equations that can be transformed into something like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=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-2%29%5E2%2F9+-+%28y%2B2%29%5E2%2F4+=+1 ALIGN=MIDDLE  ALT=\"+%28x-2%29%5E2%2F9+-+%28y%2B2%29%5E2%2F4+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2F3%5E2-%28y-%28-2%29%29%5E2%2F2%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2F3%5E2-%28y-%28-2%29%29%5E2%2F2%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is already it that form (no need to transform it much).<BR>\n" );
document.write( "Its axes of symmetry are the lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > .<BR>\n" );
document.write( "It is centered at (2,-2).<BR>\n" );
document.write( "Your other example takes a little more work to get it into that form,<BR>\n" );
document.write( "but you may not need to do the transformation.<BR>\n" );
document.write( "If the terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > have coefficients with opposite sign when they are on the same side of the equal sign, suspect a hyperbola.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+y%5E2+-+7x+%2B+2+=+0+ ALIGN=MIDDLE  ALT=\"3x%5E2+-+y%5E2+-+7x+%2B+2+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x%5E2+-+7x%2F3%29%2B7%5E2%2F6%5E2+-+y%5E2+=+-2+ ALIGN=MIDDLE  ALT=\"3%28x%5E2+-+7x%2F3%29%2B7%5E2%2F6%5E2+-+y%5E2+=+-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x%5E2-7x%2F3%2B%287%2F6%29%5E2-49%2F36%29+-+y%5E2+=+-2+ ALIGN=MIDDLE  ALT=\"3%28x%5E2-7x%2F3%2B%287%2F6%29%5E2-49%2F36%29+-+y%5E2+=+-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x%5E2-7x%2F3%2B%287%2F6%29%5E2%29-3%2849%2F36%29+-+y%5E2+=+-2+ ALIGN=MIDDLE  ALT=\"3%28x%5E2-7x%2F3%2B%287%2F6%29%5E2%29-3%2849%2F36%29+-+y%5E2+=+-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-7%2F6%29%5E2-49%2F12+-+y%5E2+=+-2+ ALIGN=MIDDLE  ALT=\"3%28x-7%2F6%29%5E2-49%2F12+-+y%5E2+=+-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-7%2F6%29%5E2+-+y%5E2+=+49%2F12-2+ ALIGN=MIDDLE  ALT=\"3%28x-7%2F6%29%5E2+-+y%5E2+=+49%2F12-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-7%2F6%29%5E2+-+y%5E2+=+25%2F12+ ALIGN=MIDDLE  ALT=\"3%28x-7%2F6%29%5E2+-+y%5E2+=+25%2F12+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now, I divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2F12 ALIGN=MIDDLE  ALT=\"25%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or multiply times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F25 ALIGN=MIDDLE  ALT=\"12%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" > , same thing) to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2812%2F25%29%28x-7%2F6%29%5E2+-+%2812%2F25%29y%5E2+=+%2812%2F25%29%2825%2F12%29+ ALIGN=MIDDLE  ALT=\"3%2812%2F25%29%28x-7%2F6%29%5E2+-+%2812%2F25%29y%5E2+=+%2812%2F25%29%2825%2F12%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2836%2F25%29%28x-7%2F6%29%5E2+-+%2812%2F25%29y%5E2+=+1 ALIGN=MIDDLE  ALT=\"%2836%2F25%29%28x-7%2F6%29%5E2+-+%2812%2F25%29y%5E2+=+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-7%2F6%29%5E2%2F%2825%2F36%29+-+y%5E2%2F%2825%2F12%29+=+1 ALIGN=MIDDLE  ALT=\"%28x-7%2F6%29%5E2%2F%2825%2F36%29+-+y%5E2%2F%2825%2F12%29+=+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-7%2F6%29%5E2%2F%285%2F6%29%5E2+-+y%5E2%2F%2825%2F12%29+=+1 ALIGN=MIDDLE  ALT=\"%28x-7%2F6%29%5E2%2F%285%2F6%29%5E2+-+y%5E2%2F%2825%2F12%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is really a hyperbola, because teachers usually do not try to trick you.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "If the teacher wanted to trick you, he/she could give you something that had an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y%5E2 ALIGN=MIDDLE  ALT=\"-y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but transformed into something like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2-%28y%2B2%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2-%28y%2B2%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2B2%29%5E2=%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"%28y%2B2%29%5E2=%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%2B2=x-2%2C%22or%22%2Cy%2B2=-%28x-2%29%29 ALIGN=MIDDLE  ALT=\"system%28y%2B2=x-2%2C%22or%22%2Cy%2B2=-%28x-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=x-4%2C%22or%22%2Cy=-x%29 ALIGN=MIDDLE  ALT=\"system%28y=x-4%2C%22or%22%2Cy=-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that graphs as just two lines that intersect,<BR>\n" );
document.write( "but you can call that a degenerate hyperbola,<BR>\n" );
document.write( "a hyperbola that degenerated into those two lines.<BR>\n" );
document.write( "You could also have something that may have an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with positive coefficients on the same side of the equal sign,<BR>\n" );
document.write( "but when transformed turned out to be a \"degenerate ellipse,\"<BR>\n" );
document.write( "that could be a single point (as for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2%7D%5E2%2F4%2By%2B2%292%2F9=0 ALIGN=MIDDLE  ALT=\"x-2%7D%5E2%2F4%2By%2B2%292%2F9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),<BR>\n" );
document.write( "or no point at all (as for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2%7D%5E2%2F4%2By%2B2%292%2F9=-1 ALIGN=MIDDLE  ALT=\"x-2%7D%5E2%2F4%2By%2B2%292%2F9=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );