document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=135709>Question 135709</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2.  Check for symmetries, find intercepts and asymptotes, analyze the behavior near<BR>\n" );
document.write( "the graph of: 4y^2 + x – xy^2 – 1 = 0 (first solve for y^2) Is this thee graph of a        function? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #99454 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=99454\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2+%2B+x-xy%5E2-1+=+0 ALIGN=MIDDLE  ALT=\"4y%5E2+%2B+x-xy%5E2-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2-xy%5E2=+1-x ALIGN=MIDDLE  ALT=\"4y%5E2-xy%5E2=+1-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract x from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%284-x%29=+1-x ALIGN=MIDDLE  ALT=\"y%5E2%284-x%29=+1-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=++%281-x%29%2F%284-x%29+ ALIGN=MIDDLE  ALT=\"y%5E2+=++%281-x%29%2F%284-x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-x ALIGN=MIDDLE  ALT=\"4-x\"   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=y%5E2+=++%281-0%29%2F%284-0%29+ ALIGN=MIDDLE  ALT=\"y%5E2+=++%281-0%29%2F%284-0%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the y-intercept, plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=++1%2F4+ ALIGN=MIDDLE  ALT=\"y%5E2+=++1%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++0%2B-sqrt%281%2F4%29+ ALIGN=MIDDLE  ALT=\"y+=++0%2B-sqrt%281%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F2 ALIGN=MIDDLE  ALT=\"y=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1%2F2 ALIGN=MIDDLE  ALT=\"y=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "So the y-intercepts are (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   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=y%5E2+=++%281-x%29%2F%284-x%29+ ALIGN=MIDDLE  ALT=\"y%5E2+=++%281-x%29%2F%284-x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the original equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5E2+=++%281-x%29%2F%284-x%29+ ALIGN=MIDDLE  ALT=\"0%5E2+=++%281-x%29%2F%284-x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the x-intercept, plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   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=0+=++%281-x%29%2F%284-x%29+ ALIGN=MIDDLE  ALT=\"0+=++%281-x%29%2F%284-x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 0 to get 0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%284-x%29+=++1-x+ ALIGN=MIDDLE  ALT=\"0%284-x%29+=++1-x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 4-x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=++1-x+ ALIGN=MIDDLE  ALT=\"0+=++1-x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<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;\" > Add x to both sides\r<BR>\n" );
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document.write( "So the x-intercept is (1,0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-x=0 ALIGN=MIDDLE  ALT=\"4-x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the vertical asymptote(s), simply set the denominator equal to zero\r<BR>\n" );
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document.write( "<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;\" > Solve for x\r<BR>\n" );
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document.write( "So the vertical asymptote is at <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;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=++%281-x%29%2F%284-x%29+ ALIGN=MIDDLE  ALT=\"y%5E2+=++%281-x%29%2F%284-x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the original equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=++%28-x%2B1%29%2F%28-x%2B4%29+ ALIGN=MIDDLE  ALT=\"y%5E2+=++%28-x%2B1%29%2F%28-x%2B4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms\r<BR>\n" );
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document.write( "Notice how the coefficient of \"x\" for the numerator and denominator is -1. So the horizontal asymptote is the ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F-1=1 ALIGN=MIDDLE  ALT=\"-1%2F-1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since we are dealing with a square in <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;\" >, this means that the final equation looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++0%2B-sqrt%28%281-x%29%2F%284-x%29%29 ALIGN=MIDDLE  ALT=\"y+=++0%2B-sqrt%28%281-x%29%2F%284-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So there are two final parts <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++sqrt%28%281-x%29%2F%284-x%29%29 ALIGN=MIDDLE  ALT=\"y+=++sqrt%28%281-x%29%2F%284-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++-sqrt%28%281-x%29%2F%284-x%29%29 ALIGN=MIDDLE  ALT=\"y+=++-sqrt%28%281-x%29%2F%284-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This means that there is symmetry with respect with the x-axis and that there are two horizontal asymptotes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice if we graph both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++sqrt%28%281-x%29%2F%284-x%29%29 ALIGN=MIDDLE  ALT=\"y+=++sqrt%28%281-x%29%2F%284-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++-sqrt%28%281-x%29%2F%284-x%29%29 ALIGN=MIDDLE  ALT=\"y+=++-sqrt%28%281-x%29%2F%284-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can visually verify our answer\r<BR>\n" );
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document.write( "Notice how that if you pass a vertical line through the graph, the line will intersect with the graph more than once. So this tells us that this graph is <b>not</b> a function since it fails the vertical line test. </SPAN>\n" );
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