document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=927655>Question 927655</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please solve the symmetry and odd even or neither for the following equation: <BR>\n" );
document.write( "x^3 - 9x / 4x^2 - 4x - 80 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #563135 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=563135\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To prove an function is odd, we need to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+=+-f%28x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So let's first find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29\"   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=f%28-x%29+=+%28%28-x%29%5E3+-+9%28-x%29%29%2F%284%28-x%29%5E2+-+4%28-x%29+-+80%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+%28%28-x%29%5E3+-+9%28-x%29%29%2F%284%28-x%29%5E2+-+4%28-x%29+-+80%29\"   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=f%28-x%29+=+%28-x%5E3+%2B+9%29%2F%284x%5E2+%2B+4x+-+80%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+%28-x%5E3+%2B+9%29%2F%284x%5E2+%2B+4x+-+80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29\"   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=f%28x%29+=+%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29\"   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=-f%28x%29+=+-%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29 ALIGN=MIDDLE  ALT=\"-f%28x%29+=+-%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29\"   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=-f%28x%29+=+%28-x%5E3+%2B+9x%29%2F%284x%5E2+-+4x+-+80%29 ALIGN=MIDDLE  ALT=\"-f%28x%29+=+%28-x%5E3+%2B+9x%29%2F%284x%5E2+-+4x+-+80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29%3C%3E-f%28x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29%3C%3E-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice how the +4x and -4x in the denominators don't match).\r<BR>\n" );
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document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT an odd function.\r<BR>\n" );
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document.write( "To show that an function is even, we need to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+=+f%28x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the last part, we found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+=+%28-x%5E3+%2B+9%29%2F%284x%5E2+%2B+4x+-+80%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+%28-x%5E3+%2B+9%29%2F%284x%5E2+%2B+4x+-+80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > but this is not equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "The numerators don't match at all in terms of their signs and the denominators are off too a bit (again the +4x and -4x).\r<BR>\n" );
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document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT an even function.\r<BR>\n" );
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document.write( "Summary: Above we've shown that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is neither even nor odd. \r<BR>\n" );
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document.write( "Because of this, it does NOT have y axis symmetry (it's not even) and it does NOT have origin symmetry (it's not odd).\r<BR>\n" );
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document.write( "The graph visually sums it all up for us without having to say a single word. It looks like the graph has origin symmetry, but it's approximately symmetric for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%3C=x%3C=3 ALIGN=MIDDLE  ALT=\"-3%3C=x%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Any x outside this interval has this symmetry breaking down.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+0%2C%28x%5E3+-+9x%29%2F%284x%5E2+-+4x+-+80%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From <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;\" > to <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;\" >, this graph looks pretty symmetric (with respect to the origin), but as x goes beyond +4, the curve does not go beyond -4 to reciprocate which is where the symmetry starts to break down.\r<BR>\n" );
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document.write( "A more basic and quick indication is to notice how the vertical asymptotes <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5 ALIGN=MIDDLE  ALT=\"x+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > are not equal in magnitude. Ie, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-4%29+%3C%3E+abs%285%29 ALIGN=MIDDLE  ALT=\"abs%28-4%29+%3C%3E+abs%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so this also tells us f(x) does not have origin symmetry.\r<BR>\n" );
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document.write( "Jim\r<BR>\n" );
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