document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089798>Question 1089798</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Analyze the graph of the function. R(x)= x^2+6x-27 /x-9\r<BR>\n" );
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document.write( "a) What is the domain of R(x)?<BR>\n" );
document.write( "b) What is the equation of the vertical asymptote(s) of R(x)? x=<BR>\n" );
document.write( "c) What is the equation of the horizontal or oblique asymptote of R(x)? y= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #704206 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=704206\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29=+%28x%5E2%2B6x-27%29+%2F%28x-9+%29 ALIGN=MIDDLE  ALT=\"R%28x%29=+%28x%5E2%2B6x-27%29+%2F%28x-9+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "vertical asymptote: \r<BR>\n" );
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document.write( "Vertical Asymptotes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+p%28x%29+%2F+q%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+p%28x%29+%2F+q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%28x%29 ALIGN=MIDDLE  ALT=\"q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters. \r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-9+%29=0 ALIGN=MIDDLE  ALT=\"%28x-9+%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Vertical asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The location of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=horizontal ALIGN=MIDDLE  ALT=\"horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote is determined by looking at the degrees of the numerator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and denominator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Cm ALIGN=MIDDLE  ALT=\"n%3Cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x-axis, <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;\" > is the horizontal asymptote.<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m ALIGN=MIDDLE  ALT=\"n=m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=an%2Fbm ALIGN=MIDDLE  ALT=\"y=an%2Fbm\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the horizontal asymptote. That is, the ratio of the leading coefficients.<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Em ALIGN=MIDDLE  ALT=\"n%3Em\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal asymptote. <BR>\n" );
document.write( "However, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m%2B1 ALIGN=MIDDLE  ALT=\"n=m%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=oblique ALIGN=MIDDLE  ALT=\"oblique\"   DISABLED_style= \"max-width:90%; height:auto;\" > or slant asymptote.\r<BR>\n" );
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document.write( "in your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1 ALIGN=MIDDLE  ALT=\"m=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Em ALIGN=MIDDLE  ALT=\"n%3Em\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=horizontal ALIGN=MIDDLE  ALT=\"horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote\r<BR>\n" );
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document.write( "To find the equation of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=oblique ALIGN=MIDDLE  ALT=\"oblique\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29+=+%28x%5E2+%2B+6+x+-+27%29%2F%28x+-+9%29+ ALIGN=MIDDLE  ALT=\"R%28x%29+=+%28x%5E2+%2B+6+x+-+27%29%2F%28x+-+9%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  -------x+15<BR>\n" );
document.write( "(x - 9) |x^2 + 6 x - 27<BR>\n" );
document.write( "---------x^2-9x<BR>\n" );
document.write( "----------0+15x<BR>\n" );
document.write( "-------------15x-27<BR>\n" );
document.write( "-------------15x-135<BR>\n" );
document.write( "----------------0+108\r<BR>\n" );
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document.write( "is asymptotic to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+15 ALIGN=MIDDLE  ALT=\"x+%2B+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Oblique asymptote: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%29+=x+%2B+15 ALIGN=MIDDLE  ALT=\"R%28x%29+=x+%2B+15\"   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=drawing%28+600%2C+600%2C+-25%2C+25%2C+-25%2C+25%2C%0D%0Aline%289%2C25%2C9%2C-25%29%2C%0D%0A+graph%28+600%2C+600%2C+-25%2C+25%2C+-25%2C+25%2Cx+%2B+15+%2C%28x%5E2+%2B+6+x+-+27%29%2F%28x+-+9%29+%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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