document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=985083>Question 985083</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do i analyze a rational function?\r<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 #605928 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=605928\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Analyzing the graph of a rational function <BR>\n" );
document.write( "step 1:  Find the domain of the rational function<BR>\n" );
document.write( "step 2:  Write R in lowest terms (simplify the rational function if possible)<BR>\n" );
document.write( "step 3:  Locate the intercepts of the graph.<BR>\n" );
document.write( "step 4:  Test for symmetry<BR>\n" );
document.write( "step 5:  Locate the vertical asymptotes<BR>\n" );
document.write( "step 6:  Locate the horizontal or oblique asymptotes<BR>\n" );
document.write( "step 7:  Determine points, if any, where the graph crosses the asymptotes<BR>\n" );
document.write( "(horizontal or oblique)\r<BR>\n" );
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document.write( "here is an example:\r<BR>\n" );
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document.write( "analyze the graph of a rational function\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-1%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"R%28x%29=%28x-1%29%2F%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "step 1:  Find the domain of the rational function\r<BR>\n" );
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document.write( "domain is:\r<BR>\n" );
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document.write( "{ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >| <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E2 ALIGN=MIDDLE  ALT=\"x%3C%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E-2 ALIGN=MIDDLE  ALT=\"x%3C%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "step 2:  Write R in lowest terms (simplify the rational function if possible)\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-1%29%2F%28%28x%2B2%29%28x-2%29%29+ ALIGN=MIDDLE  ALT=\"R%28x%29=%28x-1%29%2F%28%28x%2B2%29%28x-2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "step 3:  Locate the intercepts of the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%280%29=%280-1%29%2F%28%280%2B2%29%280-2%29%29+ ALIGN=MIDDLE  ALT=\"R%280%29=%280-1%29%2F%28%280%2B2%29%280-2%29%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=R%280%29=%28-1%29%2F%282%28-2%29%29+ ALIGN=MIDDLE  ALT=\"R%280%29=%28-1%29%2F%282%28-2%29%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=R%280%29=%28-1%29%2F%28-4%29+ ALIGN=MIDDLE  ALT=\"R%280%29=%28-1%29%2F%28-4%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=R%280%29=1%2F4+ ALIGN=MIDDLE  ALT=\"R%280%29=1%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> y-intercept is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   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=%28x-1%29%2F%28%28x%2B2%29%28x-2%29%29+ ALIGN=MIDDLE  ALT=\"0=%28x-1%29%2F%28%28x%2B2%29%28x-2%29%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=0%28%28x%2B2%29%28x-2%29%29+=%28x-1%29+ ALIGN=MIDDLE  ALT=\"0%28%28x%2B2%29%28x-2%29%29+=%28x-1%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=0+=x-1 ALIGN=MIDDLE  ALT=\"0+=x-1\"   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=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> x-intercept is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "step 4:  Test for symmetry\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28-x%29+=+R%28x%29 ALIGN=MIDDLE  ALT=\"R%28-x%29+=+R%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the function has symmetry about the y­-axis <BR>\n" );
document.write( "find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28-x%29 ALIGN=MIDDLE  ALT=\"R%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28-x%29=%28-x-1%29%2F%28%28-x%29%5E2-4%29 ALIGN=MIDDLE  ALT=\"R%28-x%29=%28-x-1%29%2F%28%28-x%29%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28-x%29=-%28x%2B1%29%2F%28x%5E2-4%29+ ALIGN=MIDDLE  ALT=\"R%28-x%29=-%28x%2B1%29%2F%28x%5E2-4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28-x%29+%3C+%3E+R%28x%29 ALIGN=MIDDLE  ALT=\"R%28-x%29+%3C+%3E+R%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > symmetry\r<BR>\n" );
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document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28-x%29%3C%3E-R%28x%29 ALIGN=MIDDLE  ALT=\"R%28-x%29%3C%3E-R%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the function has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > symmetry about the origin \r<BR>\n" );
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document.write( "step 5:  Locate the vertical asympthote\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are  the vertical asympthotes\r<BR>\n" );
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document.write( "for next step:<BR>\n" );
document.write( "Horizontal and Oblique Asympthote reminder(the degree of the numerator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the degree of the denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "1.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+%3C+m ALIGN=MIDDLE  ALT=\"n+%3C+m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a proper fraction and will have the horizontal asympthote <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;\" >.<BR>\n" );
document.write( "2.<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;\" >, then R is improper and long division is used.<BR>\n" );
document.write( "(a)<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;\" >, the quotient obtained will be a number, and the line y = is a horizontal asymptote.<BR>\n" );
document.write( "(b)<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+m+%2B+1 ALIGN=MIDDLE  ALT=\"n+=+m+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the quotient obtained is of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax+%2B+b ALIGN=MIDDLE  ALT=\"ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >(a polynomial of degree 1), and the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax+%2B+b ALIGN=MIDDLE  ALT=\"y+=+ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an oblique asymptote.<BR>\n" );
document.write( "(c)<BR>\n" );
document.write( "If<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n+%3E+m+%2B+1 ALIGN=MIDDLE  ALT=\"+n+%3E+m+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the quotient obtained is a polynomial of degree 2 or higher and R has <BR>\n" );
document.write( "neither a horizontal nor an oblique asymptote.\r<BR>\n" );
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document.write( " Horizontal asymptote:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%2F%28%28x-2%29+%28x%2B2%29%29-%3E0 ALIGN=MIDDLE  ALT=\"%28x-1%29%2F%28%28x-2%29+%28x%2B2%29%29-%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >   as   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >->±<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so, horizontal asymptote is <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( "no oblique asymptotes found \r<BR>\n" );
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document.write( "step 7:  Determine points, if any, where the graph crosses the asymptotes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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;\" >\r<BR>\n" );
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document.write( "graph:\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+-10%2C+10%2C+-10%2C+10%2C%0D%0Aline%28-2%2C10%2C-2%2C-10%29%2Cline%282%2C10%2C2%2C-10%29%2C%0D%0A+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28x-1%29%2F%28x%5E2-4%29%2C0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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