document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=165403>Question 165403</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all horizontal/vertical asymptotes and all x-intercepts:<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 #121908 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=121908\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x-intercepts:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%5E2-5x%2B6%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"y=%28x%5E2-5x%2B6%29%2F%28x-1%29%29\"   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=0=%28x%5E2-5x%2B6%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"0=%28x%5E2-5x%2B6%29%2F%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in y=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5x%2B6=0 ALIGN=MIDDLE  ALT=\"x%5E2-5x%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the numerator equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"%28x-3%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)\r<BR>\n" );
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document.write( "Now set each factor 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=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <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;\" >    Now solve for x in each case\r<BR>\n" );
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document.write( "So our answers are \r<BR>\n" );
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document.write( "So the x-intercepts are (3,0) and (2,0) \r<BR>\n" );
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document.write( "Asymptotes:\r<BR>\n" );
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document.write( "Looking at the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5x%2B6 ALIGN=MIDDLE  ALT=\"x%5E2-5x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that the degree is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the highest exponent of the numerator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For the denominator <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;\" >, we can see that the degree is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the highest exponent of the denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<b> Oblique Asymptote: </b>\r<BR>\n" );
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document.write( "Since the degree of the numerator (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is greater than the degree of the denominator (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >), there is no horizontal asymptote. In this case, there's an oblique asymptote.\r<BR>\n" );
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document.write( "To find the oblique asymptote, simply use polynomial long division to find it.\r<BR>\n" );
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document.write( "So the quotient is <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;\" >, this means that the oblique asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-4 ALIGN=MIDDLE  ALT=\"y=x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<b> Vertical Asymptote: </b>\r<BR>\n" );
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document.write( "To find the vertical asymptote, just set the denominator equal to zero and solve for x\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;\" > 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=0%2B1 ALIGN=MIDDLE  ALT=\"x=0%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add 1 to both sides\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;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "So the vertical asymptote is <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( "Notice if we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%5E2-5x%2B6%29%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"y=%28x%5E2-5x%2B6%29%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can visually verify our answers:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28x%5E2-5x%2B6%29%2F%28x-1%29%2C100%2Cx-4%29%2C%0D%0Agreen%28line%281%2C-20%2C1%2C20%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%5E2-5x%2B6%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"y=%28x%5E2-5x%2B6%29%2F%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  with the x-intercepts (3,0) and (2,0), the oblique asymptote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-4 ALIGN=MIDDLE  ALT=\"y=x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (blue line)  and the vertical asymptote <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;\" >  (green line)<BR>\n" );
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