document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131510>Question 131510</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are the asymptotes and intercepts of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%28x%29=%282x-4%29%2F%28x%5E2%2Bx-2%29 ALIGN=MIDDLE  ALT=\"s%28x%29=%282x-4%29%2F%28x%5E2%2Bx-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >? Show it on graph. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #95995 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=95995\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The equation of a vertical asymptote is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=a ALIGN=MIDDLE  ALT=\"x=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a value that would make the denominator go to zero.  \r<BR>\n" );
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document.write( "To find horizontal asymptotes, compare the degree of the numerator and denominator polynomials.\r<BR>\n" );
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document.write( "If the degree of the denominator polynomial is greater than the degree of the numerator polynomial, the 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;\" >, in other words, the x-axis.  That's because, as x gets larger, the denominator is going to get larger much faster than the numerator and the entire fraction will tend to zero.\r<BR>\n" );
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document.write( "If the degrees of the numerator and denominator are the same, then, as x gets very large, the only significant difference between the numerator and denominator will be the lead coefficients.  Therefore, the horizontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%5Bn%5D%2Fa%5Bd%5D ALIGN=MIDDLE  ALT=\"y=a%5Bn%5D%2Fa%5Bd%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the lead coefficient on the numerator polynomial and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bd%5D ALIGN=MIDDLE  ALT=\"a%5Bd%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the lead coefficient on the denominator polynomial.\r<BR>\n" );
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document.write( "If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.  There is a straight line slant or skew asymptote if the degrees differ by 1 that is found by performing polynomial long division of the denominator into the numerator.  The equation of the skew asymptote is the quotient obtained excluding any possible remainder.\r<BR>\n" );
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document.write( "The x-intercepts are given by setting the numerator equal to zero and solving.  Any zero that is also a zero of the denominator must be excluded.  This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb=0 ALIGN=MIDDLE  ALT=\"a%2Fb=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0 ALIGN=MIDDLE  ALT=\"a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3C%3E0 ALIGN=MIDDLE  ALT=\"b%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "y-intercepts are found by evaluating <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29 ALIGN=MIDDLE  ALT=\"f%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now, for your problem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%28x%29=%282x-4%29%2F%28x%5E2%2Bx-2%29 ALIGN=MIDDLE  ALT=\"s%28x%29=%282x-4%29%2F%28x%5E2%2Bx-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "Vertical asymptotes: You need to find the zeros of the denominator polynomial:\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-1%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the zeros are at 1 and -2, making the equations of the asymptotes <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;\" > 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;\" >\r<BR>\n" );
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document.write( "Horizontal asymptotes:  The degree of the denominator is 2 and the degree of the numerator is 1, therefore the horizontal asymptote is at <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( "x-intercepts:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4=0 ALIGN=MIDDLE  ALT=\"2x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<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;\" >\r<BR>\n" );
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document.write( "And the x-intercept is at (2,0)\r<BR>\n" );
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document.write( "y=intercept:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%280%29=-4%2F-2=2 ALIGN=MIDDLE  ALT=\"s%280%29=-4%2F-2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the y-intercept is at (0,2)\r<BR>\n" );
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