document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172180>Question 172180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine the horizontal asymptotes of the function \r<BR>\n" );
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document.write( "f(x)=(3x^2+4x-12/x^2-16) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127201 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=127201\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283x%5E2%2B4x-12%29%2F%28x%5E2-16%29%29 ALIGN=MIDDLE  ALT=\"y=%283x%5E2%2B4x-12%29%2F%28x%5E2-16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\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=3x%5E2%2B4x-12 ALIGN=MIDDLE  ALT=\"3x%5E2%2B4x-12\"   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%5E2-16 ALIGN=MIDDLE  ALT=\"x%5E2-16\"   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 denominator 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;\" >.\r<BR>\n" );
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document.write( "<b> Horizontal Asymptote: </b>\r<BR>\n" );
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document.write( "Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:\r<BR>\n" );
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document.write( "To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.\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=3x%5E2%2B4x-12 ALIGN=MIDDLE  ALT=\"3x%5E2%2B4x-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the leading coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-16 ALIGN=MIDDLE  ALT=\"x%5E2-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the leading coefficient 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( "So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%2F%281%29=3 ALIGN=MIDDLE  ALT=\"%283%29%2F%281%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the horizontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   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%5E2-16=0 ALIGN=MIDDLE  ALT=\"x%5E2-16=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%5E2=0%2B16 ALIGN=MIDDLE  ALT=\"x%5E2=0%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add 16 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%5E2=16 ALIGN=MIDDLE  ALT=\"x%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B-sqrt%2816%29 ALIGN=MIDDLE  ALT=\"x=0%2B-sqrt%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%2816%29 ALIGN=MIDDLE  ALT=\"x=sqrt%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%2816%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Break up the \"plus/minus\" to get two equations\r<BR>\n" );
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document.write( "<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;\" > or <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;\" >  Take the square root of 16 to get 4\r<BR>\n" );
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document.write( "So the vertical asymptotes are the equations <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=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "           \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=%283x%5E2%2B4x-12%29%2F%28x%5E2-16%29 ALIGN=MIDDLE  ALT=\"y=%283x%5E2%2B4x-12%29%2F%28x%5E2-16%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%283x%5E2%2B4x-12%29%2F%28x%5E2-16%29%29%2C%0D%0Ablue%28line%28-20%2C3%2C20%2C3%29%29%2C%0D%0Agreen%28line%284%2C-20%2C4%2C20%29%29%2C%0D%0Agreen%28line%28-4%2C-20%2C-4%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=%283x%5E2%2B4x-12%29%2F%28x%5E2-16%29%29 ALIGN=MIDDLE  ALT=\"y=%283x%5E2%2B4x-12%29%2F%28x%5E2-16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  with the horizontal asymptote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (blue line)  and 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=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (green lines)<BR>\n" );
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