document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=688433>Question 688433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the rational function f(x)=8x-3/ 4x +12 explain in detail how to find the equations of the horizontal and vertical asymptotes.  Be sure to conclude your explanation by giving the equations of the lines for the horizontal<BR>\n" );
document.write( "and vertical asymptotes.\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 #425633 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=425633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> DISCLAIMER: I was never a high math school teacher (just a physics substitute), and do not know the exact answer expected. I just know what makes sense.<BR>\n" );
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document.write( "I believe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=(8x-3)/(4x +12)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288x-3%29%2F%284x+%2B12%29 ALIGN=MIDDLE  ALT=\"%288x-3%29%2F%284x+%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is your function.<BR>\n" );
document.write( "(Without those brackets we would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=8x-3/ 4x +12 =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x-3%2F+4x+%2B12 ALIGN=MIDDLE  ALT=\"8x-3%2F+4x+%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
document.write( "The long horizontal fraction line in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%288x-3%29%2F%284x+%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%288x-3%29%2F%284x+%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > carries those hidden brackets).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%288x-3%29%2F%284x+%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%288x-3%29%2F%284x+%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%288x%2B24-3-24%29%2F%284x+%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%288x%2B24-3-24%29%2F%284x+%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%288x%2B24-27%29%2F%284x+%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%288x%2B24-27%29%2F%284x+%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%288x%2B24%29%2F%284x+%2B12%29-27%2F%284x+%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%288x%2B24%29%2F%284x+%2B12%29-27%2F%284x+%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2-27%2F%284%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=2-27%2F%284%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29 ALIGN=MIDDLE  ALT=\"abs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > grows without bounds, the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-27%2F%284%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"-27%2F%284%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches zero, so the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2-27%2F%284%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=2-27%2F%284%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=2%29 ALIGN=MIDDLE  ALT=\"highlight%28y=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the horizontal asymptote.<BR>\n" );
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document.write( "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;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-27%2F%284%28x%2B3%29%29%29 ALIGN=MIDDLE  ALT=\"abs%28-27%2F%284%28x%2B3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > grows without bounds,<BR>\n" );
document.write( "making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the vertical asymptote.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-18%2C12%2C-8%2C12%2C%288x-3%29%2F%284x+%2B12%29%2C2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-18%2C12%2C-8%2C12%2C%288x-3%29%2F%284x+%2B12%29%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "DISCLAIMER BOOKEND:<BR>\n" );
document.write( "Your teacher/textbook may require using certain buzzwords/symbols (e.g. for limits), and/or referencing certain theorems or paint-by-number rules that I cannot guess about.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );