document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=631227>Question 631227</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Finding rational functions off of graphs:\r<BR>\n" );
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document.write( "Hi! I don't understand what to do with the horizontal asymptote when I am finding the graph function. For example:\r<BR>\n" );
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document.write( "Zero's of the function: -2 and 3<BR>\n" );
document.write( "Vertical asymptote: -1<BR>\n" );
document.write( "Horizontal Asymptote: y=2x-4\r<BR>\n" );
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document.write( "Thank you very much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #397449 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=397449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "WARNING! Danger, Will Robinson.  Terminology Error!  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\ =\ 2x\ -\ 4\"> is NOT a horizontal asymptote.  Horizontal asymptotes are horizontal, hence the name.  Linear functions with a non-zero (2 is not zero) slope are NOT horizontal.  Asymptotes that are slanted or oblique are called Slant Asymptotes or Oblique Asymptotes.\r<BR>\n" );
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document.write( "The fact that the function has zeros at -2 and 3 tells us that the factors of the numerator polynomial are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x\ +\ 2)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x\ -\ 3)\">, and some constant <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\"> (because all polynomial equations <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\left(p(x)\right)\ =\ 0\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p(x)\"> is a polynomial with degree <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ \neq\ 0\">have identical solution sets)\r<BR>\n" );
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document.write( "Hence, the numerator is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE kx^2\ -\ kx\ -\ 6k\">\r<BR>\n" );
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document.write( "The fact that the function has a vertical asymptote of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -1\"> means that the denominator polynomial has a zero at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -1\">, therefore the denominator polynomial must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ +\ 1\">.\r<BR>\n" );
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document.write( "If a rational function has a numerator that is one degree greater than the degree of the denominator, then the function will have a slant asymptote equal to the quotient of a polynomial long division of the numerator by the denominator.\r<BR>\n" );
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document.write( "Perform the polynomial long division of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE kx^2\ -\ kx\ -\ 6k\ \div\ x\ +\ 1\">.\r<BR>\n" );
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document.write( "Your quotient will have a factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\"> in it, but if you set the quotient equal to the given slant asymptote <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x\ -\ 4\">, you will very quickly see the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\">.\r<BR>\n" );
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document.write( "Then it is simply a matter of constructing your function from the derived numerator and denominator.\r<BR>\n" );
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document.write( "Go to <a  rel=nofollow HREF=\"http://www.purplemath.com/modules/polydiv2.htm\">Purple Math Polynomial Long Division</a> if you need a refresher on polynomial long division.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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