document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1045870>Question 1045870</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A function $f$ has horizontal asymptote of $y = -4,$ a vertical asymptote of $x = 3,$ and an $x$-intercept at $(1,0).$ <BR>\n" );
document.write( "Part (b): Let $f$ be of the form<BR>\n" );
document.write( "$$f(x) = \frac{rx+s}{2x+t}.$$<BR>\n" );
document.write( "Find an expression for $f(x).$  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #661408 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=661408\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Igor's input box doesn't understand TeX or LaTeX.  Your dollar signs just make your input difficult to read.  Use standard plain text notation when posting here, please.  See <a  rel=nofollow HREF=\"http://www.purplemath.com/modules/mathtext.htm\">Formatting Math as Text</a> for more information about how to communicate math expressions in plain text that everyone understands.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If there is a horizontal asymptote, then the degree of the numerator and denominator polynomials must be equal (which is true in this case) and the equation of the asymptote must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ \frac{p}{q}\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the lead coefficient of the numerator polynomial and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large q\"> is the lead coefficient of the denominator polynomial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So if your function is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{rx\ +\ s}{2x\ +\ t}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "and if the equation of the horizontal asymptote is\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{r}{2}\ =\ -4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "and\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ -8\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A vertical asymptote exists where the denominator polynomial is equal to zero, so if the vertical asymptote is\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "then\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2(3)\ -\ t\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which is to say\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ 6\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In order for the function value to be zero, the numerator must be zero.  So, given an x-intercept of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (1,0)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -8(1)\ +\ s\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "or\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ s\ =\ 8\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Putting it all together,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ -\frac{8x\ -\ 8}{2x\ -\ 6}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );