document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=849547>Question 849547</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a rational function with x-intercept at (0,0), two points at (1,1) and (-2,4), horizontal asymptote at y=2, and vertical asymptote at x=-1.  Sketch the graph and write an equation for the function.\r<BR>\n" );
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document.write( "I have plotted the given points and asymptotes.  I pretty much know how the graph is going, however I can't seem to figure out the equation.  I know that the numerator should have 2x somewhere in it, but not sure where or what else goes with it.  I know that the denominator is going to be x+1. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #511746 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=511746\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the solution is:\r<BR>\n" );
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document.write( "(2x) / (x+1)\r<BR>\n" );
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document.write( "the graph that is generated looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C%282x%29%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C%282x%29%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the table that could be generated looks like this:\r<BR>\n" );
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document.write( "        x         y = (2x) / (x+1)\r\n" );
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document.write( "        -2             4\r\n" );
document.write( "        -1             undefined\r\n" );
document.write( "         0             0\r\n" );
document.write( "         1             1\r\n" );
document.write( "         2             4/3\r\n" );
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document.write( "since the vertical asymptote was at -1, the denominator of the graph had to be some for of x + 1 because then -1 + 1 = 0 and the graph is undefined at that point.\r<BR>\n" );
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document.write( "the numerator of the graph had to be the same level as the denominator, because then the graph has a horizontal asymptote at whatever level of y the numerator was at when the value of x goes to infinity.\r<BR>\n" );
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document.write( "a recommended method that i learned is that you divide the numerator and the denominator by the highest level exponent.   This makes everything that is not important cancel out.\r<BR>\n" );
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document.write( "For example:\r<BR>\n" );
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document.write( "Assuming the numerator is 2x and the denominator is (x+1), and you want to find the horizontal asymptote, factor out the greatest exponent term.\r<BR>\n" );
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document.write( "2x factors out an x.\r<BR>\n" );
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document.write( "(x+1) factors out an x.\r<BR>\n" );
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document.write( "you are left with:\r<BR>\n" );
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document.write( "x * (2) / (x * (1 + 1/x)\r<BR>\n" );
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document.write( "now, as x goes to infinity, the numerator stays at 2 and the denominator stays at 1, so the limit of the expression as x goes to infinity is equal to 2.\r<BR>\n" );
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document.write( "the solution is part luck and part detective work in finding what the expression could be.   \r<BR>\n" );
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document.write( "not sure if i would be successful every time, but it worked this time.\r<BR>\n" );
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document.write( "some tutorials on the web that may be helpful are:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm\" target = \"_blank\">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut40_ratgraph.htm\" target = \"_blank\">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut40_ratgraph.htm</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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