document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=912332>Question 912332</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Help with rational functions<BR>\n" );
document.write( "these are confusing me: \r<BR>\n" );
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document.write( "y = 3(2^x)-1 \r<BR>\n" );
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document.write( "r(x) = ((2x+1)x)/(4-x^2) \r<BR>\n" );
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document.write( "How would I get the vertical/horizontal asymptotes along with the x and y intercepts from these rational functions? <BR>\n" );
document.write( "Please explain \r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #553733 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39618)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=553733\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The first equation or function is not a rational one, but is a polynomial equation or function.  The function r(x) IS a rational function.  You need to learn what is a polynomial function and then what is a rational function before you can effectively make the sense of these in the way that you are asking.  They ARE related but they are not the same thing.\r<BR>\n" );
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document.write( "Moving directly to the r(x), the RATIONAL function,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=%28x%282x%2B1%29%29%2F%28%282-x%29%282%2Bx%29%29 ALIGN=MIDDLE  ALT=\"r%28x%29=%28x%282x%2B1%29%29%2F%28%282-x%29%282%2Bx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > when factored.\r<BR>\n" );
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document.write( "No factor common to both numerator and denominator, meaning no holes.\r<BR>\n" );
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document.write( "Undefined for x at 2 and -2, so r(x) has vertical asymptote at x=-2 and at x=2.\r<BR>\n" );
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document.write( "x-intercepts for x=0 because it is a factor in the numerator, and for x=-1/2 which makes the binomial 2x+1=0; so x-intercepts  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "y-intercept:  Let x=0, and evaluate r(0). </SPAN>\n" );
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