document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=787302>Question 787302</A>: <I><SPAN STYLE=\"word-break: break-all;\"> sketch the graph of each rational function:<BR>\n" );
document.write( "3x - 4 / 10x^2 - 21x - 10\r<BR>\n" );
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document.write( "find vert. asymptote <BR>\n" );
document.write( "horiz. asymptote<BR>\n" );
document.write( "y-intercept<BR>\n" );
document.write( "x-intercepts </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #478167 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=478167\">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( "Vertical asymptotes occur where the denominator function has a real zero.  Set the denominator equal to zero and find all real number zeros.  Let's say the set of real number zeros for your denominator is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{\psi_1,\,\psi_2,\,.\,.\,.\,,\psi_n\}\">, then you will have a vertical asymptote at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \psi_1,\ \ \"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ x\ =\ \psi_2\ \ \"> and so on up through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \psi_n\">.\r<BR>\n" );
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document.write( "Horizontal asymptote:\r<BR>\n" );
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document.write( "If the degree of the numerator polynomial is less than the degree of the denominator polynomial, there is a horizontal asymptote at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ 0\">, which is to say, the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-axis.\r<BR>\n" );
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document.write( "If the degree of the numerator polynomial is equal to the degree of the denominator polynomial, then there is a horizontal asymptote at <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 on the numerator polynomial and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large q\"> is the lead coefficient on the denominaor polynomial.\r<BR>\n" );
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document.write( "If the degree of the numerator polynomial is greater than the degree of the denominator polynomial then there is no horizontal asymptote.  If the difference in degree is 1, then there is a slant or oblique asymptote whose equation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> equal to the result of polynomial long division of the numerator by the denominator excluding any remainder.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept.  Find the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> coordinate.\r<BR>\n" );
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document.write( "Set <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> equal to the numerator polynomial.  Substitute 0 for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, then solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y_0\">, the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept.  The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept is the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,\,y_0)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts.  Find all <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinates of all <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts\r<BR>\n" );
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document.write( "Set the numerator polynomial equal to zero.  Solve for all real-number zeros. Assume the set of real number zeros of the numerator polynomial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{\varphi_1,\,\varphi_2,\,.\,.\,.\,,\varphi_n\}\">.  Then the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts are the points:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (\varphi_1,\,0),\,(\varphi_2,\,0)\ .\,.\,.\,(\varphi_n,\,0)\">\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( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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