document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102598>Question 1102598</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help!! I need a rational function that crosses the x-axis at 3, touches the x-axis at -2, has vertical asymptotes at x=1 and x=-4, has a hole at x=5 and has a horizontal asymptote at y=2\r<BR>\n" );
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document.write( "I kind of figured out how to get it to do all except touching the x-axis at 2 and the hole at x=5. what am I doing wrong? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717299 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717299\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Suppose a rational function is written with  numerator consisting of a constant coefficient and linear factors of the form (x-a), and with the denominator consisting of linear factors of the form (x-a).  Then...<BR>\n" );
document.write( "(1) Each factor (x-a) that appears the same number of times in both numerator and denominator will produce a hole in the graph at x=a;<BR>\n" );
document.write( "(2) Each factor (x-b) that appears only in the denominator will produce a vertical asymptote at x=b;<BR>\n" );
document.write( "(3) Each factor (x-c) that appears only once (or any odd number of times) in the numerator will produce a root at x=c, with the graph crossing the x-axis at that point; and<BR>\n" );
document.write( "(4) Each factor (x-d) that appears twice (or any even number of times) in the numerator will produce a root at x=d, with the graph just touching the x-axis at that point.<br><BR>\n" );
document.write( "(5a) The graph of the rational function will have a horizontal asymptote of y=0 if the degree of the numerator is less than the degree of the denominator;<BR>\n" );
document.write( "(5b) The graph will have a horizontal asymptote of y=k if the degrees of the numerator and denominator are equal and the constant coefficient is k.<br><BR>\n" );
document.write( "Here is a rational function that shows most of these features, along with its graph (red) and the horizontal asymptote (green).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%28x-2%29%28x%2B2%29%5E2%28x-1%29%29%2F%28%28x%2B1%29%28x-1%29%28x-3%29%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"%284%28x-2%29%28x%2B2%29%5E2%28x-1%29%29%2F%28%28x%2B1%29%28x-1%29%28x-3%29%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C400%2C-8%2C8%2C-20%2C20%2C4%28x-2%29%28x%2B2%29%5E2%28x-1%29%2F%28x%2B1%29%28x-1%29%28x-3%29%28x%2B3%29%2C4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The constant coefficient 4, along with the fact that both numerator and denominator are of degree 4, produces the horizontal asymptote y=4 (item 5b above).<br><BR>\n" );
document.write( "The identical factors of (x-1) in the numerator and denominator produce a hole in the graph at x=1 (item 1 above).  The hole is not apparent in the graph produced by the software used on this site.<br><BR>\n" );
document.write( "(If you graph this function on a good graphing calculator, with a small window either side of x=1, you should be able to see the hole.)<br><BR>\n" );
document.write( "The factors (x+3), (x+1), and (x-3) in the denominator produce the vertical asymptotes at x=-3, x=-1, and x=3 (item 2 above).<br><BR>\n" );
document.write( "The single factor (x-2) in the numerator produces a zero at x=2, with the graph crossing the x-axis at that point (item 3 above).<br><BR>\n" );
document.write( "And the double factor of (x+2) in the numerator produces a double root at x=-2, with the graph just touching the x-axis at that point (item 4 above).<br><br><BR>\n" );
document.write( "Use that example and the preceding discussion to determine the factors that are required for your example.<br><BR>\n" );
document.write( "Notice that the requirements in your problem require...<BR>\n" );
document.write( "(1) a single root  (1 factor in the numerator);<BR>\n" );
document.write( "(2) a double root  (2 factors in the numerator);<BR>\n" );
document.write( "(3) two vertical asymptotes  (2 factors in the denominator); and<BR>\n" );
document.write( "(4) a hole (1 factor each in numerator and denominator)<br><BR>\n" );
document.write( "With only factors causing those features, the numerator has 4 linear factors while the denominator has only 3.  If those were the only factors, then there would be no horizontal asymptote, as required.  To get the required horizontal asymptote y=2, you need another factor in the denominator (producing a third vertical asymptote) to make the degrees of the numerator and denominator the same; and you need a constant coefficient 2 in the numerator.<br><BR>\n" );
document.write( "That is why my example is similar to yours but has three vertical asymptotes instead of two.<br>\r<BR>\n" );
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