document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104537>Question 1104537</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Give an example of a rational function that fulfills<BR>\n" );
document.write( "the description.<BR>\n" );
document.write( "a. A rational function that has a vertical asymptote<BR>\n" );
document.write( "at -5 and a hole at 7.\r<BR>\n" );
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document.write( "b. A rational function that has no vertical asymptote<BR>\n" );
document.write( "and does have a slant asymptote </SPAN>\n" );
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document.write( "a. A rational function that has a vertical asymptote at -5 and a hole at 7.<br><BR>\n" );
document.write( "The graph of a rational function has a hole at x=a if there are linear factors of (x-a) in both numerator and denominator.<br><BR>\n" );
document.write( "The graph of a rational function has a vertical asymptote at x=a if there is a linear factor of x-a in the denominator but not in the numerator.<br><BR>\n" );
document.write( "So to get a vertical asymptote at x=-5 there must be a factor of (x+5) in the denominator but not in the numerator; to get a hole at x=7 there must be a factor of (x-7) in both numerator and denominator .<br><BR>\n" );
document.write( "The simplest rational function with those factors is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-7%29%2F%28%28x%2B5%29%28x-7%29%29 ALIGN=MIDDLE  ALT=\"%28x-7%29%2F%28%28x%2B5%29%28x-7%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "b. A rational function that has no vertical asymptote and does have a slant asymptote<br><BR>\n" );
document.write( "The graph of a rational function has a slant asymptote if the degree of the numerator is 1 greater than the degree of the denominator.<br><BR>\n" );
document.write( "Assuming for simplicity that there are no holes in the graph, caused by identical linear factors in both numerator and denominator, then if the graph of a rational function has NO vertical asymptotes, then there are no linear factors in the denominator.  That means the denominator has no real zeros.<br><BR>\n" );
document.write( "So a rational function that has a slant asymptote but no vertical asymptote must have a denominator with no real zeros, and a numerator with degree 1 greater than the degree of the denominator.<br><BR>\n" );
document.write( "A simple example of such a function is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2F%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"x%5E3%2F%28x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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