document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166001>Question 1166001</A>: <I><SPAN STYLE=\"word-break: break-all;\">  I am a rational function having a vertical asymptote at<BR>\n" );
document.write( "the lines x = 3 and x = -3, and a horizontal asymptote y = 1.<BR>\n" );
document.write( "If my only x-intercept is 5, and y-intercept is -5/9, What<BR>\n" );
document.write( "function am I? </SPAN>\n" );
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document.write( "As we will see, there are too many constraints; there is no rational function that meets all the requirements.<br><BR>\n" );
document.write( "(a) Vertical asymptotes at x=3 and x=-3<br><BR>\n" );
document.write( "This requires factors of (x-3) and (x+3) (and no others) in the denominator:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+a%2F%28%28x%2B3%29%28x-3%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+a%2F%28%28x%2B3%29%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(b) Only x-intercept at x=5<br><BR>\n" );
document.write( "This requires a factor of (x-5) in the numerator, and no other factors in the numerator except a constant:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+a%28x-5%29%2F%28%28x%2B3%29%28x-3%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+a%28x-5%29%2F%28%28x%2B3%29%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(c) Horizontal asymptote at y=1<br><BR>\n" );
document.write( "This requires that the degree of the numerator be equal to the degree of the denominator; it also requires that the constant be a=1.<br><BR>\n" );
document.write( "The only factors in the denominator are (x+3) and (x-3), and the only factor in the numerator is (x-5).  For the degrees of the numerator and denominator to be equal, the factor in the numerator needs to be there twice:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x-5%29%5E2%2F%28%28x%2B3%29%28x-3%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x-5%29%5E2%2F%28%28x%2B3%29%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "At this point, there are no unknowns in the function; it is completely determined by the horizontal and vertical asymptotes and the x-intercept.<br><BR>\n" );
document.write( "A graph showing the vertical asymptotes and the single x-intercept; the horizontal asymptote doesn't show up well because the function value gets close to 1 only for very large positive or very large negative values of x....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C%28x-5%29%5E2%2F%28%28x%2B3%29%28x-3%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C%28x-5%29%5E2%2F%28%28x%2B3%29%28x-3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(d) y-intercept -5/9<br><BR>\n" );
document.write( "This is problematic.  The function as it currently stands has a y-intercept of -25/9.<br><BR>\n" );
document.write( "If we were to introduce a factor of 1/5 in the function to get a y-intercept of -5/9, then the horizontal asymptote would be y=1/5 instead of y=1.<br><BR>\n" );
document.write( "So we can't find a rational function that satisfies all of the given conditions.<br><BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );