document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=81824>Question 81824</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Make up a rational function that has the following characteristics: crosses the x-axis at 3; touches the x-axis at -2; has a vertical asymptote at x= 1 and at x= -4; has a hole at x=5; has a horizontal asymptote at y= 2.\r<BR>\n" );
document.write( "\n" );
document.write( "Please do try to help me! If you can't figure out the whole thing, then it would be a HUGE help in itself if you could tell me what \"touches the x-axis at -2\" means. What is the difference between that and crossing the x-axis? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58625 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=58625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Make up a rational function that has the following characteristics: <BR>\n" );
document.write( "crosses the x-axis at 3; <BR>\n" );
document.write( "Has (x-3) in the numerator.<BR>\n" );
document.write( "--------------------- <BR>\n" );
document.write( "touches the x-axis at -2;<BR>\n" );
document.write( "Has (x+2)^2 in the numerator.<BR>\n" );
document.write( "-------------------- <BR>\n" );
document.write( "has a vertical asymptote at x= 1 and at x= -4;<BR>\n" );
document.write( "Has (x-1) and (x+4) in the denominator<BR>\n" );
document.write( "--------------- <BR>\n" );
document.write( "has a hole at x=5;<BR>\n" );
document.write( "Has (x-5) in the numerator and in the denominator.<BR>\n" );
document.write( "-------------------- <BR>\n" );
document.write( "has a horizontal asymptote at y= 2.<BR>\n" );
document.write( "At this point the degrees of the numerator is 4<BR>\n" );
document.write( "and the degree of the denominator is 3<BR>\n" );
document.write( "so you need a factor of \"2\" in the numerator and a factor of \"x\" in the <BR>\n" );
document.write( "denominator.<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "f(x) = [2(x-3)(x+2)^2(x-5)]/[x(x-1)(x+4)(x-5)]<BR>\n" );
document.write( "==================<BR>\n" );
document.write( "What does \"touches the x-axis\" mean?<BR>\n" );
document.write( "Example y=x is a line that \"passes thru\" the x-axis at x=0.<BR>\n" );
document.write( "y=x^2 is a parabola that \"touches\" the x-axis at x=0<BR>\n" );
document.write( "This is what \"touches\" looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );