document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1059284>Question 1059284</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to 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; and has a horizontal asymptote at y=2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #674351 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39623)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=674351\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <b>NOTE:  Because of the \"touch at x=-2\" part of the description, another tutor may need to discuss this exercise for a better solution.</b>\r<BR>\n" );
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document.write( "Only some GUIDANCE...\r<BR>\n" );
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document.write( "The numerator may control all your x-intercepts.  Having a horizontal asymtptote means degree of numerator and degree of denominator are the same.  Your vertical asymptotes tell you where along the x-axis the \"roots\" for the numerator are.\r<BR>\n" );
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document.write( "That should help you very much.\r<BR>\n" );
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document.write( "--According to your personal note, the guidance above was not clear enough for you or you did not seem ready for that guidance.\r<BR>\n" );
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document.write( "Look at the specifications for your rational function:<BR>\n" );
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document.write( "Zeros at x for 3 and -2<BR>\n" );
document.write( "Vertical asymptotes at x for 1 and -4<BR>\n" );
document.write( "Hole at x=5<BR>\n" );
document.write( "Horizontal asymptote for y=2<BR>\n" );
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document.write( "Two things may stand out to you right-away.<BR>\n" );
document.write( "Horizontal Asymptote, telling you degree of numerator and denominator are same; and the hole and the vertical asymptotes tell you three x-values for which the function is undefined.\r<BR>\n" );
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document.write( "Give the function a factor, k.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%28x-a%29%28x-b%29%28x-5%29%2F%28%28x-1%29%28x-%28-4%29%29%28x-5%29%29 ALIGN=MIDDLE  ALT=\"y=k%28x-a%29%28x-b%29%28x-5%29%2F%28%28x-1%29%28x-%28-4%29%29%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Look and see that:<BR>\n" );
document.write( "(*) Degree of numerator and denominator are equal.<BR>\n" );
document.write( "(*) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-5 ALIGN=MIDDLE  ALT=\"x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor in both the numerator AND the denominator.\r<BR>\n" );
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document.write( "Good so far?\r<BR>\n" );
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document.write( "Degree of numerator is 3 and degree of denominator is ALSO 3.<BR>\n" );
document.write( "Formula for y contains the factor   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%2F%28x-5%29 ALIGN=MIDDLE  ALT=\"%28x-5%29%2F%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, obviously an expression meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  but makes the formula undefined at x=5.\r<BR>\n" );
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document.write( "A minor simplification of the formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28k%28x-3%29%28x%2B2%29%28x-5%29%29%2F%28%28x-1%29%28x%2B4%29%28x-5%29%29 ALIGN=MIDDLE  ALT=\"y=%28k%28x-3%29%28x%2B2%29%28x-5%29%29%2F%28%28x-1%29%28x%2B4%29%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice this also written to account for the two zeros at x=3 and x=-2.\r<BR>\n" );
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document.write( "The horizontal asymptote:<BR>\n" );
document.write( "Need y=2 when x goes unbound toward either negative or positive infinity.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%28x-3%29%28x%2B2%29%28x-5%29%29%2F%28%28x-1%29%28x%2B4%29%28x-5%29%29 ALIGN=MIDDLE  ALT=\"y=%282%28x-3%29%28x%2B2%29%28x-5%29%29%2F%28%28x-1%29%28x%2B4%29%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that this solution attempt may still be incomplete, because I am unsure how to establish that y crosses x-axis at 3 but TOUCHES the x-axis at -2.  Another tutor may need to help with finishing for that.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-6%2C6%2C-12%2C12%2C%282%28x-3%29%28x%2B2%29%28x-5%29%29%2F%28%28x-1%29%28x%2B4%29%28x-5%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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