document.write( "Question 1185294: I am a rational
\n" ); document.write( "function having a vertical asymptote at
\n" ); document.write( "the lines x = 3 and x = -3, and a
\n" ); document.write( "horizontal asymptote at y = 1 . If my
\n" ); document.write( "only x-intercept is 5, and my y-intercept
\n" ); document.write( "is - 5/9 , what function am I? \n" ); document.write( "

Algebra.Com's Answer #816117 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "Greenestamps didn't think of the sneaky trick of eliminating an x-intercept by\r\n" );
document.write( "putting an extra factor in the numerator and denominator to create a hole in the\r\n" );
document.write( "curve instead of having an x-intercept.  I'll use that trick.\r\n" );
document.write( "\r\n" );
document.write( "To have those vertical asymptotes, the denominator, when set = 0, must have\r\n" );
document.write( "roots 3 and -3, so the denominator must contain (x-3)(x+3), which has degree 2\r\n" );
document.write( "and leading coefficient 1.  That might be enough for the denominator.  If so,\r\n" );
document.write( "the numerator must also be of degree 2 and have leading coefficient 1 in order\r\n" );
document.write( "to have horizontal asymptote y = 1.  So let's try this version:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To have x-intercept 5, it must go through (5,0).  So we substitute \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by 16\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To have y-intercept -5/9, it must go through (0,-5/9).  So we substitute \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides by -9\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Substituting in\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So let's substitute those values for A and B in\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let's graph it and see what we have:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Oh darn! That has an extra x-intercept at (1,0).  Aha, but I can play the trick!\r\n" );
document.write( "I'll put a hole in the curve at (1,0) by putting in a factor of (x-1) in\r\n" );
document.write( "the numerator and the denominator:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the graph has a hole at (1,0) instead of an x-intercept there.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The un-factored form of the rational function is:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );