document.write( "Question 1186817: write a rational function f with the following properties.\r
\n" ); document.write( "\n" ); document.write( "f has a vertical asymptote x=2, hole at x=-6, x-intercept (1,0), and end behavior model q(x)=x+7 \n" ); document.write( "
Algebra.Com's Answer #817790 by greenestamps(13215)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "(1) vertical asymptote at x=2: there is a factor of (x-2) in the denominator but not in the numerator

\n" ); document.write( "(2) hole at x=-6: there are factors of (x+6) in both numerator and denominator

\n" ); document.write( "(3) x-intercept (1,0): there is a factor of (x-1) in the numerator but not in the denominator

\n" ); document.write( "At this point, the parts of the function we have are these:

\n" ); document.write( "\"f%28x%29=%28%28x-1%29%28x%2B6%29%29%2F%28%28x-2%29%28x%2B6%29%29\"

\n" ); document.write( "That function has a horizontal asymptote at y=1; we need a slant asymptote of y=x+7. To get a slant asymptote, we need an additional factor (x-a) in the numerator such that

\n" ); document.write( "\"f%28x%29=%28%28x-1%29%28x%2B6%29%28x-a%29%29%2F%28%28x-2%29%28x%2B6%29%29\"

\n" ); document.write( "has quotient (x+7) (and we don't care about the remainder)

\n" ); document.write( "We can determine the constant a using synthetic division of (x-1)(x-a) = x^2+(-a-1)x+a by x-2:
\r\n" );
document.write( "\r\n" );
document.write( "  2  |  1  -a-1  a\r\n" );
document.write( "     |        2 ...\r\n" );
document.write( "     +---------------\r\n" );
document.write( "        1  -a+1 ...

\n" ); document.write( "Since we want the asymptote to be y=x+7, we need to have

\n" ); document.write( "\"-a%2B1=7\"
\n" ); document.write( "\"-a=6\"
\n" ); document.write( "\"a=-6\"

\n" ); document.write( "The additional factor we need in the numerator is (x-(-6)) = (x+6).

\n" ); document.write( "ANSWER: \"f%28x%29=%28%28x-1%29%28x%2B6%29%28x%2B6%29%29%2F%28%28x-2%29%28x%2B6%29%29\"

\n" ); document.write( "A graph, showing the slant asymptote...

\n" ); document.write( "

\n" ); document.write( " \n" ); document.write( "
\n" );