document.write( "Question 1090285: Help please \r
\n" ); document.write( "\n" ); document.write( "Find the slant asymptote,the vertical asymptote,the x and the y intercepts and then sketch the graph of
\n" ); document.write( "f(x)=(x^3+x^2)/x^2-4
\n" ); document.write( "

Algebra.Com's Answer #704728 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
IF YOU MEANT \"f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29\" :
\n" ); document.write( "Factoring, we get
\n" ); document.write( "\"f%28x%29=%28x%2B1%29x%5E2%2F%28%28x%2B2%29%28x-2%29%29\"
\n" ); document.write( "The function is undefined for \"x=-2\" and \"x=2\" ,
\n" ); document.write( "and \"highlight%28x=-2%29\" and \"highlight%28x=2%29\" are the equations of the vertical asymptotes.
\n" ); document.write( "The value of the function is zero for \"x=0\" and \"x=-1\" ,
\n" ); document.write( "so \"highlight%28x=-1%29\" and \"highlight%28x=0%29\" are the x-intercepts, where \"y=0\" ,
\n" ); document.write( "and \"y=0\" is the y-intercept, where \"x=0\" .
\n" ); document.write( "In other words, we know the graph goes though (0,0) and (-1,0).
\n" ); document.write( "The numerator of the function, \"%28x%2B1%29x%5E2\" , changes sign only at \"x=-1\" .
\n" ); document.write( "The denominator of the function, \"%28x%2B2%29%28x-2%29\" , changes sign at \"x=-2\" and \"x=2\" .
\n" ); document.write( "All four factors are positive for \"x%3E2%3E-1%3E-2\" , nd so is \"f%28x%29\" .
\n" ); document.write( "As you cross \"x=2\" , the fucntion changes sign,
\n" ); document.write( "so \"f%28x%29%3C=0\" for \"-1%3Cx%3C2\" , with \"f%280%29=0\" for \"x=0\" .
\n" ); document.write( "For \"-2%3Cx%3C-1\" , \"f%28x%29%3E0\" ,
\n" ); document.write( "and for \"x%3C-2\" , \"f%28x%29%3C0\" .
\n" ); document.write( "As for a slant asymptote, doing the division we find that
\n" ); document.write( "\"%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29=x%2B1%2B%284x%2B4%29%28x%5E2-4%29\" ,
\n" ); document.write( "so \"highlight%28y=x%2B1%29\" is the slant asymptote.
\n" ); document.write( "This is what the graph of the function and the slant asymptote looks like:
\n" ); document.write( "\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29%2Cx%2B1%29\" .
\n" ); document.write( "With the vertical asymptotes added, it would look like this:
\n" ); document.write( " .
\n" ); document.write( "
\n" );