document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090285>Question 1090285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Help please \r<BR>\n" );
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document.write( "Find the slant asymptote,the vertical asymptote,the x and the y intercepts and then sketch the graph of <BR>\n" );
document.write( "f(x)=(x^3+x^2)/x^2-4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #704730 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=704730\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+p%28x%29+%2F+q%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+p%28x%29+%2F+q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "you are given a rational function written as the ratio of two polynomials where the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%28x%29 ALIGN=MIDDLE  ALT=\"q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > isn't zero\r<BR>\n" );
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document.write( "Vertical Asymptotes:<BR>\n" );
document.write( "An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%28x%29 ALIGN=MIDDLE  ALT=\"q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Completely ignore the numerator when looking for vertical asymptotes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=only ALIGN=MIDDLE  ALT=\"only\"   DISABLED_style= \"max-width:90%; height:auto;\" > the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=denominator ALIGN=MIDDLE  ALT=\"denominator\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+matters ALIGN=MIDDLE  ALT=\"+matters\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4=0 ALIGN=MIDDLE  ALT=\"x%5E2-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=4 ALIGN=MIDDLE  ALT=\"x%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%284%29 ALIGN=MIDDLE  ALT=\"x=sqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+2 ALIGN=MIDDLE  ALT=\"x=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, vertical asymptotes: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+2 ALIGN=MIDDLE  ALT=\"x=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Horizontal Asymptotes:\r<BR>\n" );
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document.write( "The location of the horizontal asymptote is determined by looking at the degrees of the numerator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and denominator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Cm ALIGN=MIDDLE  ALT=\"n%3Cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x-axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=horizontal ALIGN=MIDDLE  ALT=\"horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote.<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m ALIGN=MIDDLE  ALT=\"n=m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=an+%2F+bm ALIGN=MIDDLE  ALT=\"y=an+%2F+bm\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=horizontal ALIGN=MIDDLE  ALT=\"horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote. That is, the ratio of the leading coefficients.<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Em ALIGN=MIDDLE  ALT=\"n%3Em\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal asymptote. <BR>\n" );
document.write( "However, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m%2B1 ALIGN=MIDDLE  ALT=\"n=m%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=oblique ALIGN=MIDDLE  ALT=\"oblique\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slant ALIGN=MIDDLE  ALT=\"slant\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote.\r<BR>\n" );
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document.write( "in your case,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2 ALIGN=MIDDLE  ALT=\"m=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >; so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Em ALIGN=MIDDLE  ALT=\"n%3Em\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal asymptote\r<BR>\n" );
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document.write( "Oblique Asymptotes:\r<BR>\n" );
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document.write( "When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique (slant) asymptote. \r<BR>\n" );
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document.write( "in your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2 ALIGN=MIDDLE  ALT=\"m=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m%2B1 ALIGN=MIDDLE  ALT=\"n=m%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >; so, you have a slant asymptote<BR>\n" );
document.write( "To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%2B1%29 ALIGN=MIDDLE  ALT=\"highlight%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"+%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >/<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Bx%5E2 ALIGN=MIDDLE  ALT=\"x%5E3%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-4x ALIGN=MIDDLE  ALT=\"x%5E3-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "----------------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2Bx%5E2 ALIGN=MIDDLE  ALT=\"0%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---------------------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is asymptotic to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=highlight%28x+%2B+1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=highlight%28x+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Agreen%28line%282%2C10%2C2%2C-10%29%29%2Cgreen%28line%28-2%2C10%2C-2%2C-10%29%29%2C%0D%0A+graph%28600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29%2Cx%2B1%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );