document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113820>Question 113820</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5.	Determine the behavior of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Domain :	<BR>\n" );
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document.write( "Hole(s) :	                     <BR>\n" );
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document.write( "Vertical Asymptote(s) :	          <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82829 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=82829\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <b>Domain:</b> Simply set the denominator equal to zero and solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-4x=0 ALIGN=MIDDLE  ALT=\"x%5E3-4x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%5E2-4%29=0 ALIGN=MIDDLE  ALT=\"x%28x%5E2-4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out an x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B2%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"x%28x%2B2%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <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;\" > using the difference of squares\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2=0 ALIGN=MIDDLE  ALT=\"x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set each factor equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" > Solve for x in each case\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" > makes the denominator zero, we must take these values out of the domain.\r<BR>\n" );
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document.write( "So our domain is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \textrm{\left{x|x\in\mathbb{R} x\neq-2 x\neq0 x\neq2\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: x is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E-2 ALIGN=MIDDLE  ALT=\"x%3C%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E0 ALIGN=MIDDLE  ALT=\"x%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E2 ALIGN=MIDDLE  ALT=\"x%3C%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<b>x-intercept(s):</b> To find the x-intercepts, let f(x)=0 and solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29 ALIGN=MIDDLE  ALT=\"0=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-4x%5E2-16x=0 ALIGN=MIDDLE  ALT=\"2x%5E3-4x%5E2-16x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the numerator equal to zero. Remember the denominator <b>cannot</b> equal zero. So the numerator \r<BR>\n" );
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document.write( "can only equal zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%282x%5E2-4x-16%29=0 ALIGN=MIDDLE  ALT=\"2x%282x%5E2-4x-16%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out 2x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%28x-4%29%28x%2B2%29=0 ALIGN=MIDDLE  ALT=\"2x%28x-4%29%28x%2B2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-8 ALIGN=MIDDLE  ALT=\"x%5E2-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x-4%29%28x%2B2%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=2x=0 ALIGN=MIDDLE  ALT=\"2x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4=0 ALIGN=MIDDLE  ALT=\"x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2=0 ALIGN=MIDDLE  ALT=\"x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set each factor equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   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;\" > Solve for x in each case\r<BR>\n" );
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document.write( "So the possible x-intercepts are: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   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( "However, since x=-2 and x=0 is not in the domain, the only x-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer: The x-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<b>Holes:</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%282x%5E2-4x-16%29%2F%28x%28x%5E2-4%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=x%282x%5E2-4x-16%29%2F%28x%28x%5E2-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Factor out an x out of the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=cross%28x%29%282x%5E2-4x-16%29%2F%28cross%28x%29%28x%5E2-4%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=cross%28x%29%282x%5E2-4x-16%29%2F%28cross%28x%29%28x%5E2-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Cancel like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E2-4x-16%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E2-4x-16%29%2F%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28x-4%29%28x%2B2%29%29%2F%28%28x-2%29%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28x-4%29%28x%2B2%29%29%2F%28%28x-2%29%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the numerator and denominator \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28x-4%29cross%28%28x%2B2%29%29%29%2F%28%28x-2%29cross%28%28x%2B2%29%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28x-4%29cross%28%28x%2B2%29%29%29%2F%28%28x-2%29cross%28%28x%2B2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28x-4%29%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28x-4%29%29%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "So here we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28x-4%29%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28x-4%29%29%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . However, we must \r<BR>\n" );
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document.write( "keep in mind to exclude x=-2, x=0, and x=2 (remember the domain). Since we can plug in  x=-2 and x=0 into \r<BR>\n" );
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document.write( "at x=-2 and x=0. \r<BR>\n" );
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document.write( "Answer: holes at x=-2 and x=0\r<BR>\n" );
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document.write( "<b>y-intercept(s):</b> To find the y-intercepts, let x=0 and solve for y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28x-4%29%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28x-4%29%29%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the simplified equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%280-4%29%29%2F%280-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%280-4%29%29%2F%280-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28-4%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28-4%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28-8%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28-8%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "So when x=0, f(x) (which is y) is equal to 4. \r<BR>\n" );
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document.write( "Answer: So the y intercept is (0,4)\r<BR>\n" );
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document.write( "<b>Vertical Asymptote(s):</b>\r<BR>\n" );
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document.write( "Vertical asymptotes occur when the denominator is equal to zero. In other words, at the value of x that is not in the domain.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%28x-4%29%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%28x-4%29%29%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the simplified equation\r<BR>\n" );
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document.write( "Answer: So the vertical asymptote is <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( "<b>Horizontal Asymptote(s):</b>\r<BR>\n" );
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document.write( "To find the horizontal asymptote(s), simply take the leading coefficient of the numerator (which is 2) and divide it by the leading coefficient of the denominator (which is 1)\r<BR>\n" );
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document.write( "So simply divide 2 by 1 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F1=2 ALIGN=MIDDLE  ALT=\"2%2F1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the horizontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we graph the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can visually verify all of our answers:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E3-4x%5E2-16x%29%2F%28x%5E3-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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