document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=134056>Question 134056</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph (x+3) (x-2)^2=0     (use maxs, mins, x and y intercepts) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #98085 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=98085\">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=%28x%2B3%29+%28x-2%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x%2B3%29+%28x-2%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > doesn't actually have a graph other than the three values (-3, 2, and 2) on the number line that satisfy the equation.  You can, however, graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%2B3%29%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%2B3%29%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and I suspect that is what you really meant.  (I sincerely hope your instructor didn't pose the question that way.  If s/he did, go get your money back)\r<BR>\n" );
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document.write( "Zeros, or x-intercepts at -3, 2, and 2  (we should have an extrema at 2)\r<BR>\n" );
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document.write( "Expand:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3-x%5E2-8x%2B12 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3-x%5E2-8x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Extreme points where f'(x) = 0, so:\r<BR>\n" );
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document.write( "f'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-2x-8 ALIGN=MIDDLE  ALT=\"3x%5E2-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Set the first derivative equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-2x-8=0 ALIGN=MIDDLE  ALT=\"3x%5E2-2x-8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence critical points are at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%2F3%29 ALIGN=MIDDLE  ALT=\"f%28-4%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29 ALIGN=MIDDLE  ALT=\"f%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=0 ALIGN=MIDDLE  ALT=\"f%282%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we have a critical point at (2,0).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%2F3%29=%28%28-4%2F3%29%2B3%29%28%28-4%2F3%29-2%29%5E2 ALIGN=MIDDLE  ALT=\"f%28-4%2F3%29=%28%28-4%2F3%29%2B3%29%28%28-4%2F3%29-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%2F3%29=%285%2F3%29%28-10%2F3%29%5E2 ALIGN=MIDDLE  ALT=\"f%28-4%2F3%29=%285%2F3%29%28-10%2F3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%2F3%29=%285%2F3%29%28100%2F9%29 ALIGN=MIDDLE  ALT=\"f%28-4%2F3%29=%285%2F3%29%28100%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%2F3%29=%28500%2F27%29 ALIGN=MIDDLE  ALT=\"f%28-4%2F3%29=%28500%2F27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And we have an extreme point at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2F27 ALIGN=MIDDLE  ALT=\"500%2F27\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "f\"(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x-2 ALIGN=MIDDLE  ALT=\"6x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "f\"(2)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-2=6%3E0 ALIGN=MIDDLE  ALT=\"8-2=6%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > => (2,f(2)) is a local minimum<BR>\n" );
document.write( "f\"(-4/3)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%28-4%2F3%29-2=-8-2=-10%3C0 ALIGN=MIDDLE  ALT=\"6%28-4%2F3%29-2=-8-2=-10%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > => (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%2F3%29 ALIGN=MIDDLE  ALT=\"f%28-4%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is a local maximum\r<BR>\n" );
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document.write( "The y-intercept is found by evaluating f(0):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=0%5E3-0%5E2-80%2B12=12 ALIGN=MIDDLE  ALT=\"f%280%29=0%5E3-0%5E2-80%2B12=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > hence the y-intercept is (0,12)\r<BR>\n" );
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document.write( "Using this information, we can make a rough sketch of the graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C600%2C-20%2C20%2C-20%2C20%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28600%2C600%2C-20%2C20%2C-20%2C20%2Cx%5E3-x%5E2-8x%2B12%29%2C%0D%0Acircle%282%2C0%2C.3%29%2C%0D%0Acircle%28-4%2F3%2C500%2F27%2C.3%29%2C%0D%0Acircle%28-3%2C0%2C.3%29%2C%0D%0Acircle%280%2C12%2C.3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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