document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=821326>Question 821326</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the x- and y- axis of f(x)=(x-4)(x+2)^2(x-2)<BR>\n" );
document.write( "Also, does the graoh of f touch or cross the x-axis at the x-intercept? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #494061 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=494061\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Read the zeros directly from the function definition and use them as critical points which define intervals along the x-axis.  \r<BR>\n" );
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document.write( "The zeros are at -2, 2, and 4.  You want to check a value for sign in each of the intervals:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-2 ALIGN=MIDDLE  ALT=\"x%3C-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3Cx%3C2 ALIGN=MIDDLE  ALT=\"-2%3Cx%3C2\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%3Cx%3C4 ALIGN=MIDDLE  ALT=\"2%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3Cx ALIGN=MIDDLE  ALT=\"4%3Cx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Intuitively, you would well guess that f(x) will be positive in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-2 ALIGN=MIDDLE  ALT=\"x%3C-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3Cx ALIGN=MIDDLE  ALT=\"4%3Cx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Try 0 within <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3Cx%3C2 ALIGN=MIDDLE  ALT=\"-2%3Cx%3C2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280-4%29%280%2B2%29%5E2%280-2%29=%28-4%29%284%29%28-2%29%3E0 ALIGN=MIDDLE  ALT=\"%280-4%29%280%2B2%29%5E2%280-2%29=%28-4%29%284%29%28-2%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so positive.\r<BR>\n" );
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document.write( "Try 3 within <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3Cx ALIGN=MIDDLE  ALT=\"4%3Cx\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-4%29%283%2B2%29%5E2%283-2%29=%28-1%29%2825%29%281%29%3C0 ALIGN=MIDDLE  ALT=\"%283-4%29%283%2B2%29%5E2%283-2%29=%28-1%29%2825%29%281%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, negative.\r<BR>\n" );
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document.write( "You should sketch a graph based on these and you then can find where f touches or crosses the x-axis.  Note what happens around <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;\" >; positive on both sides near this, so f not cross the x-axis, but TOUCHES it only.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-25%2C30%2C%28x-4%29%28x%2B2%29%5E2%28x-2%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-25%2C30%2C%28x-4%29%28x%2B2%29%5E2%28x-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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