document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=550135>Question 550135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the function f(x)=x(x-2)^2(x-4)<BR>\n" );
document.write( "I. find x and y intercept<BR>\n" );
document.write( "II.Determine whether the graph crosses or touches the x-axis at each intercept<BR>\n" );
document.write( "III. Graph the function </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #358515 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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=358515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the function f(x)=x(x-2)^2(x-4)<BR>\n" );
document.write( "I. find x and y intercept<BR>\n" );
document.write( "II.Determine whether the graph crosses or touches the x-axis at each intercept<BR>\n" );
document.write( "III. Graph the function<BR>\n" );
document.write( "**<BR>\n" );
document.write( "I. x-intercept:<BR>\n" );
document.write( "set y=0, then solve for x<BR>\n" );
document.write( "x(x-2)^2(x-4=0<BR>\n" );
document.write( "x-intercepts=0,2(mult. 2), and 4<BR>\n" );
document.write( "..<BR>\n" );
document.write( "y-intercept<BR>\n" );
document.write( "set x=0, then solve for y<BR>\n" );
document.write( "y-intercept=x(x-2)(x-2)(x-4)<BR>\n" );
document.write( "..<BR>\n" );
document.write( "II. Number line:<BR>\n" );
document.write( "<....+....0....-....2...-.....4...+.....><BR>\n" );
document.write( "For x>4, f>0<BR>\n" );
document.write( "Moving to the left, the sign of each interval switches each time we go thru a zero of odd multiplicity like 1, 3, 5, etc.  The sign of each in interval does not change if we go thru a zero of even multiplicity like 2, 4, 6, etc. Accordingly, the graph crosses the x-axis at x= 0 and 4, but at x=2, only touches but does not cross the x-axis.<BR>\n" );
document.write( "..<BR>\n" );
document.write( "III.See graph below as a visual check on the above:<BR>\n" );
document.write( "..<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cx%5E4-8x%5E3%2B20x%5E2-16x%29%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cx%5E4-8x%5E3%2B20x%5E2-16x%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );