document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189458>Question 1189458</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the polynomial Function : F(x)=-2(x + 1/2) (x+4)^2\r<BR>\n" );
document.write( "\n" );
document.write( "a)List each real zero and its multiplicity:<BR>\n" );
document.write( "b)Determine whether graph crosses or touches the x-axis at each x-intercept:<BR>\n" );
document.write( "c)Determine the behavior of the graph near each x-intercept(zero):<BR>\n" );
document.write( "d)Determine the maximum number of turning point on the graph:<BR>\n" );
document.write( "e)Determine the end behavior, that is finding the power function that the graph of <BR>\n" );
document.write( "  f resembles for large values of |x|: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820849 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=820849\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
document.write( "\n" );
document.write( "For the polynomial Function : \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-2%28x+%2B+1%2F2%29+%28x%2B4%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=-2%28x+%2B+1%2F2%29+%28x%2B4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a)List each real zero and its multiplicity:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%28x+%2B+1%2F2%29+%28x%2B4%29%5E2=0 ALIGN=MIDDLE  ALT=\"-2%28x+%2B+1%2F2%29+%28x%2B4%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+1%2F2%29=0 ALIGN=MIDDLE  ALT=\"%28x+%2B+1%2F2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,..... multiplicity 1<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%5E2=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;\" >,..... multiplicity 2\r<BR>\n" );
document.write( "\n" );
document.write( "b)Determine whether graph crosses or touches the x-axis at each x-intercept:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For zeros with<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+even ALIGN=MIDDLE  ALT=\"+even\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiplicities, the graphs <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=touch ALIGN=MIDDLE  ALT=\"touch\"   DISABLED_style= \"max-width:90%; height:auto;\" > or are tangent to the x-axis at these x-values. <BR>\n" );
document.write( "For zeros with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=odd ALIGN=MIDDLE  ALT=\"odd\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiplicities, the graphs <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross ALIGN=MIDDLE  ALT=\"cross\"   DISABLED_style= \"max-width:90%; height:auto;\" > or intersect the x-axis at these x-values. \r<BR>\n" );
document.write( "\n" );
document.write( "<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;\" > have an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+even ALIGN=MIDDLE  ALT=\"+even\"   DISABLED_style= \"max-width:90%; height:auto;\" >  multiplicity => the graph will <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=touch ALIGN=MIDDLE  ALT=\"touch\"   DISABLED_style= \"max-width:90%; height:auto;\" > the x-axis \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > have an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=odd ALIGN=MIDDLE  ALT=\"odd\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiplicity=> the graph will <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross ALIGN=MIDDLE  ALT=\"cross\"   DISABLED_style= \"max-width:90%; height:auto;\" >   the x-axis\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c)Determine the behavior of the graph near each x-intercept(zero):\r<BR>\n" );
document.write( "\n" );
document.write( "Since the leading term of the polynomial (the term in the polynomial which contains the highest power of the variable) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E3 ALIGN=MIDDLE  ALT=\"-2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the degree is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, i.e. even, and the leading coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, i.e. negative.\r<BR>\n" );
document.write( "\n" );
document.write( "This means that f(x)→ ∞ as x→ -∞ , f(x)→ -∞ as x→ ∞\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "d)Determine the maximum number of turning point on the graph:\r<BR>\n" );
document.write( "\n" );
document.write( " The maximum number of turning points of a polynomial function is always one less than the degree of the function.<BR>\n" );
document.write( "This function f is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >th degree polynomial function and has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > turning points. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "e)Determine the end behavior, that is finding the power function that the graph of<BR>\n" );
document.write( "f resembles for large values of |<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >|:\r<BR>\n" );
document.write( "\n" );
document.write( "If we expand we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-2+x%5E3+-+17+x%5E2+-+40+x+-+16 ALIGN=MIDDLE  ALT=\"f%28x%29+=+-2+x%5E3+-+17+x%5E2+-+40+x+-+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-2x%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+-2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .....  the dominating term\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );