document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1078936>Question 1078936</A>: <I><SPAN STYLE=\"word-break: break-all;\">  For the sinusoidal function below, give its vertical shift, amplitude, phase shift, period, and range. The, graph two periods of the function, labeling five consecutive key points.<BR>\n" );
document.write( "y= -cos(x/4-&#960;/2)<BR>\n" );
document.write( "Can someone help me to solve this?<BR>\n" );
document.write( "Thanks  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #693325 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693325\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Definitions:\r<BR>\n" );
document.write( "\n" );
document.write( "y = A cos(Bx + C) + D<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "amplitude = A<BR>\n" );
document.write( "period = 2&#960;/B<BR>\n" );
document.write( "phase shift = &#8722;C/B<BR>\n" );
document.write( "vertical shift = D<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+-cos%28+x%2F4+-+pi%2F2+%29+ ALIGN=MIDDLE  ALT=\"+y+=+-cos%28+x%2F4+-+pi%2F2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+D+=+0+ ALIGN=MIDDLE  ALT=\"+D+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so, the vertical shift is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+ ALIGN=MIDDLE  ALT=\"+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+-1+ ALIGN=MIDDLE  ALT=\"+A+=+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so, the amplitude is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1+ ALIGN=MIDDLE  ALT=\"+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "( same as the regular amplitude, except inverted )<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-C%2FB+=+%28-%28-pi%2F2%29%29%2F%281%2F4%29+ ALIGN=MIDDLE  ALT=\"+-C%2FB+=+%28-%28-pi%2F2%29%29%2F%281%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-C%2FB+=+4%2A%28pi%2F2%29+ ALIGN=MIDDLE  ALT=\"+-C%2FB+=+4%2A%28pi%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-C%2FB+=+2%2Api+ ALIGN=MIDDLE  ALT=\"+-C%2FB+=+2%2Api+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the phase shift is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2Api+ ALIGN=MIDDLE  ALT=\"+2%2Api+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is 360 degrees,<BR>\n" );
document.write( "which means there is no phase shift at all.<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2Api%29%2FB+=+%282%2Api%29%2F%281%2F4%29+ ALIGN=MIDDLE  ALT=\"+%282%2Api%29%2FB+=+%282%2Api%29%2F%281%2F4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2Api%29%2FB+=+8%2Api+ ALIGN=MIDDLE  ALT=\"+%282%2Api%29%2FB+=+8%2Api+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so, the period is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+8%2Api+ ALIGN=MIDDLE  ALT=\"+8%2Api+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "The range is from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1+ ALIGN=MIDDLE  ALT=\"+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2B1+ ALIGN=MIDDLE  ALT=\"+%2B1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > which 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;\" ><BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "Here is the plot over 2 periods, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+16%2Api+ ALIGN=MIDDLE  ALT=\"+16%2Api+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-30%2C+30%2C+-2%2C+2%2C+-cos%28+x%2F4+-+pi%2F2+%29+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+400%2C+400%2C+-30%2C+30%2C+-2%2C+2%2C+-cos%28+x%2F4+-+pi%2F2+%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "( note that a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+8%2Api+=+25.13+ ALIGN=MIDDLE  ALT=\"+8%2Api+=+25.13+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is about what I see on the plot )<BR>\n" );
document.write( " </SPAN>\n" );
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