document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=403781>Question 403781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm having trouble with the following problem:  Find the domain, range, amplitude, period, phase shift, and verticle shift of y=3cos1/2(x-2pi) +1.<BR>\n" );
document.write( "I know the equation is of the general form y=a*cos(bx+c) + d, but the 1/2 behind the cos has me stumped.  What effect does the 1/2 have?\r<BR>\n" );
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document.write( "Thanks,\r<BR>\n" );
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document.write( "DS </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #285669 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=285669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm having trouble with the following problem: Find the domain, range, amplitude, period, phase shift, and verticle shift of y=3cos1/2(x-2pi) +1.<BR>\n" );
document.write( "I know the equation is of the general form y=a*cos(bx+c) + d, but the 1/2 behind the cos has me stumped. What effect does the 1/2 have? \r<BR>\n" );
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document.write( "..\r<BR>\n" );
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document.write( "The standard form of the cos function is y=Acos(Bx-C)<BR>\n" );
document.write( "For the given function, <BR>\n" );
document.write( "y=3cos1/2(x-2pi) +1 =3cos(x/2-pi)+1<BR>\n" );
document.write( "The 1/2 after the Cos is B factored out of the expression,(x/2-pi)<BR>\n" );
document.write( "For example,(Bx-C)=B(x-C/B) As you can see, B and the phase shift,C/B, can be readily identified with this change. <BR>\n" );
document.write( "phase shift =2pi<BR>\n" );
document.write( "period=2pi/B=2pi/(1/2)=4pi<BR>\n" );
document.write( "(1/4) period = (4pi/4)=pi<BR>\n" );
document.write( "without vertical or horizontal shifting, (x,y)coordinates would be as follows:<BR>\n" );
document.write( "(0,3),(1/4pi,0),(1/2pi,-3),(3/4pi,0),(4pi,3)<BR>\n" );
document.write( "with a horizontal shift of 2pi,(x,y)coordinates would be as follows:<BR>\n" );
document.write( "(2pi,3),(9/4pi,0),(5/2pi,-3),(11/4pi,0),(6pi,3)<BR>\n" );
document.write( "with a vertical shift shift of +1,(x,y)coordinates would finally be as follows:<BR>\n" );
document.write( "(2pi,4),(9/4pi,1),(5/2pi,-2),(11/4pi,1),(6pi,4)<BR>\n" );
document.write( "..<BR>\n" );
document.write( "Domain [2pi,6pi] (1-cycle)<BR>\n" );
document.write( "Range  [4,-2]<BR>\n" );
document.write( "Amplitude 4max,2min<BR>\n" );
document.write( "Period 4pi<BR>\n" );
document.write( "Hor. Phase Shift 2pi (to the right)<BR>\n" );
document.write( "Vert Shift 1 up<BR>\n" );
document.write( "..<BR>\n" );
document.write( "See the graph below for the final translated configuration<BR>\n" );
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