document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=957213>Question 957213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the equation for a sin function\r<BR>\n" );
document.write( "\n" );
document.write( "Amplitude: 5, vertical shift: -4, phase shift: 3pie/4, and period is 8pie </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #584874 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=584874\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> general form of formula is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = a * sin(b * (x-c)) + d\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a is the amplitude<BR>\n" );
document.write( "b is the frequency<BR>\n" );
document.write( "c is the horizontal shift<BR>\n" );
document.write( "d is the vertical shift\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "phase shift is the same as horizontal shift.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you have:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "amplitude = 5 = a<BR>\n" );
document.write( "vertical shift = -4 = d<BR>\n" );
document.write( "phase shift / horizontal shift = (3/4)*pi = c<BR>\n" );
document.write( "period = 8*pi\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "frequency is equal to 2*pi / period<BR>\n" );
document.write( "period = 8*pi.<BR>\n" );
document.write( "frequency = 2*pi/(8*pi) = 4 = b\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 5 * sin(4*(x-(3/4)*pi)) - 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the amplitude and the vertical shift and the frequency / period are fairly easy to show.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the phase shift / horizontal shift is a little harder.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to show the horizontal shift, i created 2 graphs.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the first graph is without the horizontal shift and the second graph is with the horizontal shift so you can the effects of the horizontal shift.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the vertical shift moves the center line of the graph the number of units indicated.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's why the graphs shows the center line at y = -4 rather than at y = 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the amplitude shows the maximum / minimum height of the graph in relationship to the center line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since the amplitude of the sine wave is normally 1, and the amplitude was shown as 5, then the amplitude of the sine wave is 5 * 1 = 5 instead of 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the graphs, you will see that the maximum value of the sine wave is y = 1 and the minimum value of the sine wave is - 9.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 1 is 5 units above the center line at y = -4 and y = -9 is 5 units below the center line at y = -4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the frequency is equal to 2 * pi divided by the period which becomes 2*pi / (8*pi) which then becomes 1/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the frequency is 1/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the period is equal to 2 * pi divided by the frequency which becomes 2 * pi / (1/4) which then becomes 8 * pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the graph without the horizontal shift, you can see that one full cycle of the sine wave goes from 0 to 8 * pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "8*pi - 0*pi results in a period of 8*pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the graph with the horizontal shift, you can see that one full cycle of the sine wave goes from 3/4 * pi to 35/4 * pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "35/4 * pi - 3/4 * pi is equal to 32/4 * pi which is equal to 8 * pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the period in both graphs is the same.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the starting point of the sine wave in the second graph has been shifted to the right by 3/4 * pi radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that difference is your phase shift.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can see that the second full cycle of the sine wave has been shifted to the right by 3/4 * pi radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's the graphs.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/032001.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/032002.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );