document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1135312>Question 1135312</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Finding the number of cycles in 2 pi, the horizontal length of each of the 4 quadrants of each cycle =, vertical shift, horizontal shift of each cycle, and determining a reflection over the x axis. \r<BR>\n" );
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document.write( "f(theta)=4sin(3theta+theta/4)-10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #752979 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=752979\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this graph is a little different.\r<BR>\n" );
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document.write( "i use x for theta and y for f(x), so the formula in my language is:\r<BR>\n" );
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document.write( "y = 4 * sin(3x + x/4) - 10\r<BR>\n" );
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document.write( "it's the same formula except it's easier to graph this way using the graphing calculator at desmos.com.\r<BR>\n" );
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document.write( "within the parentheses, 3x + x/4 is equivalent to 12x/4 + x/4; which is equal to 13x/4.\r<BR>\n" );
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document.write( "that translates to the frequency being 13/4 * x.\r<BR>\n" );
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document.write( "there is no horizontal shift.\r<BR>\n" );
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document.write( "it's just a frequency change.\r<BR>\n" );
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document.write( "if the frequency is 13/4, then the period of the sine function is equal to 2 * pi / (13/4) which is equal to 2 * pi * 4 / 13 which is equal to 8/13 * pi.\r<BR>\n" );
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document.write( "your sine function of y = 4 * sin(3x + x/4) - 10 translates to y = 4 * sin(13/4 * x) - 10.\r<BR>\n" );
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document.write( "the amplitude is 4.<BR>\n" );
document.write( "the vertical displacement is -10.<BR>\n" );
document.write( "the frequency is 13/4<BR>\n" );
document.write( "the period is 8/13 * pi.\r<BR>\n" );
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document.write( "here's the graph from 0 radians to 8pi/13 radians.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/022305.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "here's the graph from 8pi/13 radians to 16pi/13 radians.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/022306.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the amplitude is 4.<BR>\n" );
document.write( "the center line is at y = -10<BR>\n" );
document.write( "that means the high point of the graph is y = -10 + 4 which becomes y = -6.<BR>\n" );
document.write( "that means the low point of the graph is y = -10 - 4 which becomes y = -14.<BR>\n" );
document.write( "the period is 8pi/13 radians.<BR>\n" );
document.write( "that means one cycle goes from 0 to 8pi/13 and from 8pi/13 to 16pi/13.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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