document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1142058>Question 1142058</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the period of y = 3cos ( 4x + π/2) + 5? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #762703 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=762703\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> period = 2 * pi / frequency\r<BR>\n" );
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document.write( "frequency = 2 * pi / period\r<BR>\n" );
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document.write( "your equation is y = 3 * cos (4x + pi/2) + 5\r<BR>\n" );
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document.write( "since the general form of the equation is y = a * cos(b * (x - c)) + d, you can rewrite the equation to be:\r<BR>\n" );
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document.write( "y = 3 * cos(4 * (x + pi/8) + 5\r<BR>\n" );
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document.write( "your frequency is 4.\r<BR>\n" );
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document.write( "that makes your period equal to 2 * pi / 4 = pi / 2\r<BR>\n" );
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document.write( "there is a vertical shift of 5.\r<BR>\n" );
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document.write( "this makes the center line equal to y = 5 on the graph.\r<BR>\n" );
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document.write( "the amplitube is 3.\r<BR>\n" );
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document.write( "that makes the graph go from y = 8 to y = 2.\r<BR>\n" );
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document.write( "the horizontal shift is pi/8.\r<BR>\n" );
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document.write( "i'll get to that later.\r<BR>\n" );
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document.write( "your equation, without the horizontal shift, is:\r<BR>\n" );
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document.write( "y = 3 * cos(4 * x) + 5\r<BR>\n" );
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document.write( "the graph of that equation looks like this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/060908.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "with the horizontal shift, the equation is:\r<BR>\n" );
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document.write( "y = 3 * cos(4 * (x - pi/8)) + 5\r<BR>\n" );
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document.write( "the graph of that equation looks like this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/060909.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "before the shift, the high point of the cosine function was at x = 0.\r<BR>\n" );
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document.write( "after the shift, the high point of the cosine function was at x = -pi/8.\r<BR>\n" );
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document.write( "this confirms that the graph of the equation was shifted to the left by pi/8.\r<BR>\n" );
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document.write( "if you try to find the value of y using your calculator, make sure that the calculator is set to radians.\r<BR>\n" );
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document.write( "the equation, and the graph, assumes x is in radians\r<BR>\n" );
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document.write( "this probably way more than you needed to know, but hopefully it is instructive.\r<BR>\n" );
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document.write( "the solution to your problem is:\r<BR>\n" );
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document.write( "frequency = 4.\r<BR>\n" );
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document.write( "period = 2 * pi / 4.\r<BR>\n" );
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document.write( "that makes period equal to pi/2, or 1/2 * pi, whichever way you want to show it.\r<BR>\n" );
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document.write( "a period of 1/2 * pi means you get 4 full cycles of the cosine function in the normal period of 2 * pi.\r<BR>\n" );
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document.write( "the graph of that looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/060910.jpg\" alt=\"$$$\" >\r<BR>\n" );
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