document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202955>Question 1202955</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A given sinusoidal function has a period of 3, an amplitude of 7, and a maximum at (0, 2). Represent the function with a sine equation and a cosine equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838178 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=838178\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the general form of the sine and cosine function is y = a * sin(b * (x - c)) + d or y = a * cos(b * (x - c)) + d.\r<BR>\n" );
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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" );
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document.write( "since the amplitude is 7, the general form becomes y = 7 * sin(b * (x - c)) + d or y = 7 * cos(b * (x - c)) + d. \r<BR>\n" );
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document.write( "when d = 0, the horizontal center line of the graph is at y = 0.<BR>\n" );
document.write( "if the horizontal center line of the graph is at y = 0, then the max value of the graph will be at y = 7 and the min value of the graph will be at y = -7.<BR>\n" );
document.write( "since the max value of the graph is at (0,2), then the max value of the graph is at y = 2.<BR>\n" );
document.write( "subtract 7 from that to get a horizontal center line at y = -5.<BR>\n" );
document.write( "this means that d must be equal to -5, since d gives you the vertical shift of the graph horizontal center line.\r<BR>\n" );
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document.write( "the general form of the graph now becomes y = 7 * sin(b * (x - c)) - 5 or y = 7 * cos(b * (x - c)) - 5.\r<BR>\n" );
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document.write( "the period of the graph is 3.<BR>\n" );
document.write( "since the frequency of the graph is equal to 360 / period, then the frequency of the graph must be equal to 120.<BR>\n" );
document.write( "this means that you will get 120 full cycles of the sine or cosine graph in the normal period of 360 degrees.\r<BR>\n" );
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document.write( "the general form of the graph now becomes y = 7 * sin(120 * (x - c)) - 5 or y = 7 * cos(120 * (x - c)) - 5.\r<BR>\n" );
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document.write( "you are given that the graph has a maximum at (0,2).<BR>\n" );
document.write( "this means that the high peak of the graph is at 2 when the value of x is at 0.<BR>\n" );
document.write( "since the cosine function normal cycle has a peak at x = 0, then there is no horixontal shift of the cosine function.\r<BR>\n" );
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document.write( "the general form of the graph now becomes y = 7 * sin(120 * (x - c)) - 5 or y = 7 * cos(120 * x) - 5.\r<BR>\n" );
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document.write( "the c disappears in the cosine graph because the shift is equal to 0.\r<BR>\n" );
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document.write( "the sine function peaks at x = 90 degrees when the period is 360 degrees.<BR>\n" );
document.write( "to make the sine function pack at 0 degrees, the graph of the sine function must be shifted to the left 90 degrees.<BR>\n" );
document.write( "that is when the peiod is 360 degrees.<BR>\n" );
document.write( "when the period is 3 degrees, the shift is equal to 90 / 120 = .75 degrees.\r<BR>\n" );
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document.write( "the general form of the graph now becomes y = 7 * sin(120 * (x - .75)) - 5 or y = 7 * cos(120 * x) - 5.\r<BR>\n" );
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document.write( "these are the final forms of the graph and should satisfy the reqirements of the problem.\r<BR>\n" );
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document.write( "the graph is shown below.\r<BR>\n" );
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document.write( "both sine and cosine graphs are shown.<BR>\n" );
document.write( "since they are equivalent, they both draw the same image on the graph.<BR>\n" );
document.write( "3 periods of both functions are shown.<BR>\n" );
document.write( "the main, or base period is from x = 0 to x = 3<BR>\n" );
document.write( "the period before is from x = -3 to x = 0<BR>\n" );
document.write( "the period after is from x = 3 to x = 6\r<BR>\n" );
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document.write( "the horizontal center line is at y = -5<BR>\n" );
document.write( "the high value of the graph is at y = 2<BR>\n" );
document.write( "the low value of the grpah is at y = -12\r<BR>\n" );
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document.write( " </SPAN>\n" );
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