document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204997>Question 1204997</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph shows part of a sine function of the form y = A sin B(x + C) + D. Determine the values of A, B, C, and D.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841583 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=841583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
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document.write( "One of the sine templates is<BR>\n" );
document.write( "y = A*sin( B(x + C) ) + D<BR>\n" );
document.write( "which is what your teacher gave you. <BR>\n" );
document.write( "Another template is <BR>\n" );
document.write( "y = A*sin( Bx + C ) + D<BR>\n" );
document.write( "which is slightly different. I'll stick to the first template.\r<BR>\n" );
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document.write( "The key parameters are:<BR>\n" );
document.write( "|A| = amplitude<BR>\n" );
document.write( "B = used to determined the period<BR>\n" );
document.write( "C = used to determine horizontal phase shift<BR>\n" );
document.write( "D = used to determine vertical shift\r<BR>\n" );
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document.write( "The highest and lowest points occur when y = 7 and y = -3 respectively.<BR>\n" );
document.write( "The gap between those y values is 10 units. <BR>\n" );
document.write( "Count the spaces, or subtract and use absolute value, to determine this gap.\r<BR>\n" );
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document.write( "The curve is 10 units tall. Half of which is 10/2 = 5, and this is the amplitude. <BR>\n" );
document.write( "It's the vertical distance from the horizontal midline to either peak or valley.<BR>\n" );
document.write( "This leads to |A| = 5.<BR>\n" );
document.write( "So either A = 5 or A = -5.<BR>\n" );
document.write( "Either value will work. Let's go with <font color=red>A = 5</font> for simplicity.\r<BR>\n" );
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document.write( "One of the lowest points on this curve is at (0,-3)<BR>\n" );
document.write( "One max point occurs at (2pi, 7)<BR>\n" );
document.write( "Going from valley to neighboring peak is a horizontal distance of 2pi units. <BR>\n" );
document.write( "This represents 1/2 of the full period, so the full period is 2*2pi = 4pi units.<BR>\n" );
document.write( "The curve repeats itself every 4pi units along the x axis. \r<BR>\n" );
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document.write( "T = 4pi = period<BR>\n" );
document.write( "T = 2pi/B<BR>\n" );
document.write( "B = 2pi/T<BR>\n" );
document.write( "B = (2pi)/(4pi)<BR>\n" );
document.write( "<font color=red>B = 1/2</font>\r<BR>\n" );
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document.write( "The parent sine function y = sin(x) maxes out when x = pi/2. <BR>\n" );
document.write( "According to this graph, one max is when x = 2pi.<BR>\n" );
document.write( "This is a gap of 2pi - pi/2 = 3pi/2<BR>\n" );
document.write( "So what happened is that the parent sine curve had been shifted 3pi/2 units to the right.\r<BR>\n" );
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document.write( "Let's return to the idea that the max and min occur when y = 7 and y = -3 respectively.  <BR>\n" );
document.write( "Add those y coordinates up and divide by 2 so we can get the midpoint.<BR>\n" );
document.write( "(7+(-3))/2 = 4/2 = 2<BR>\n" );
document.write( "The midline is y = 2 which leads to <font color=red>D = 2</font><BR>\n" );
document.write( "This is how much we shift the parent function upward. \r<BR>\n" );
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document.write( "The last thing we need is C.<BR>\n" );
document.write( "Let's plug in those previous values along with the coordinates of the min (0,-3) so we can solve for C.<BR>\n" );
document.write( "y = A*sin( B(x + C) ) + D<BR>\n" );
document.write( "-3 = 5*sin( 0.5(0 + C) ) + 2<BR>\n" );
document.write( "-3 = 5*sin(0.5C) + 2<BR>\n" );
document.write( "-3-2 = 5*sin(0.5C)<BR>\n" );
document.write( "-5 = 5*sin(0.5C)<BR>\n" );
document.write( "sin(0.5C) = -5/5<BR>\n" );
document.write( "sin(0.5C) = -1<BR>\n" );
document.write( "0.5C = arcsin(-1)<BR>\n" );
document.write( "0.5C = -pi/2<BR>\n" );
document.write( "(1/2)C = -pi/2<BR>\n" );
document.write( "C = 2*(-pi/2)<BR>\n" );
document.write( "<font color=red>C = -pi</font>\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "<font color=red>A = 5</font><BR>\n" );
document.write( "<font color=red>B = 1/2</font><BR>\n" );
document.write( "<font color=red>C = -pi</font><BR>\n" );
document.write( "<font color=red>D = 2</font>\r<BR>\n" );
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document.write( "Confirmation using Desmos which is a graphing tool.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/yuzs46tz03\">https://www.desmos.com/calculator/yuzs46tz03</a><BR>\n" );
document.write( "GeoGebra is another graphing tool that I recommend. <BR>\n" );
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