document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202196>Question 1202196</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the equation of a cosine function for the following graph of a sinusoidal function:<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 #836858 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=836858\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>y = 2.5*cos(2x)-1</font><BR>\n" );
document.write( "This is when you are in degree mode.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "The image from the question is this<BR>\n" );
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document.write( "The template for cosine is <BR>\n" );
document.write( "y = A*cos(B(x-C))+D\r<BR>\n" );
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document.write( "The variables are:<BR>\n" );
document.write( "|A| = amplitude<BR>\n" );
document.write( "B = helps determine the period, more on that later<BR>\n" );
document.write( "C = phase shift, which is the side to side shift<BR>\n" );
document.write( "D = up and down shift, useful to determine midline\r<BR>\n" );
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document.write( "The goal is to determine the values of A, B, C, and D.\r<BR>\n" );
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document.write( "Let's start with A.\r<BR>\n" );
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document.write( "The amplitude is half of the height of the sine wave.<BR>\n" );
document.write( "The peak occurs when y = 1.5 and the valley is when y = -3.5; these represent the largest and smallest possible y values respectively.\r<BR>\n" );
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document.write( "Subtract and use absolute value to find the vertical distance between peak and valley.<BR>\n" );
document.write( "|peak - valley| = |1.5 - (-3.5)|<BR>\n" );
document.write( "|peak - valley| = |1.5 + 3.5|<BR>\n" );
document.write( "|peak - valley| = |5|<BR>\n" );
document.write( "|peak - valley| = 5\r<BR>\n" );
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document.write( "Or you could say<BR>\n" );
document.write( "|valley - peak| = |-3.5 - 1.5|<BR>\n" );
document.write( "|valley - peak| = |-5|<BR>\n" );
document.write( "|valley - peak| = 5\r<BR>\n" );
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document.write( "The vertical distance from peak to valley is 5 units.<BR>\n" );
document.write( "Half of this is 5/2 = 2.5 and it is the amplitude.\r<BR>\n" );
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document.write( "|A| = 2.5 leads to either A = 2.5 or A = -2.5<BR>\n" );
document.write( "We use A = 2.5 because the cosine graph is above the x axis when x = 0.\r<BR>\n" );
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document.write( "Now onto the variable B.\r<BR>\n" );
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document.write( "This variable is tied to the period.<BR>\n" );
document.write( "The period is the length of each cycle, aka wavelength.\r<BR>\n" );
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document.write( "One peak occurs at x = 0<BR>\n" );
document.write( "The next neighboring peak to the right is when x = 180<BR>\n" );
document.write( "This is a gap of 180-0 = 180 units<BR>\n" );
document.write( "The period is 180 because the curve repeats itself every 180 units.\r<BR>\n" );
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document.write( "T = 180 = period<BR>\n" );
document.write( "B = 2pi/T<BR>\n" );
document.write( "B = 2pi/180<BR>\n" );
document.write( "B = pi/90<BR>\n" );
document.write( "That would be the value of B if we are in radian mode.\r<BR>\n" );
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document.write( "However, the next set of steps show a much nicer result for B when in degree mode.<BR>\n" );
document.write( "B = 360/T<BR>\n" );
document.write( "B = 360/180<BR>\n" );
document.write( "B = 2\r<BR>\n" );
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document.write( "Because the x axis shows 90,180,270,etc it appears your teacher might want degree mode.\r<BR>\n" );
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document.write( "The phase shift is C = 0 because cosine has x = 0 lead to a peak. We haven't shifted the peak left or right off the y axis. \r<BR>\n" );
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document.write( "I recommend using a graphing tool like Desmos to look at y = cos(x) to see what I mean. <BR>\n" );
document.write( "GeoGebra is another handy graphing tool I use all the time.\r<BR>\n" );
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document.write( "The midline is y = D to represent the horizontal line cutting through the middle of this cosine curve.\r<BR>\n" );
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document.write( "Apply the midpoint of the peak and valley<BR>\n" );
document.write( "(peak+valley)/2 = (1.5+(-3.5))/2 = -2/2 = -1\r<BR>\n" );
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document.write( "The midline is y = -1 and we get D = -1\r<BR>\n" );
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document.write( "Summary <BR>\n" );
document.write( "A = 2.5<BR>\n" );
document.write( "B = 2 (when in degree mode)<BR>\n" );
document.write( "C = 0<BR>\n" );
document.write( "D = -1\r<BR>\n" );
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document.write( "Then<BR>\n" );
document.write( "y = A*cos(B(x-C))+D<BR>\n" );
document.write( "y = 2.5*cos(2(x-0))+(-1)<BR>\n" );
document.write( "<font color=red>y = 2.5*cos(2x)-1</font><BR>\n" );
document.write( "This is when your calculator is in degree mode.\r<BR>\n" );
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document.write( "Graph using Desmos<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/ju2jkpfbkq\">https://www.desmos.com/calculator/ju2jkpfbkq</a><BR>\n" );
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