document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205033>Question 1205033</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph represents a sine function in the form<BR>\n" );
document.write( "y = A sin B(t + C) + D or a cosine function in the form y = A cos B(t + C) + D. Write both equations for each.\r<BR>\n" );
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document.write( "Answer key: y = 4 sin πt, y = 4 cos π(t  -  1/2)<BR>\n" );
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document.write( "By the way, is there an edit button or similar function to make changes to a post after I've already posted it in case I discover later on I've made a typo? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841633 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=841633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Students aren't able to edit, but tutors can. So please be sure to review your questions carefully. If you do notice a mistake after the fact, then you can mention it in the thank you note to the tutor. A possibly better alternative is to make a new post. \r<BR>\n" );
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document.write( "The highest and lowest points occur when y = 4 and y = -4 respectively.<BR>\n" );
document.write( "Use the midpoint formula to find that y = 0 is the midpoint. <BR>\n" );
document.write( "This leads us to <font color=red>D = 0</font> which represents the midline.<BR>\n" );
document.write( "This applies to both sine and cosine templates.\r<BR>\n" );
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document.write( "The height of this curve is 8 units. <BR>\n" );
document.write( "Half of which is 4, which is the value of A.<BR>\n" );
document.write( "<font color=red>A = 4</font><BR>\n" );
document.write( "This applies to both sine and cosine templates.<BR>\n" );
document.write( "Technically we could go for A = -4, but I'll stick with the positive version since that's what your teacher picked.\r<BR>\n" );
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document.write( "The neighboring peak points occur when x = 0.5 and x = 2.5<BR>\n" );
document.write( "The gap is 2.5 - 0.5 = 2 units which is the period.<BR>\n" );
document.write( "The curve repeats itself every 2 units along the x axis.\r<BR>\n" );
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document.write( "T = period = 2<BR>\n" );
document.write( "T = 2pi/B<BR>\n" );
document.write( "B = 2pi/T<BR>\n" );
document.write( "B = 2pi/2<BR>\n" );
document.write( "<font color=red>B = pi</font><BR>\n" );
document.write( "This applies to both sine and cosine templates.\r<BR>\n" );
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document.write( "So far we found that:<BR>\n" );
document.write( "<font color=red>A = 4</font><BR>\n" );
document.write( "<font color=red>B = pi</font><BR>\n" );
document.write( "<font color=red>D = 0</font><BR>\n" );
document.write( "and they work for both sine and cosine templates mentioned.<BR>\n" );
document.write( "Side note: Cosine is a phase shifted version of sine. \r<BR>\n" );
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document.write( "Let's plug those values in. We'll also plug in one of the points on the curve.<BR>\n" );
document.write( "I'll plug in (0.5, 4)<BR>\n" );
document.write( "This will allow us to solve for C.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "y = A*sin( B(x+C) ) + D<BR>\n" );
document.write( "4 = 4*sin( pi(0.5+C) ) + 0<BR>\n" );
document.write( "4 = 4*sin( pi(0.5+C) )<BR>\n" );
document.write( "sin( pi(0.5+C) ) = 4/4<BR>\n" );
document.write( "sin( pi(0.5+C) ) = 1<BR>\n" );
document.write( "pi(0.5+C) = arcsin(1)<BR>\n" );
document.write( "pi(0.5+C) = pi/2<BR>\n" );
document.write( "0.5+C = 1/2<BR>\n" );
document.write( "1/2+C = 1/2<BR>\n" );
document.write( "C = 1/2 - 1/2<BR>\n" );
document.write( "<font color=red>C = 0</font>\r<BR>\n" );
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document.write( "Summary for the sine template<BR>\n" );
document.write( "<font color=red>A = 4</font><BR>\n" );
document.write( "<font color=red>B = pi</font><BR>\n" );
document.write( "<font color=red>C = 0</font><BR>\n" );
document.write( "<font color=red>D = 0</font><BR>\n" );
document.write( "This will mean we go from y = A*sin(B(t+C)) + D to <font color=red>y = 4*sin(pi*t)</font>\r<BR>\n" );
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document.write( "Cosine will look almost identical in terms of A,B,C,D values. <BR>\n" );
document.write( "A,B,D will be the same.<BR>\n" );
document.write( "C is going to be different.\r<BR>\n" );
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document.write( "Let's plug the known A,B,D values into the cosine template. <BR>\n" );
document.write( "Also let's plug in (0.5, 4) so we can solve for C.<BR>\n" );
document.write( "y = A*cos(B(x+C)) + D<BR>\n" );
document.write( "4 = 4*cos(pi(0.5+C)) + 0<BR>\n" );
document.write( "4 = 4*cos(pi(0.5+C))<BR>\n" );
document.write( "cos(pi(0.5+C)) = 1<BR>\n" );
document.write( "pi(0.5+C) = arccos(1)<BR>\n" );
document.write( "pi(0.5+C) = 0<BR>\n" );
document.write( "0.5+C = 0<BR>\n" );
document.write( "C = -0.5<BR>\n" );
document.write( "<font color=red>C = -1/2</font>\r<BR>\n" );
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document.write( "For cosine we have<BR>\n" );
document.write( "<font color=red>A = 4</font><BR>\n" );
document.write( "<font color=red>B = pi</font><BR>\n" );
document.write( "<font color=red>C = -1/2</font><BR>\n" );
document.write( "<font color=red>D = 0</font><BR>\n" );
document.write( "which explains how we arrive at <font color=red>y = 4*cos(pi(t - 1/2))</font>\r<BR>\n" );
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document.write( "Confirmation using Desmos graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/uatjjocx1m\">https://www.desmos.com/calculator/uatjjocx1m</a><BR>\n" );
document.write( "I'm using x in place of t much of the time, but you get the idea.<BR>\n" );
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