document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=962524>Question 962524</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I missed something in lecture somewhere about finding the functions for graphs such as the image shows. What is the method for solving these. I know what all the standard graphs look like sin,cos,tan,sec,csc,cot and their asymptotes. \r<BR>\n" );
document.write( "\n" );
document.write( "Please explain how these are solved/figured out. \r<BR>\n" );
document.write( "\n" );
document.write( "THANKS!\r<BR>\n" );
document.write( "\n" );
document.write( "http://i.imgur.com/QQS7VCY.png<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #588142 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=588142\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> picture submitted by student.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://i.imgur.com/QQS7VCY.png\" target = \"_blank\">http://i.imgur.com/QQS7VCY.png</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the standard form of a trigonometric function is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = a * (sin or cos or tan or sec or csc or cot)(b * (x - c)) + d\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a is the amplitude<BR>\n" );
document.write( "b is the frequency<BR>\n" );
document.write( "c is the horizontal displacement.<BR>\n" );
document.write( "d is the vertical displacement.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "period is equal to 360 / frequency in degrees and 2*pi / frequency in radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "frequency is equal to 360 / period in degrees and 2*pi / period in radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your first picture looks like a cotangent function.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "one complete cycle goes from -4 * pi to 4 * pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's a period of 8 * pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the frequency is therefore equal to (2 * pi) divided by (8 * pi) which is equal to 1/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the function of y = a * cot(b * (x - c)) + d becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = cot(1/4 * x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a is equal to 1 and is therefore not shown.<BR>\n" );
document.write( "b is equal to 1/4<BR>\n" );
document.write( "c is equal to 0 and is therefore not shown.<BR>\n" );
document.write( "d is equal to 0 and is therefore not shown.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of y = cot(1/4 * x) looks like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/041205.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your second picture looks like an upside sine function or a sine function that has been horizontally displaced by 180 degrees (equivalent to pi radians).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "either assumption should yield the same graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "assuming it's an upside down sine function, then we'll check the other data to determine what value to place there.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the basic function would be y = a * sin(b * (x - c)) + d\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a is equal to -8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is because the normal amplitude is 1 and this one is 8, so a has to be equal to 8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since it's upside down, then a also has to be negative, so a will be equal to -8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the center of the graph is on the x-axis, so d is equal to 0 and therefore not shown.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "one complete cycle goes from 0 to 2*pi/3, so the period is equal to 2pi/3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the frequency is equal to (2 * pi) divided by (2 * pi / 3) which is equal to 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equation looks like it should be:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -8 * sin(3*x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "there is no horizontal displacement (c = 0 and therefore not shown) and there is no vertical displacement (d = 0and therefore not shown).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of y = -8 * sin(3*x) looks like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/041206.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "assuming it's a sine function that is horizontally displaced by 180 degrees (equivalent to pi radians) rather than an upside sine wave , then you would tackle it as follows:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the amplitude is still 8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it is not negative, however, since we are not looking at an upside down sine wave anymore.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the frequency is still 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the horizontal displacement, however, is equal to pi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is because pi radians is equivalent to 180 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the general formula of y = a * sin(b * (x-c)) + d becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 8 * sin(3 * (x - pi)).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of y = 8 * sin(3 * (x - pi)) looks like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/041207.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "as you can see, the graph of y = -8 * sin(3x) looks exactly the same as the graph of y = 8 * sin(3 * (x-pi)).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "they're identical.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can further confirm that this is true by taking your calculator and solving for any random value of x using both formulas.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for example, assume x is equal to 15.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "remember, we're dealing in radians and not degrees.  your calculator should be set to radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -8 * sin(3 * 15) = -6.807228196\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 8 * sin(3 * (15 - pi)) = -6.807228196\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "they're the same.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this and the graph provides pretty conclusive proof that the equations are identical.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can graph these yourself using the online calculator found at:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );