document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=920439>Question 920439</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the amplitude, Period, Phase Shift, Graph and max/min and x and y intercepts of:\r<BR>\n" );
document.write( "\n" );
document.write( "g(t) = -2sin(&#960;t + &#960;/3)\r<BR>\n" );
document.write( "\n" );
document.write( "Please give a detailed answer for this. I think I understand the amplitude being the height from the negative or positive of the y axis. And therefore it would be from 2 to -2? And it's always the absolute of the first number.\r<BR>\n" );
document.write( "\n" );
document.write( "The period I think is the number of times It goes around and I think that is the &#960;/3 part\r<BR>\n" );
document.write( "\n" );
document.write( "The phase shift not sure\r<BR>\n" );
document.write( "\n" );
document.write( "and how to find the max and min, not sure<BR>\n" );
document.write( "or the x and y intercepts.\r<BR>\n" );
document.write( "\n" );
document.write( "Please help<BR>\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #558393 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=558393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the general formula is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = a*(sin(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 shift<BR>\n" );
document.write( "d is the vertical shift.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "replace t with x and your equation is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -2 * sin( pi * x + pi/3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this can also be written as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -2 * sin( pi * (x + 1/3))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the general equation is, once again:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y =  a * (sin( b * (x-c)) + d\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in your equation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "|a| = 2<BR>\n" );
document.write( "b = pi<BR>\n" );
document.write( "c = 1/3<BR>\n" );
document.write( "d = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your amplitude is equal to 2<BR>\n" );
document.write( "your frequency is equal to pi<BR>\n" );
document.write( "your horizontal shift is equal to -1/3<BR>\n" );
document.write( "your vertical shift is equal to 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your period is equal to 2*pi divided by your frequency.<BR>\n" );
document.write( "this makes your period equal to 2*pi / pi = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "everything is in radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equation 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/2014/1105_0002.jpg\" alt=\"picture 1\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the period is equal to 2 and goes from -1/3 to 5/3.<BR>\n" );
document.write( "your amplitude is equal to 2 so the graph goes from a high of 2 to a low of -2.<BR>\n" );
document.write( "there is no vertical shift because d = 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "take away the horizontal shift and your graph 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/2014/1105_0004.jpg\" alt=\"picture 2\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "everything else remains the same except there is no more horizontal shift so the period goes from 0 to 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your y intercept is the value of y when x = 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equation is, once again:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -2 * sin( pi * (x + 1/3))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when x = 0, y = -2 * sin(pi/3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "pi/3 = 60 degrees and sin (60) = sqrt(3)/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-2*sqrt(3)/2) = -sqrt(3) which is equivalent to -1.73.....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can see that on the following graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the value of y when x = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/1105_0005.jpg\" alt=\"picture 3\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the x-intercept is the value of x when y = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "set your equation and solve for x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "0 = -2sin(pi(x+1/3))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "divide both sides by -2 and you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sin(pi(x+1/3) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is true if and only if arc-sine(0) = pi(x+1/3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "arc-sin(0) = 0 so pi(x+1/3) = 0 which results in x = -1/3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the value of y is 0 when x = -1/3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/1105_0006.jpg\" alt=\"picture 4\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's a good reference on the general formula used.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.regentsprep.org/regents/math/algtrig/ATT7/sinusoidal.htm\" target = \"_blank\">http://www.regentsprep.org/regents/math/algtrig/ATT7/sinusoidal.htm</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( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );