document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=968758>Question 968758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help, I posted this question earlier only to realise I had put 1&#960; instead of 1/2&#960; like i should have.<BR>\n" );
document.write( "Consider the trigonometric function f (t) = &#8722;1 + 4 sin( 1/2 &#960;(t &#8722; 1)).<BR>\n" );
document.write( "(a) What is the amplitude of f(t)? (b) What is the period of f(t)?<BR>\n" );
document.write( "(c) What are the maximum and minimum values attained by f(t)? (d) Sketch the graph of f (t) for t &#8712; [&#8722;1, 3]. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #591947 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=591947\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the general formula for the sine function is:\r<BR>\n" );
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document.write( "y = a * sin(b * (x-c)) + d\r<BR>\n" );
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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" );
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document.write( "the normal period of the sine function is 2*pi.\r<BR>\n" );
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document.write( "the formula for the period is:\r<BR>\n" );
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document.write( "period = (2 * pi) / freqauency.\r<BR>\n" );
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document.write( "when a or b or c or d are not shown, their default values are:<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 1<BR>\n" );
document.write( "c = 0<BR>\n" );
document.write( "d = 0\r<BR>\n" );
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document.write( "when d = 0, the center line of the graph is at y = 0.<BR>\n" );
document.write( "when d = -1, the center line of the graph is at y = -1.\r<BR>\n" );
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document.write( "when b = 1, the period = 2 * pi divided by 1 which is equal to 2 * pi.\r<BR>\n" );
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document.write( "your formula brecomes:\r<BR>\n" );
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document.write( "f(t) = 4 * sin(pi/2 * x) - 1\r<BR>\n" );
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document.write( "your amplitude = 4<BR>\n" );
document.write( "your frequency = pi/2<BR>\n" );
document.write( "your horizontal shift = 0<BR>\n" );
document.write( "your vertical shift = -1\r<BR>\n" );
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document.write( "the center line of your graph is at y = -1<BR>\n" );
document.write( "the maximum value of your graph is -1 + 4 = 3<BR>\n" );
document.write( "the minimum value of your graph is -1 - 4 = -5\r<BR>\n" );
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document.write( "your period is equal to 2 * pi divided by pi/2.<BR>\n" );
document.write( "this makes your period equal to 5.\r<BR>\n" );
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document.write( "your horizontal shift is one unit to the right.<BR>\n" );
document.write( "this one is a litle tricky to figure out, so we'll go through a calculation to show you what happens.\r<BR>\n" );
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document.write( "your questions were:\r<BR>\n" );
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document.write( "a) What is the amplitude of f(t)? \r<BR>\n" );
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document.write( "the amplitude is equal to 4\r<BR>\n" );
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document.write( "(b) What is the period of f(t)?\r<BR>\n" );
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document.write( "the period is 4 radians\r<BR>\n" );
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document.write( "(c) What are the maximum and minimum values attained by f(t)? \r<BR>\n" );
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document.write( "the maximum value is f(T) = 3<BR>\n" );
document.write( "the minimum value is f(t) = -5\r<BR>\n" );
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document.write( "(d) Sketch the graph of f (t) for t &#8712; [&#8722;1, 3].\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/050602.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the period from x = -1 to x = 3 is marked between vertical dashed lines.\r<BR>\n" );
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document.write( "the center line of the graph is at y = -1\r<BR>\n" );
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document.write( "the maximum value is at y = 3 and the minimum value is at y = -5.\r<BR>\n" );
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document.write( "you can see that the graph crosses the center line at x = -1, x = 1, x = 3 and x = 5.\r<BR>\n" );
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document.write( "the center line is at y = -1.\r<BR>\n" );
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document.write( "take away the vertical displacement of -1 and the center line would have been at the x-axis.\r<BR>\n" );
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document.write( "you would analyze this graph as follows:\r<BR>\n" );
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document.write( "when x = 3, the equation of the graph becomes:\r<BR>\n" );
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document.write( "y = 4 * sin(pi/2 * (3-1)) - 1\r<BR>\n" );
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document.write( "this becomes y = 4 * sin(pi/2 * 2) - 1\r<BR>\n" );
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document.write( "this becomes y = 4 * sin(pi) - 1\r<BR>\n" );
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document.write( "sin(pi) is equal to 0.\r<BR>\n" );
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document.write( "the equation becomes y = 4 * 0 - 1\r<BR>\n" );
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document.write( "it finally becomes y = -1.\r<BR>\n" );
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document.write( "you can see on the graph that, when x = 3, y = -1.\r<BR>\n" );
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document.write( "this agrees with the equation.\r<BR>\n" );
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document.write( "we'll do one more.\r<BR>\n" );
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document.write( "when x = -1, the equation of y = 4 * sin(pi/2 * (x-1)) - 1 becomes:\r<BR>\n" );
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document.write( "y = 4 * sin(pi/2 * (-1-1)) - 1 which becomes:\r<BR>\n" );
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document.write( "y = 4 * sin(pi/2 * -2) - 1 which becomes:\r<BR>\n" );
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document.write( "y = 4 * sin(-pi) - 1 which becomes:\r<BR>\n" );
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document.write( "y = 4 * 0 - 1 which becomes:\r<BR>\n" );
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document.write( "y = -1.\r<BR>\n" );
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document.write( "when x = -1, the value of y is equal to -1.\r<BR>\n" );
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document.write( "the graph confirms that.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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