document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122862>Question 1122862</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.  Write a cosine function, f(x), such that on one period a local maximum is at (- pi/3, 1) followed by a local maximum at (2pi/3, -7).  F(x)= \r<BR>\n" );
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document.write( "2.  Write a tangent function, h(x), such that the halfway points on one period are (- pi/2, -2) and (pi/4, 4).  H(x)=\r<BR>\n" );
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document.write( "3.  Rewrite g(x)= 5cos (2x - pi/4) -3 as, k(x), a function of sine. I got:  k(x)= 5sin (3 pi/4). Is it right? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #739052 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=739052\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "1. Clearly there can't be a local maximum at (-pi/3,1) and another at (2pi/3,-7).  The maximum values have to be the same.<br><BR>\n" );
document.write( "So I will assume that (2pi/3,-7) is the local minimum that follows the local maximum at (-pi/3,1).<br><BR>\n" );
document.write( "We want a function in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Asin%28b%28x-c%29%29%2Bd ALIGN=MIDDLE  ALT=\"a%2Asin%28b%28x-c%29%29%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where a is the vertical stretch, b defines the period, c is the phase shift, and d is the vertical shift.<br><BR>\n" );
document.write( "From the local maximum to the next local minimum is half a period; in this example that is from -pi/3 to 2pi/3, a difference of pi.  That means the period of the function is 2pi; so there is no horizontal compression of the graph; b = 1.<br><BR>\n" );
document.write( "The difference between the maximum and minimum values is 8, so the vertical stretch is 4: a = 4.<br><BR>\n" );
document.write( "The center line is halfway between the maximum 1 and the minimum -7, at -3; d = -3.<br><BR>\n" );
document.write( "The basic cosine function has its maximum at x = 0; since this function has its maximum at x = -pi/3, the phase shift is -pi/3: c = -pi/3.<br><BR>\n" );
document.write( "We have all the pieces; the function with a=4, b=1, c=-pi/3, and d=-3 is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Acos%28x%2Bpi%2F3%29-3 ALIGN=MIDDLE  ALT=\"4%2Acos%28x%2Bpi%2F3%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph, showing a local maximum at (-pi/3,1) and the following local minimum at (2pi/3,-7)...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C200%2C-pi%2C2pi%2C-8%2C2%2C4cos%28x%2Bpi%2F3%29-3%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C200%2C-pi%2C2pi%2C-8%2C2%2C4cos%28x%2Bpi%2F3%29-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "2. We need a function in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=atan%28b%28x-c%29%29%2Bd%29 ALIGN=MIDDLE  ALT=\"atan%28b%28x-c%29%29%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where a is the vertical stretch, b defines the period, c is the phase shift, and d is the vertical shift.<br><BR>\n" );
document.write( "The halfway points are (-pi/2,-2) and (pi/4,4).<br><BR>\n" );
document.write( "The difference in x values between the halfway points is 3pi/4, so the period is 3pi/2.  Since the period of the basic tangent function is pi, the graph is stretched horizontally by a factor of 3/2.  That means b = (pi)/(3(pi)/2)) = 2/3.<br><BR>\n" );
document.write( "The difference between the y values of the midpoints is 6.  Since the difference between the y values at the midpoints of the basic tangent function is 2, the vertical stretch is 3: a = 3.<br><BR>\n" );
document.write( "The center line is halfway between the y values at the midpoints, which is 1: d = 1.<br><BR>\n" );
document.write( "The center of the period is halfway between the two midpoints, at x = -pi/8.  Since the basic tangent function has the center of its period at x=0, the phase shift is -pi/8: b = -pi/8.<br><BR>\n" );
document.write( "Again we have all the pieces.  The tangent function with a=3, b=2/3, c=-pi/8, and d=1 is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3tan%28%282%2F3%29%28x%2Bpi%2F8%29%29%2B1 ALIGN=MIDDLE  ALT=\"3tan%28%282%2F3%29%28x%2Bpi%2F8%29%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph, showing (-pi/2,-2) and (pi/4,4) as the halfway points of a period....<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-pi%2Cpi%2C-8%2C8%2C3tan%28%282%2F3%29%28x%2Bpi%2F8%29%29%2B1%2C-2%2C4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "3. Your answer can't be right; the \"2x\" still has to be part of the argument for the sine function.<br><BR>\n" );
document.write( "The given function is<br><BR>\n" );
document.write( "5cos(2x-pi/4)-3<br><BR>\n" );
document.write( "To find an equivalent sine function, you only need to know that cos(x) = sin(x+pi/2).  Note that is easy to see if you see that the maximum value of the basic cosine function is at x = 0 while the maximum value of the basic sine function is at x = pi/2.<br><BR>\n" );
document.write( "So all you need to do for this question is add pi/2 to the angle in the given formula:<br><BR>\n" );
document.write( "5sin(2x+pi/4)-3<br><BR>\n" );
document.write( "A graph of the two functions, with the sine function (green) shifted up 1 unit so you can see the two graphs...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C0%2C2pi%2C-12%2C6%2C5cos%282x-pi%2F4%29-3%2C5sin%282x%2Bpi%2F4%29-2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C0%2C2pi%2C-12%2C6%2C5cos%282x-pi%2F4%29-3%2C5sin%282x%2Bpi%2F4%29-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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