document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190402>Question 1190402</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch, on separate diagrams, the following graphs for 0° ≤ x ≤ 360°. For each function, write down<BR>\n" );
document.write( "the amplitude, the period, the coordinates of the maximum and minimum points and the corresponding<BR>\n" );
document.write( "range of y.<BR>\n" );
document.write( "a)y = 4sin3x + 2<BR>\n" );
document.write( "b)y =3|cos2x| </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849192 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849192\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Unfortunately, I can't directly sketch graphs here. However, I can provide you with all the information you need to sketch them yourself, including the key features and transformations.\r<BR>\n" );
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document.write( "**a) y = 4sin3x + 2**\r<BR>\n" );
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document.write( "* **Amplitude:** 4 (the coefficient of the sine function)<BR>\n" );
document.write( "* **Period:** (360°) / 3 = 120° (the period of the basic sine function divided by the coefficient of x)<BR>\n" );
document.write( "* **Vertical Shift:** 2 units upward (the constant term)<BR>\n" );
document.write( "* **Maximum Points:**  The maximum value of the sine function is 1. So, the maximum value of y is 4(1) + 2 = 6. This occurs when sin3x = 1, which happens at x = 30°, 150°, 270° within the given range.  The maximum points are (30°, 6), (150°, 6), and (270°, 6).<BR>\n" );
document.write( "* **Minimum Points:** The minimum value of the sine function is -1. So, the minimum value of y is 4(-1) + 2 = -2. This occurs when sin3x = -1, which happens at x = 90°, 210°, 330° within the given range. The minimum points are (90°, -2), (210°, -2), and (330°, -2).<BR>\n" );
document.write( "* **Range:** -2 ≤ y ≤ 6\r<BR>\n" );
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document.write( "**To sketch the graph:**\r<BR>\n" );
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document.write( "1. Start with the basic sine graph.<BR>\n" );
document.write( "2. Compress it horizontally by a factor of 3 (so that it completes one cycle within 120°).<BR>\n" );
document.write( "3. Stretch it vertically by a factor of 4.<BR>\n" );
document.write( "4. Shift the entire graph 2 units upward.\r<BR>\n" );
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document.write( "**b) y = 3|cos2x|**\r<BR>\n" );
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document.write( "* **Amplitude:** 3 (the coefficient in front of the absolute value)<BR>\n" );
document.write( "* **Period:** (360°) / 2 = 180° (the period of the basic cosine function divided by the coefficient of x)<BR>\n" );
document.write( "* **Absolute Value:** The absolute value ensures that the function is always non-negative.<BR>\n" );
document.write( "* **Maximum Points:** The maximum value of |cos2x| is 1. So, the maximum value of y is 3(1) = 3. This occurs when cos2x = 1 or -1, which happens at x = 0°, 90°, 180°, 270°, 360° within the given range. The maximum points are (0°, 3), (90°, 3), (180°, 3), (270°, 3), and (360°, 3).<BR>\n" );
document.write( "* **Minimum Points:** The minimum value of |cos2x| is 0. So, the minimum value of y is 3(0) = 0. This occurs when cos2x = 0, which happens at x = 45°, 135°, 225°, 315° within the given range. The minimum points are (45°, 0), (135°, 0), (225°, 0), and (315°, 0).<BR>\n" );
document.write( "* **Range:** 0 ≤ y ≤ 3\r<BR>\n" );
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document.write( "**To sketch the graph:**\r<BR>\n" );
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document.write( "1. Start with the basic cosine graph.<BR>\n" );
document.write( "2. Compress it horizontally by a factor of 2 (so that it completes one cycle within 180°).<BR>\n" );
document.write( "3. Reflect any parts of the graph that are below the x-axis to above the x-axis (because of the absolute value).<BR>\n" );
document.write( "4. Stretch the graph vertically by a factor of 3. </SPAN>\n" );
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