document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1135335>Question 1135335</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how to find the number of cycles in 2pi, the horizontal length of each of the 4 quadrants of each cycle, the horizontal shift of each cycle\r<BR>\n" );
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document.write( "f(x)=-tan(2x)-3<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #752976 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=752976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y = -tan(2x)-3\r<BR>\n" );
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document.write( "the normal period for the tan function is pi, or 180 degrees.\r<BR>\n" );
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document.write( "to convert between radians and degrees, here's the formula.\r<BR>\n" );
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document.write( "degrees = radians * 180 / pi.\r<BR>\n" );
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document.write( "raedians = degrees * pi / 180.\r<BR>\n" );
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document.write( "pi radians * 180 / pi is equal to 180 degrees.\r<BR>\n" );
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document.write( "180 degrees * pi / 180 is equal to pi radians.\r<BR>\n" );
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document.write( "pi/2 radians * 180 / pi is equal to 90 degrees.\r<BR>\n" );
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document.write( "90 degrees * pi / 180 is equal to pi/2 radians.\r<BR>\n" );
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document.write( "the relationship between frequency and period is:\r<BR>\n" );
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document.write( "frequency = pi / period.\r<BR>\n" );
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document.write( "period = pi / frequency\r<BR>\n" );
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document.write( "the general form of the equation is y = a * tan(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 displacement<BR>\n" );
document.write( "d is the vertical displacement.\r<BR>\n" );
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document.write( "here's the graph of the normal tan function of y = tan(x).\r<BR>\n" );
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document.write( "here's the graph of y = tan(2x).\r<BR>\n" );
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document.write( "the normal cycle for the tan function is pi because the tan function repeats every pi radians, unlike the sine and cosine functions, which repeater every 2pi radians.\r<BR>\n" );
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document.write( "here's a graph of y = tan(x).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/022301.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "as you can see, the pattern repeats every pi radians.\r<BR>\n" );
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document.write( "here's a graph of y = tan(2x).\r<BR>\n" );
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document.write( "as you can see, the pattern repeats every pi/2 radians.\r<BR>\n" );
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document.write( "here's a graph of y = -tan(2x).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/022303.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the pattern repeats every pi/2 radians and the pattern is reversed from top to bottom because the frequency is minus 1 rather than plus one.\r<BR>\n" );
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document.write( "this flips the graph vertically.\r<BR>\n" );
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document.write( "a full cycle of the tan function is between the vertical dashed lines in all the graphs.\r<BR>\n" );
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document.write( "here's a good reference that explains it all about as good as any.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/grphtrig.htm\" target = \"_blank\">https://www.purplemath.com/modules/grphtrig.htm</a>\r<BR>\n" );
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document.write( "here's the final graph.\r<BR>\n" );
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document.write( "it's the same as the last graph except there has been a vertical displacment down 3 units.\r<BR>\n" );
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document.write( "this means the center line of the graph is at y = -3 rather than at y = 0 as it normally is.\r<BR>\n" );
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document.write( "here's your final graph of the function y = -tan(2x) - 3.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/022304.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the general form is y = a * tan(b * (x - c)) + d.\r<BR>\n" );
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document.write( "the amplitude is reversed because a is equal to -1 rather than 1.\r<BR>\n" );
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document.write( "there is no horizontal shift because c is equal to 0.\r<BR>\n" );
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document.write( "the period is pi/2 because b is equal to 2 and period = pi / frequency for the tan function.\r<BR>\n" );
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document.write( "there is a vertical shift of the center line from y = 0 to y = -3 because d is equal to -3.\r<BR>\n" );
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