document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161305>Question 1161305</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the equation of a sine function that has a range of {y ∈ R|−7 ≤ y ≤ 2}, the first minimum value at −30° and the first maximum value at 60°. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #784809 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=784809\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Asin%28b%28x-c%29%29%2Bd ALIGN=MIDDLE  ALT=\"y+=+a%2Asin%28b%28x-c%29%29%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "a is the amplitude<BR>\n" );
document.write( "b determines, or is determined by, the period<BR>\n" );
document.write( "c is the horizontal (phase) shift<BR>\n" );
document.write( "d is the vertical shift (determines the center line of the oscillation)<br><BR>\n" );
document.write( "The range is from -7 to +2, a difference of 9.  The amplitude a is half of that, 4.5.<br><BR>\n" );
document.write( "The center line is halfway between the minimum and maximum; d = -2.5.<br><BR>\n" );
document.write( "The function has a minimum at -30 degrees and a next maximum at 60 degrees; that difference of 90 degrees is half the period, so the period is 180 degrees.  That is half the period of the basic sine function, so b = 360/180 = 2.<br><BR>\n" );
document.write( "At this point our function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+4.5%2Asin%282%28x-c%29%29-2.5 ALIGN=MIDDLE  ALT=\"y+=+4.5%2Asin%282%28x-c%29%29-2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "When the angle is 0 degrees, the value of sine is 0 and increasing.  In this example, the angle is 0 when x=c.<br><BR>\n" );
document.write( "With a minimum at -30 degrees and a maximum at 60 degrees, this function (before the vertical shift) is 0 and increasing halfway between -30 degrees and +60 degrees -- at +15 degrees.<br><BR>\n" );
document.write( "So c = 15 (degrees), and the function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+4.5%2Asin%282%28x-15%29%29-2.5 ALIGN=MIDDLE  ALT=\"y+=+4.5%2Asin%282%28x-15%29%29-2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph, showing the sine function and the constants representing the minimum and maximum values and the center line.<br><BR>\n" );
document.write( "Note the minimum at -30 degrees, the maximum at +60 degrees, and the function value at the center line and increasing at +15 degrees.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28900%2C400%2C-90%2C90%2C-10%2C6%2C4.5%2Asin%282%28pi%2F180%29%28x-15%29%29-2.5%2C-7%2C-2.5%2C2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );