document.write( "Question 1038320: Find a trig function given the max of (8am,20) and the min of (2am,10) \n" ); document.write( "
Algebra.Com's Answer #653006 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the function be\r\n" );
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document.write( "\"y+=+A%2Asin%28bx%2Bc%29%2Bd\" \r\n" );
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document.write( "[We use capital A here to avoid conflicting notation with\r\n" );
document.write( "small \"a\" used here for some value that has nothing to do\r\n" );
document.write( "with amplitude, which is usually |a| in other problems but \r\n" );
document.write( "here the amplitude will be |A|.  I have no idea why the \r\n" );
document.write( "author of your problem would have put in the \"am\" anyway,\r\n" );
document.write( "since it just clutters the problem.  He should have given \r\n" );
document.write( "points (8,20) and (2,10).  But I'll clutter the problem with\r\n" );
document.write( "the nonsensical \"am\", since he did.]\r\n" );
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document.write( "The function has max at point where \"bx%2Bc+=+pi%2F2\" \r\n" );
document.write( "                                    \"bx+=pi%2F2-c\"\r\n" );
document.write( "                                    \"x+=+%28pi%2F2-c%29%2Fb\"\r\n" );
document.write( "                                    \"x+=+%28pi-2c%29%2F%282b%29\"\r\n" );
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document.write( "Since we are given the max at (8am,20), that's where x = 8am,\r\n" );
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document.write( "So    \"8am+=+%28pi-2c%29%2F%282b%29\"\r\n" );
document.write( "      \"16abm+=+pi-2c\"\r\n" );
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document.write( "and where y = 20, that's where\r\n" );
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document.write( "\"y+=+A%2Asin%28bx%2Bc%29%2Bd\"\r\n" );
document.write( "\"20+=+A%2Asin%28pi%2F2%29%2Bd\"\r\n" );
document.write( "\"20+=+A%2A1%2Bd\"\r\n" );
document.write( "\"A%2Bd=20\"\r\n" );
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document.write( "========================\r\n" );
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document.write( "\"y+=+A%2Asin%28bx%2Bc%29%2Bd\" [Use capital A to avoid conflicting notation]\r\n" );
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document.write( "has min at point where \"bx%2Bc+=+3pi%2F2\" \r\n" );
document.write( "                       \"bx+=3pi%2F2-c\"\r\n" );
document.write( "                       \"x+=+%283pi%2F2-c%29%2Fb\"\r\n" );
document.write( "                       \"x+=+%283pi-2c%29%2F%282b%29\"\r\n" );
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document.write( "Since given the min at (2am,10), that's where x = 2am,\r\n" );
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document.write( "So    \"2am+=+%283pi-2c%29%2F%282b%29\"\r\n" );
document.write( "      \"4abm+=+3pi-2c\"\r\n" );
document.write( "     \r\n" );
document.write( "and where y = 10, that's where\r\n" );
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document.write( "\"y+=+A%2Asin%28bx%2Bc%29%2Bd\"\r\n" );
document.write( "\"10+=+A%2Asin%283pi%2F2%29%2Bd\"\r\n" );
document.write( "\"10+=+A%2A%28-1%29%2Bd\"\r\n" );
document.write( "\"A-d=10\"\r\n" );
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document.write( "So we have the two systems of equations:\r\n" );
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document.write( "\"system%2816abm+=+pi-2c%2C4abm+=+3pi-2c%29\" \r\n" );
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document.write( "and\r\n" );
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document.write( "\"system%28A%2Bd=20%2CA-d=10%29\"\r\n" );
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document.write( "From the first system:\r\n" );
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document.write( "\"system%2816abm+=+pi-2c%2C4abm+=+3pi-2c%29\"\r\n" );
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document.write( "Multiply the second by -4\r\n" );
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document.write( "\"system%2816abm+=+pi-2c%2C-16abm+=+-12pi%2B8c%29\"\r\n" );
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document.write( "Adding the equations gives;\r\n" );
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document.write( "\"0=4pi%2B6c\"\r\n" );
document.write( "\"-6c=4pi\"\r\n" );
document.write( "\"c=-4pi%2F6\"\r\n" );
document.write( "\"c=-2pi%2F3\"\r\n" );
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document.write( "Substituting in 4abm = 3pi-2c\r\n" );
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document.write( "\"4abm+=+3pi-2c\"\r\n" );
document.write( "\"4abm+=+3pi-2%282pi%2F3%29\"\r\n" );
document.write( "\"4abm+=+3pi-4pi%2F3\"\r\n" );
document.write( "\"12abm+=+9pi-4pi\"\r\n" );
document.write( "\"12abm=5pi\"\r\n" );
document.write( "\"b=5pi%2F%2812am%29\"\r\n" );
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document.write( "From the second system:\r\n" );
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document.write( "\"system%28A%2Bd=20%2CA-d=10%29\"\r\n" );
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document.write( "Adding them gives\r\n" );
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document.write( "\"2A=30\"\r\n" );
document.write( "\"A=15\"\r\n" );
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document.write( "Subtracting them gives\"\r\n" );
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document.write( "\"2d=10\"\r\n" );
document.write( "\"d=5\"\r\n" );
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document.write( "So the equation\r\n" );
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document.write( "\"y+=+A%2Asin%28bx%2Bc%29%2Bd\"\r\n" );
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document.write( "becomes\r\n" );
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document.write( "\"y+=+15%2A%28expr%28sin%285pi%2F%2812am%29%29x%2Bexpr%282pi%2F3%29%5E%22%22%29%29%2B5\"\r\n" );
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document.write( "Edwin

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