document.write( "Question 32014: compare the graph of y2 =4sin[3(x+3pi/4)]-3 with the graph of y1=sin x \n" ); document.write( "
Algebra.Com's Answer #18540 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
Compare the graph of \r\n" );
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document.write( "y2 = 4sin[3(x + 3p/4)] - 3 with the graph of y1 = sin x \r\n" );
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document.write( "--------------------\r\n" );
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document.write( "To transform the graph of\r\n" );
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document.write( "y1 = sin x\r\n" );
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document.write( "into the graph of \r\n" );
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document.write( "y2 = A·sin[B(x + C) + D]\r\n" );
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document.write( "requires this set of 5 transformations in this order\r\n" );
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document.write( "#1. A horizontal stretch by a factor of 2p/B if 2p/B > 1\r\n" );
document.write( "    A horizontal shrink by a factor of 2p/B if 2p/B < 0\r\n" );
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document.write( "    This involves dividing the x coordinate of any point by B. \r\n" );
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document.write( "#2. A vertical stretch by a factor of |A| if |A| > 1\r\n" );
document.write( "    A vertical shrink by a factor of |A| if |A| < 1\r\n" );
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document.write( "    This involves multiplying the y-coordinate of any point by A.   \r\n" );
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document.write( "#3. A reflection in the x-axis if A < 0, (none if A > 0)\r\n" );
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document.write( "    This involves multiplying the y-coordinate by -1 if A < 0, or doing\r\n" );
document.write( "    nothing if A > 0 (as in this case)\r\n" );
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document.write( "#4. A horizontal shift left |C| units if C > 0\r\n" );
document.write( "    A horizontal shift right |C| units if C < 0\r\n" );
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document.write( "    This involves:\r\n" );
document.write( "    subtracting |C| from the x-coordinate of any point if C positive\r\n" );
document.write( "    or\r\n" );
document.write( "    adding |C| to the x-coordinate of any point if C is negative.\r\n" );
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document.write( "#5. A vertical shift upward |D| units if D > 0\r\n" );
document.write( "    A vertical shift downward |D| units if D < 0\r\n" );
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document.write( "    This involves adding |D| to the y-coordinate of any point if D is positive\r\n" );
document.write( "    or\r\n" );
document.write( "    subtracting |D| from the y-coordinate if D is negative. \r\n" );
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document.write( "The 5 important points of the basic cycle of y1 = sin x are\r\n" );
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document.write( "1.    (0,0) , an x intercept, or \"node\"\r\n" );
document.write( "2.  (p/2,1) , a \"peak\", or \"maximum\"\r\n" );
document.write( "3.    (p,0) , an x-intercept, or \"node\"\r\n" );
document.write( "4. (3p/2,-1), a \"valley\", or \"minimum\"\r\n" );
document.write( "5.   (2p,0 ), an x-intercept, or \"node\" \r\n" );
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document.write( "In your problem:\r\n" );
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document.write( "y2 = 4sin[3(x + 3p/4)] - 3\r\n" );
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document.write( "A = 4, B = 3, C = 3p/4, D = -3 \r\n" );
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document.write( "Applying the 5 transformation to each the 5 important points of the\r\n" );
document.write( "basic cycle of y1 = sin x\r\n" );
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document.write( "                              transformation\r\n" );
document.write( "          #1          #2          #3          #4               #5\r\n" );
document.write( "   (0,0)  ->    (0,0) ->    (0,0) ->   (0,0)  ->    (-3p/4,0)  ->    (-3p/4,-3)\r\n" );
document.write( " (p/2,1)  ->  (p/6,1) ->  (p/6,4) -> (p/6,4)  -> (p/6-3p/4,4)  -> (p/6-3p/4,1)  \r\n" );
document.write( "   (p,0)  ->  (p/3,0) ->  (p/3,0) -> (p/3,0)  -> (p/3-3p/4,0)  -> (p/3-3p/4,-3)\r\n" );
document.write( "(3p/2,-1) ->  (p/2,-1)->  (p/2,-4)-> (p/2,-4) -> (p/2-3p/4,-4) -> (p/2-3p/4,-7) \r\n" );
document.write( "  (2p,0)  -> (2p/3,0) -> (2p/3,0) ->(2p/3,0)  ->(2p/3-3p/4,0)  -> (2p/3-3p/4,-3)\r\n" );
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document.write( "Simplifying the 5 important points on y2\r\n" );
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document.write( "1.  (-3p/4,-3) approximately (-2.4,-3)\r\n" );
document.write( "2.  (-7p/12,1)               (-1.8,1) \r\n" );
document.write( "3   (-5p/12,-3)              (-1.3,-3 \r\n" );
document.write( "4.  (-p/4, -7)               (-0.8,-7) \r\n" );
document.write( "5.  (-p/12, -3)              (-0.3,-3)\r\n" );
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document.write( "Plot these 5 important points of y1 and y2 and extend their graphs.\r\n" );
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document.write( "The red graph is y1 and the green graph is y2: \r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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