document.write( "Question 671343: Can you create the graph of the function y=2(x-1)(x+4)(x-5) by transforming the function y=(x-4)(x+1)(x-8)?\r
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Algebra.Com's Answer #417393 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The function that we are to start with,\r\n" );
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document.write( "The \"beginning\" finction: \r\n" );
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document.write( "y=(x-4)(x+1)(x-8)\r\n" );
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document.write( "has zeros \r\n" );
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document.write( "4, -1, 8\r\n" );
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document.write( "We put them in order smallest to largest\r\n" );
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document.write( "-1, 4, 8\r\n" );
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document.write( "The function that we are to transform it into,\r\n" );
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document.write( "the \"final\" function, \r\n" );
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document.write( "y=2(x-1)(x+4)(x-5),  \r\n" );
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document.write( "has zeros 1, -4, and 5\r\n" );
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document.write( "We put them in order:\r\n" );
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document.write( "-4, 1, 5\r\n" );
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document.write( "We compare them:\r\n" );
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document.write( "beginning function's zeros:  -1, 4, 8\r\n" );
document.write( "final function's zeros:      -4, 1, 5\r\n" );
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document.write( "We observe that \r\n" );
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document.write( "moving 3 units left from -1 gives -4\r\n" );
document.write( "moving 3 units left from  4 gives  1\r\n" );
document.write( "moving 3 units left from  8  gives 5\r\n" );
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document.write( "So shifting the beginning graph left by 3 units\r\n" );
document.write( "takes care of moving the zeros of the beginning function\r\n" );
document.write( "to the zeros of the final function.\r\n" );
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document.write( "Also we notice that the final function has a 2 factor, \r\n" );
document.write( "which involves a stretch by a factor of 2.\r\n" );
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document.write( "So yes we can create the graph of\r\n" );
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document.write( "y=2(x-1)(x+4)(x-5) \r\n" );
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document.write( "by shifting the graph of\r\n" );
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document.write( "y=(x-4)(x+1)(x-8)\r\n" );
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document.write( "3 units left and stretching it by a\r\n" );
document.write( "factor of 2.\r\n" );
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document.write( "To see it done, we start with the graph of the beginning\r\n" );
document.write( "function \r\n" );
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document.write( "y=(x-4)(x+1)(x-8), which is the red graph below:\r\n" );
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document.write( "We shift it left 3 units by replacing each x by x+3\r\n" );
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document.write( "y=(x-4)(x+1)(x-8)\r\n" );
document.write( "y=(x+3-4)(x+3+1)(x+3-8)\r\n" );
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document.write( "Simplified,\r\n" );
document.write( "the intermediate function's equation is: \r\n" );
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document.write( "y=(x-1)(x+4)(x-5)\r\n" );
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document.write( "This intermediate function is graphed below (in green). It is the\r\n" );
document.write( "beginning graph (in red) shifted 3 units left.  \r\n" );
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document.write( "Now we only need to stretch the intermediate green graph vertically \r\n" );
document.write( "by a factor of 2 to have the graph of the final function:\r\n" );
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document.write( "To stretch the intermediate green graph vertically by a factor of 2,\r\n" );
document.write( "we multiply the right side by 2, and get this equation, which is\r\n" );
document.write( "equivalent to the final function:\r\n" );
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document.write( "y = 2(x-1)(x+4)(x-5) \r\n" );
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document.write( "Think of it as if the graph were drawn on a rubber sheet and we \r\n" );
document.write( "took hold of the top and bottom and stretched it double. The green\r\n" );
document.write( "graph would become the blue graph below:\r\n" );
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document.write( ".\r\n" );
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document.write( "Edwin
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