document.write( "Question 1199730: (1,2) is a point on the graph of y=f(x). Find the corresponding point on the graph of each of the following functions.\r
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Algebra.Com's Answer #833793 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "When we perform an operation on the entire right side, it causes a VERTICAL\r\n" );
document.write( "transformation on each y-coordinate.\r\n" );
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document.write( "When we perform an operation on x, it causes a HORIZONTAL transformation on each\r\n" );
document.write( "x-coordinate.\r\n" );
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document.write( "When the entire right side is multiplied by -1, each y-coordinate is reflected\r\n" );
document.write( "VERTICALLY across the x-axis.\r\n" );
document.write( "When only x is multiplied by -1, each x-coordinate is reflected HORIZONTALLY\r\n" );
document.write( "across the y-axis. \r\n" );
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document.write( "When the operation is on the entire right side, the shift, stretch, or shrink is\r\n" );
document.write( "as you would expect. That is:\r\n" );
document.write( "Adding to the entire right side shifts VERTICALLY UPWARD. \r\n" );
document.write( "Subtracting from the entire right side shifts VERTICALLY DOWNWARD.\r\n" );
document.write( "Multiplying the entire right side by a positive number greater than 1 causes a VERTICAL\r\n" );
document.write( "STRETCH. \r\n" );
document.write( "Multiplying the entire right side by a positive number less than 1 causes a VERTICAL\r\n" );
document.write( "SHRINK.\r\n" );
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document.write( "However, when the operation is on x only, the shift, stretch, or shrink is\r\n" );
document.write( "OPPOSITE from what you would expect. \r\n" );
document.write( "Adding to x shifts LEFT. \r\n" );
document.write( "Subtracting from x shifts RIGHT.\r\n" );
document.write( "Multiplying x by a positive number greater than 1 causes a HORIZONTAL\r\n" );
document.write( "SHRINK. \r\n" );
document.write( "Multiplying x by a positive number less than 1 causes a HORIZONTAL\r\n" );
document.write( "STRETCH. \r\n" );
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document.write( "(a) y=2+f(-x)\r\n" );
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document.write( "The transforming steps are: f(x), f(-x), 2+f(-x)\r\n" );
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document.write( "We start with \r\n" );
document.write( "y=f(x)\r\n" );
document.write( "Then first we replace x by -x\r\n" );
document.write( "y=f(-x)\r\n" );
document.write( "Replacing x by -x reflects the x-coordinate of each point HORIZONTALLY across\r\n" );
document.write( "the y-axis.\r\n" );
document.write( "So (1,2) is reflected to (-1,2)\r\n" );
document.write( "Then we add 2 to the entire right side.\r\n" );
document.write( "y=2+f(-x)\r\n" );
document.write( "Adding 2 to the entire right side shifts the y-coordinate of each point 2 units\r\n" );
document.write( "VERTICALLY UPWARD.\r\n" );
document.write( "So (-1,2) is shifted VERTICALLY UPWARD to (-1,4)\r\n" );
document.write( "Answer: (-1,4)\r\n" );
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document.write( "(b) y= 3f(x-5)-1\r\n" );
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document.write( "The transforming steps are f(x), 3f(x), 3f(x-5), 3f(x-5)-1\r\n" );
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document.write( "We start with \r\n" );
document.write( "y=f(x)\r\n" );
document.write( "Then we multiply the entire right side by 3\r\n" );
document.write( "y=3f(x)\r\n" );
document.write( "Multiplying the entire right side by 3 stretches VERTICALLY the y-coordinate of\r\n" );
document.write( "each point to 3 times its distance from the x-axis.\r\n" );
document.write( "So (1,2) is shifted UPWARD to (1,6)\r\n" );
document.write( "Then we subtract 5 from x\r\n" );
document.write( "y=3f(x-5)\r\n" );
document.write( "Subtracting 5 from x shifts each point 5 units HORIZONTALLY RIGHT.\r\n" );
document.write( "So (1,6) is shifted HORIZONTALLY RIGHT to (6,6)\r\n" );
document.write( "Then we subtract 1 from the entire right side.\r\n" );
document.write( "y=3f(x-5)-1\r\n" );
document.write( "Subtracting 1 from the right side shifts the y-coordinate of each point DOWNWARD 1\r\n" );
document.write( "unit.\r\n" );
document.write( "So (6,6) is shifted DOWNWARD to (6,5)\r\n" );
document.write( "Answer (6,5)\r\n" );
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document.write( "(c) y=f(3x-5)-1\r\n" );
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document.write( "The transforming steps are f(x), f(3x), f(3x-5), which must be written as\r\n" );
document.write( "\"y=f%283%28x-5%2F3%29%5E%22%22%29\", then \"y=f%283%28x-5%2F3%29%5E%22%22%29-1\", or f(3x-5)-1 \r\n" );
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document.write( "We begin with \r\n" );
document.write( "y=f(x)\r\n" );
document.write( "Then we replace x by 3x\r\n" );
document.write( "y=f(3x)\r\n" );
document.write( "Multiplying x by 3 shrinks the x-coordinate of each point HORIZONTALLY to 1/3 of\r\n" );
document.write( "its distance from the y-axis.\r\n" );
document.write( "So the point (1,2) gets its x-coordinate shrunk to (1/3,2)\r\n" );
document.write( "Now we must calculate how much shifting we must do to x to get f(3x-5)\r\n" );
document.write( "So we factor out 3 and rewrite y=f(3x-5) as\r\n" );
document.write( "\"y=f%283%28x-5%2F3%29%5E%22%22%29\"\r\n" );
document.write( "So now we can see that x has been replaced by x-5/3.\r\n" );
document.write( "Subtracting 5/3 from x shifts the x-coordinate of each point 5/3 units\r\n" );
document.write( "HORIZONTALLY RIGHT.\r\n" );
document.write( "So (1/3,2) is shifted HORIZONTALLY RIGHT to (1/3+5/3,6) or (6/3,6) or (2,6)\r\n" );
document.write( "Finally we must subtract 1 from the entire right side which shifts each y-coordinate VERTICALLY DOWNWARD.\r\n" );
document.write( "So (2,6) is shifted VERTICALLY DOWNWARD to (2,5)\r\n" );
document.write( "Answer: (2,5)\r\n" );
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document.write( "Edwin
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