document.write( "Question 1165442: The graph of f(x) is shown below. https://latex.artofproblemsolving.com/d/b/a/dba24dce7c2734af95b18f61b73b2a32f213cd79.png
\n" ); document.write( "For each point (a,b) on the graph of y = f(x), the point ( 3a - 1, b/2) is plotted to form the graph of another function y = g(x). For example, (0,2) lies on the graph of y = f(x), so (3 * 0 - 1, 2/2) = (-1,1) lies on the graph of y = g(x).
\n" ); document.write( "(a) Plot the graph of y = g(x). Include the diagram in your solution.
\n" ); document.write( "(b) Express g(x) in terms of f(x).
\n" ); document.write( "(c) Describe the transformations that you would apply to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation might be to stretch the graph horizontally by a factor of 5.
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Algebra.Com's Answer #789964 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This is the graph of f(x):\r\n" );
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document.write( "We translate the endpoints of the line segments in f(x) to line segments\r\n" );
document.write( "in g(x), using the given point transformation (a,b) ---> (3a-1, b/2)\r\n" );
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document.write( "(-4,4) in f(x) corresponds to (-13,2) in g(x).\r\n" );
document.write( "(-1,0) in f(x) corresponds to  (-4,0) in g(x).\r\n" );
document.write( " (0,2) in f(x) corresponds to  (-1,1) in g(x)\r\n" );
document.write( "(4,-4) in f(x) corresponds to (11,-2) in g(x)\r\n" );
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document.write( "f(x) is in black, g(x) is in green.\r\n" );
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document.write( "We observe what happens when point (a,b) on f(x) becomes (3a-1,b/2) on g(x)\r\n" );
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document.write( "Each x-coordinate \"a\" becomes \"3a\", and then becomes \"3a-1\"\r\n" );
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document.write( "When \"a\" becomes \"3a\", that indicated a horizontal stretch by a factor of 3.\r\n" );
document.write( "That is accomplished by replacing x in f(x) by 1/3x,\r\n" );
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document.write( "So that gives us \r\n" );
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document.write( "\"f%28expr%281%2F3%29x%29\" \r\n" );
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document.write( "for \"a\" becoming \"3a\".\r\n" );
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document.write( "Then when \"3a\" becomes \"3a-1\", that indicates a horizontal shift by 1 unit\r\n" );
document.write( "left. That is accomplished by replacing x by x+1.\r\n" );
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document.write( "So that gives us\r\n" );
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document.write( "\"f%28expr%281%2F3%29%28x%2B1%29%29\" \r\n" );
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document.write( "for \"3a\" becoming \"3a-1\".\r\n" );
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document.write( "Now each y-coordinate \"b\" becomes \"b/2\".\r\n" );
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document.write( "When each y-coordinate \"b\" becomes \"b/2\", the graph is shrunk vertically\r\n" );
document.write( "by a factor of 1/2.  That is accomplished by multiplying the entire function\r\n" );
document.write( "by 1/2,\r\n" );
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document.write( "So that gives us\r\n" );
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document.write( "\"expr%281%2F2%29f%28expr%281%2F3%29%28x%2B1%29%29\"\r\n" );
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document.write( "for b becoming b/2.\r\n" );
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document.write( "So that now completes g(x) in terms of f(x), so we have:\r\n" );
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document.write( "\"%22g%28x%29%22+=+expr%281%2F2%29f%28expr%281%2F3%29%28x%2B1%29%29\"\r\n" );
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document.write( "You will notice that the green graph is the black graph stretched\r\n" );
document.write( "horizontally by a factor of 3, then shrunk vertically by a factor of 1/2 and\r\n" );
document.write( "shifted left by 1 unit.\r\n" );
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document.write( "Edwin

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