document.write( "Question 1204961: The graph of y = f(x) is compressed horizontally by a factor of 4/9, stretched vertically by a factor of 9/8, and reflected in the y-axis. Determine the equation of the image graph in terms of function f? \n" ); document.write( "

Algebra.Com's Answer #841526 by mccravyedwin(408) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "A vertical transformation is doing something to y, the vertical axis.\r\n" );
document.write( "A horizontal transformation is doing something to x, the horizontal axis.\r\n" );
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document.write( "When we do a vertical transformation, we do something to y, and y equals the\r\n" );
document.write( "whole right side of the equation. So we do something to the WHOLE FUNCTION,\r\n" );
document.write( "which is the WHOLE RIGHT SIDE of the equation for y.\r\n" );
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document.write( "Vertical transformations are done \"the way they seem\". \r\n" );
document.write( "Something is done to the WHOLE function, not to x only.\r\n" );
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document.write( "To shift a graph upward, we ADD to the WHOLE function, which is the WHOLE RIGHT\r\n" );
document.write( "SIDE.\r\n" );
document.write( "To shift a graph downward, we SUBTRACT from the WHOLE function, the WHOLE RIGHT\r\n" );
document.write( "SIDE.\r\n" );
document.write( "To stretch a graph vertically, we MULTIPLY the WHOLE function, the WHOLE RIGHT\r\n" );
document.write( "SIDE, by a number > 1.\r\n" );
document.write( "To compress a graph vertically, we MULTIPLY the WHOLE function, the WHOLE RIGHT\r\n" );
document.write( "SIDE, by a number < 1\r\n" );
document.write( "To reflect a graph in (or across) the x- axis, this is a vertical \r\n" );
document.write( "transformation\r\n" );
document.write( "(of flipping the graph vertically, we multiply the WHOLE function, the WHOLE\r\n" );
document.write( "RIGHT SIDE by -1.\r\n" );
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document.write( "Those seem logical.  But horizontal transformations are \"reversed from the way\r\n" );
document.write( "they seem.\"  That's because we are compensating.\r\n" );
document.write( "Something is done to x only.  That is, we replace x by something.   \r\n" );
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document.write( "To shift a graph to the RIGHT, we SUBTRACT from x only. \r\n" );
document.write( "To shift a graph LEFT, we ADD to x only.\r\n" );
document.write( "[Those two seem backward, don't they? But they're not!]\r\n" );
document.write( "To stretch a graph horizontally, we MULTIPLY x only by a number < 1. \r\n" );
document.write( "To compress a graph horizontally, we MULTIPLY x only by a number > 1.\r\n" );
document.write( "[Those two also seem backward, don't they? But they're not!]\r\n" );
document.write( "To reflect a graph in (or across) the y- axis, this is a vertical \r\n" );
document.write( "transformation (of flipping the graph horizontally, we multiply x only by -1.\r\n" );
document.write( "[This is the only one that doesn't \"seem\" reversed].

\n" ); document.write( "The graph of y = f(x) is compressed horizontally by a factor of 4/9.

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document.write( "This is a horizontal compression, so it's \"the reverse of what it seems\".\r\n" );
document.write( "So we multiply x only by a number greater than 1.  Hey, but 4/9 is less than 1.\r\n" );
document.write( "Its reciprocal of a number less than 1 is a number greater than 1, so we\r\n" );
document.write( "multiply x only by the reciprocal of 4/9, which is 9/4.  So we replace x only by\r\n" );
document.write( "9/4x\r\n" );
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document.write( "

\n" ); document.write( "stretched vertically by a factor of 9/8,

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document.write( "This is a vertical stretch, so it is \"as it seems\", we multiply the entire\r\n" );
document.write( "function by a number greater than one.  9/8 is greater than 1, so we multiply\r\n" );
document.write( "the entire function (the whole right side) by 9/8:\r\n" );
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document.write( "

\n" ); document.write( "and reflected in the y-axis.

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document.write( "To flip the graph about the y-axis is a horizontal change (swapping the right\r\n" );
document.write( "and left quadrants.) So we multiply x only by -1.\r\n" );
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document.write( "     \r\n" );
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document.write( "     <--answer\r\n" );
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document.write( "[If you would like more instruction to understand the reasoning behind why it's\r\n" );
document.write( "the opposite of \"what it seems\" for horizontal transformations, just ask me in\r\n" );
document.write( "the thank-you note form below, and I'll get back to you by email.  But most\r\n" );
document.write( "likely, you'll just take my word for it.] J\r\n" );
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document.write( "Edwin

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