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Transformations of Functions

If you start with a simple parent function y = f ( x ) and its graph, certain modifications of the function will result in easily predictable changes to the graph.

For example:

Horizontal Shift

  • Replacing f ( x ) with f ( x b ) results in the graph being shifted b units to the right.

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  • Replacing f ( x ) with f ( x + b ) results in the graph being shifted b units to the left.

Vertical Shift

  • Replacing f ( x ) with f ( x ) + c results in the graph being shifted c units up.

  • Replacing f ( x ) with f ( x ) c results in the graph being shifted c units down.

Reflection

  • Replacing x with x results in the graph being reflected across the y -axis.

  • Replacing f ( x ) with f ( x ) results in the graph being reflected across the x -axis.

Horizontal Stretch/Compression

  • Replacing x with n x results in a horizontal compression by a factor of n .

  • Replacing x with x n results in a horizontal stretch by a factor of n .

Vertical Stretch/Compression

  • Replacing f ( x ) with n f ( x ) results in a vertical stretch by a factor of n .

  • Replacing f ( x ) with f ( x ) n results in a vertical compression by a factor of n .