# 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 nx 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 nf(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.