Graph Transformations

Changes in f(x)

• Vertical shift

  • y = f(x) + c: Raise the graph of f(x) by c units, add c to the y coordinate.

  • y = f(x) - c: Lower the graph of f(x) by c units, subtract c from the y coordinate.

• Horizontal shift

  • y = f(x + c): Shift the graph f(x) to the left c units, subtract c from the x coordinate.

  • y = f(x - c): Shift the graph f(x) to the right c units, add c to the x coordinate.

• Reflection

  • About the x-axis: y = -f(x), multiply each y coordinate by -1.

  • About the y-axis: y = f(-x), multiply each x coordinate by -1.

• Vertical stretching and compression

  • y = cf(x), c > 1: Vertically stretching the graph of f(x).

  • y = cf(x), 0 < c < 1: Vertically compressing the graph of f(x).

  • Multiply each y coordinate by c.

• Horizontal stretching and compression

  • y = f(cx), c > 1: Horizontally compressing the graph of f(x).

  • y = f(cx), 0 < c < 1: Horizontally stretching the graph of f(x).

  • Divide each x coordinate by c.

• y = \frac{1}{f(x)} : Take the reciprocal of each y coordinate of f(x).

• Order of operations for transformations

  1. Horizontal shifts

  2. Stretches/compressions

  3. Reflections

  4. Vertical shifts