Vertical Translation

For the base function f (x) and a constant k, the function given by

g (x) = f (x) + k,

can be sketched by shifting f (x) k units vertically.

Horizontal Translation

For the base function f (x) and a constant k, the function given by

g(x) = f (x - k),

can be sketched by shifting f (x) k units horizontally.

Vertical Stretches and Shrinks

For the base function f (x) and a constant k > 0, the function given by

g(x) = k f (x),

can be sketched by vertically stretching f (x) by a factor of k
if k > 1

or

by vertically shrinking f (x) by a factor of k
if 0 < k < 1.

Horizontal Stretches and Shrinks

For the base function f (x) and a constant k, where k > 0 and k ≠ 1, the function given by

g(x) = f (kx),

can be sketched by horizontally shrinking f (x) by a factor of 1/k if k > 1

or

by horizontally stretching f (x) by a factor of 1/k
if 0 < k < 1.

Reflections across the x-axis

For the base function f (x) , the function given by

g(x) = - f (x),

can be sketched by reflecting f (x) across the x-axis.

Reflections across the y-axis

For the base function f (x) , the function given by

g(x) = f (-x),

can be sketched by reflecting f (x) across the y-axis.