**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* (*k**x*),

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.