Horizontal and Vertical Shifts

If

is the original function where

then the graph of f(x)+c is shifted up

units,

and the graph of

is shifted down

units

A vertical shift means that every point

on the graph of the original function

is transformed to

on the graph of the transformed function

respectively.

The graph of

is shifted left

units

The graph of

is shifted right

units

A horizontal shift means that every point

on the graph of the original function

is transformed to

on the graph of the transformed function

respectively.

Reflections

If

is the original function then

The graph of

is a reflection in the x-axis.

The graph of

is a reflection in the y-axis.

Absolute value transformation

: Every part of the graph which is below x-axis is reflected in x-axis.

: For

the graph is exactly the same as this of the original function.

For

the graph is a reflection of the graph for x≥0 in y-axis.

Stretching and Shrinking

If

is the original function,

then

The graph of

is a vertical stretch by a scale factor of

If

is the original function,

then

The graph of

is a vertical shrink by a scale factor of

.

A vertical stretch or shrink means that every point

on the graph of the original function

is transformed to

on the graph of the transformed function

.

If

is the original function,

then

The graph of

is a horizontal shrink by a scale factor of

.

If

is the original function,

then

The graph of

is a horizontal stretch by a scale factor of

.

A horizontal stretch or shrink means that every point

on the graph of the original function

is transformed to

on the graph of the transformed function

.

Order of Tranformation

When we perform multiple transformations the order of these transformations may affect the final graph. Therefore we could follow the proposed order (with some exceptions) below to avoid possible wrong final graphs.

1. Horizontal Shifts

2. Stretch / Shrink

3. Reflections

4. Vertical Shifts