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

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