### Lesson 2: Combination of linear transforms

(A)  Introduction

A combination of linear transformations is one whereby we apply several of these transformations in a specific order. If we switch the order of transformations, we may get a different equation / graph.

Example 1

Let

The graph of y = f (x) under go the following two transformations:

(I):   Translate 3 units in the negative x axis direction.

(II):   Scale parallel to the x axis with scale factor 0.5.

(a)

Write down the equation of the new graph if it undergoes transformation (I) followed by (II).

(b)

Write down the equation of the new graph if it undergoes transformation (II) followed by (I).

Solution

(a)

(b)

(B)  "AMMA" Method

If we need to perform a series of linear transformations to achieve a certain equation, it is recommended to follow the "AMMA" order:

= Addition to x : Translation along x axis.

= Multiply to x : Scaling parallel to x axis or reflection about y axis.

= Multiply to y : Scaling parallel to y axis or reflection about x axis.

= Addition to y : Translation along y axis.

Example 2

Describe a series of transformations that change

.

Solution

(I)

Replace x with x + 1.

Translate 1 unit in the negative x axis direction.

(II)

Replace x with 2x.

Scale parallel to x axis by scale factor 0.5.

(III)

Replace y with y/3.

Scale parallel to y axis by scale factor 3.

(IV)

Replace y with y + 4.

Translate 4 units in the negative y axis direction.

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