### Lesson 1: Linear Transform

Introduction

Linear transformation consists of 3 types: translation, scaling and reflection.

(A) Translation

Translation involves shifting a graph along the x or y axes direction. Translation occurs when we add or subtract values to the x or y.

(I)

Replace x with x + a

Translate a units in the negative x axis direction.

(a > 0)

Translate a units in the positive x axis direction.

(a < 0)

(II)

Replace y with y + a

Translate a units in the negative y axis direction.

Translate a units in the positive y axis direction.

(a > 0)

(a < 0)

(B) Scaling

Scaling involves stretching or compressing a graph either vertically or horizontally. Scaling occurs when we multiply x or y with a positive constant.

(I)

Replace x with ax

Scale parallel to x axis by scale factor 1/a. I.e. multiply all the x coordinates by 1/a.

(II)

Replace y with ay

Scale parallel to y axis by scale factor 1/a. I.e. multiply all the y coordinates by 1/a.

(C) Reflection

Reflection about an axis creates a mirror image of the original graph with the axis as the "mirror". Reflection occurs when we multiply '-1' to x or y.

(I)

Replace x with -x

Reflect about the y axis. I.e. horizontal reflection with y axis as the "mirror".

(II)

Replace y with -y

Reflect about the x axis. I.e. vertical reflection with x axis as the "mirror".

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