Reverse graph transformation

This question was posted by a student.

” What is the series of transformation from f(3-x/2) to f(x)?”

Here are 3 different methods to solve it.

Method 1

Let f(3-x/2) = g(x)

Let u=3-x/2

x= -2u+6

f(u)=g(-2u+6)

f(x)=g(-2x+6)

Therefore, the sequence of transformation is

Translate 6 units in the negative X direction.

Scale by a factor of 1/2 parallel to the X-axis.

Reflect about the Y-axis

Method 2a

Apply f(-2x) to f(3-0.5x).

f(-2x) = f[3-0.5(-2x)] = f(x+3)

ie. Scale parallel to the X axis by a factor of 0.5. Then reflect about the Y axis

Apply f(x-3) to f(x+3) to get f(x)

ie. Translate 3 units in the positive X direction

Method 2b

Apply f(x+6) to f(3-0.5x) to get f(-0.5x)

ie. Translate 6 units in the negative X direction.

Apply f(-2x) to f(-0.5x) = f(x)

ie. Scale parallel to the X axis by a factor of 0.5. Then reflect about the Y axis.

Method 3

f(3-x/2) is transforming f(x) by

Ａ：Translate 3 units in the negative X direction

B: Scale by a factor of 2 parallel to the X axis

C: Reflect about Y axis.

So to get back f(x), we reverse the transformation:

C’: Reflect about Y axis

B’: Scale by a factor of 1/2 parallel to the X-axis.

A’: Translate 3 units in the positive X direction

Comments

Method 1 and 2 are the forward approach. Method 3 is the reverse approach.

Method 1 is the preferred approach as it is easier and faster.