Let’s do a couple of examples using the knowledge from my last 3 posts and show how the order of transformations make a difference.

The image of the equation that follows will be generated by the following transformations:

a. Dilate along *y*-axis by factor 1/2

b. Reflect along *y*-axis (across *x*-axis)

c. Dilate along x-axis by factor 3

d. Translate along *x*-axis +4 units

e. Translate along *y*-axis -2 units

In the previous post, I showed that given an equation *y* = *f*(*x*), its image under a general transformation is given by

where*a* = dilation\reflection factor along *y*-axis (from *x*-axis)*n* = dilation\reflection factor along *x*-axis (from *y*-axis)*h* = translation along *x*-axis*k* = translation along *y*-axis

This assumes that dilations/reflections are done first. So if

\[f(x)=\sqrt{3x-4}\]then

\[f(x)\Rightarrow af\left[\frac{1}{n}(x-h)\right]+k=-\frac{1}{2}\sqrt{3\left [ \frac{1}{3}(x-4)\right]-4}-2\]This can be simplified to

\[y=-\frac{1}{2}\sqrt{ x-8}-2\]In this form, the pre-image (the equation that this one came from before the transformation) is lost. The question could be asked, what are the transformations required to go from

\[y=-\frac{1}{2}\sqrt{ x-8}-2\Rightarrow y=\sqrt{3x-4}\]One would expect that if we just do the opposite of the transformations above, we would get the original equation. Let’s see. Let’s do the following:

a. Dilate along *y*-axis by factor 2

b. Reflect along *y*-axis (across *x*-axis)

c. Dilate along x-axis by factor 1/3

d. Translate along *x*-axis -4 units

e. Translate along *y*-axis +2 units

These undo the previous transformations. Putting these in our model

\[f(x)\Rightarrow af\left[\frac{1}{n}(x-h)\right]+k\]We get

\[f(x)\Rightarrow -2f\left[3(x+4)\right]+2=\sqrt{3(x+4)-8}+4+2=\sqrt{3x+4}+6\]Not exactly what we wanted. What went wrong? Well, the model we used assumes that dilations go first. If we want to undo the previous transformations, not only do we use the values we just used, but they must be applied in reverse order as well: the translations go first then the dilations/reflections. Otherwise the dilations affect the translations before they are applied.

I’ll leave this as an exercise for the reader, but the model for transforming *y* = *f*(*x*) assuming that translations go first is

So undoing the original transformations in reverse order:

a. Translate along *y*-axis +2 units

b. Translate along *x*-axis -4 units

c. Dilate along x-axis by factor 1/3

d. Reflect along *y*-axis (across *x*-axis)

e. Dilate along *y*-axis by factor 2

gives the result

\[f(x)\Rightarrow -\frac{1}{2}f\left(3x+4\right)\Rightarrow-\frac{1}{2}\sqrt{3x+4-8}-2+2\Rightarrow\sqrt{3x-4}\]which is the original equation.

So the order of transformations steps will change the final result.

I hope this series of posts helps you better understand transformations.