# Transformations 2

In my last post, I took specific points and transformed them using dilations/reflections and translations. The goal is to transform an equation into a new equation. But before we get there, let’s again look at a single point. But this time, this will be a general point (x, y).

So starting with general point A(x, y), let’s do the same transformation as I did in my last post:

A. (x,y)
B. Dilate by factor 2 along y-axis
C. Dilate by factor 1/3 along x-axis
D. Reflect across x-axis
E. Reflect across y-axis
F. Translate 3 units up
G. Translate 3 units to the left

In this order, the new transformed points become:

A(x, y) —-> B(x, 2y) —-> C(x/3, 2y) —-> D(x/3, -2y) —->
E(-x/3, -2y) —-> F(-x/3, -2y+3) —-> G(-x/3-3, -2y+3)

From my last post, if you use the particular point A(3,1), and use these values in the result above, you get the same point G(-4,1).

This can be generalised more by using letters to represent the dilation factors and the translation amounts:

a = dilation/reflection factor along y axis
n = dilation/reflection factor along x axis
h = translation along x axis
k = translation along y axis

Note that I have combined dilations and reflections. This is because a reflection can be viewed as a negative dilation. If a is negative, then this is a reflection along the y axis (across the x axis) as well as a dilation. If n is negative, then this is a reflection along the x axis (across the y axis) as well as a dilation. This does slightly restrict the flexibility of transformations as we cannot separate the order of a dilation and a reflection, but for most problems, this is not an issue. Also, if h is negative, that is a translation to the left. If k is negative, that is a translation down.

Now I’ll apply this to the same set of transformations as above except that I will combine the dilation and reflection steps:

A. (x,y)
B. Dilate/reflect by factor a along y-axis
C. Dilate/reflect by factor n along x-axis
D. Translate k units along y-axis
E. Translate h units along x-axis

This leads to the new point:

A(x, y) —-> B(x, ay) —-> C(nx, ay) —-> D(nx, ay+k) —->
E(nx+h, ay+k)

If these transformations are reversed:

A. (x,y)
B. Translate h units along x-axis
C. Translate k units along y-axis
D. Dilate/reflect by factor n along x-axis
E. Dilate/reflect by factor a along y-axis

you get the new point:

A(x, y) —-> B(x+h, y) —-> C(x+h, y+k) —-> D[n(x+h), y+k] —->
E[n(x+h), a(y+k)]

This is different and you will generally get a different ending point for a specific point and transformation values.

Some terminology is needed here. We have an original point (x,y) and then under a set of transformations, a new point is created. In our first set of transformations, this new point was (nx+h, ay+k). Point (nx+h, ay+k) is the image of (x,y) under this transformation. Or (x,y) is the pre-image of (nx+h, ay+k).

It may help to think of this as a black box where (x,y) enters and is magically transformed to (nx+h, ay+k). Though we know it is not magic at all but just maths:

As mentioned, the goal here is to transform a set of points, usually defined by an equation, to a new set (new equation). The above mental image (pun intended) will help keep in focus what is being done.

In my next post, I will transform equations and show how that is done.