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