From my experience teaching mathematics, I think that the topic most students find the most difficult is circular functions (trigonometry). The next most difficult topic is transformations: transforming a function to another via dilations, reflections, and translations. This is the first of several posts to address this topic.

First, by way of looking at individual points in a Cartesian coordinate system, let’s do some transformations and define some terms.

Consider the following points *A*, *B*, *C*, and *D*:

The first transformation will be to *dilate* these points. This means that we will change their distance from the *x *or *y* axes by multiplying the appropriate coordinate by a number. This number is called the *dilation factor*. If we dilate along the *x-*axis, we will change the *x* coordinate of each point by multiplying it by the dilation factor. Be aware that this is also called “dilating *from* the *y*-axis” because this dilation changes the point’s distance from the *y-*axis

So let’s dilate these points along the *x*-axis by a factor of 2. This makes each point twice as far from the *y*-axis. So point *A* becomes *A*1(6,1), *B* becomes *B*1(-4,3), *C* becomes *C*1(-8,-1), and *D* becomes *D*1(2,-2):

If we had used a factor of 1/2, then the new points would be half the distance from the *y-*axis than the original points.

If the factor we use is negative, not only is the dilation happening, but the points are also *reflected* across the *y*-axis. So if the dilation factor is -2 along the *x*-axis, the new points are *A*1(-6,1), *B*1(4,3), *C*1(8,-1), and *D*1(-2,-2):

Another transformation is *translation*: moving a point left or right or up or down. So if you translate a point 3 units to the right (along the *x*-axis), then the new points are *A*1(6,1), *B*1(1,3), *C*1(-1,-1), and *D*1(4,-2):

Now all of these transformations can be done along the *y*-axis (from the *x*-axis) as well. So doing all transformations along the *y*-axis, I will dilate point *A* by a factor of 2, reflect point *B* with no dilation, translate point *C* up 3 units, and dilate point *D* by a factor of 1/2. The new points will then be *A*1(3,2), *B*1(-2,-3), *C*1(-4,2), and *D*1(1,-1):

The previous example just changed the *x* or the *y* coordinate of each point. Let’s look at point *A* and mix up these transformations. This will also show that the order of the transformations can affect the final result. Consider the following sequence of transformations of point *A*:

A. (3,1)

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

The following graph shows the sequence of these transformations .The new point’s letter refer to the result of the transformation’s letter above:

Now let’s do the previous steps in reverse:

A. (3,1)

B. Translate 3 units to the left

C. Translate 3 units up

D. Reflect across *y*-axis

E. Reflect across *x*-axis

F. Dilate by factor 1/3 along *x*-axis

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

We end up at a very different place. Also notice that if a point is on the *y*-axis, reflections across the *y*-axis and dilations along the *x*-axis, have no effect.

I will generalise this for a generic point (*x*,*y*) in my next post.