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 A1(6,1), B becomes B1(-4,3), C becomes C1(-8,-1), and D becomes D1(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 A1(-6,1), B1(4,3), C1(8,-1), and D1(-2,-2):
of factor -2 along x-axis
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 A1(6,1), B1(1,3), C1(-1,-1), and D1(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 A1(3,2), B1(-2,-3), C1(-4,2), and D1(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.