Transformations

Rotations

Rotations are, in my experience, the most difficult transformation to master. As the name suggests, they involve rotating points around a center of rotation. In most school and simple mathematics, this point will be the origin \((0,0)\). The most commonly used angles of rotation are multiples of \(90^\circ\). The rules for rotation about the origin for those are as follows:

1. For a \(90^\circ\) rotation counterclockwise, the new point is \((-y, x)\)
2. For a \(180^\circ\) rotation counterclockwise, the new point is \((-x, -y)\)
3. For a \(270^\circ\) rotation counterclockwise, the new point is \((y, -x)\)

Note that we were specific to indicate counterclockwise rotation. Unless otherwise specified, all rotations are counterclockwise. This is due to the orientation and use of the unit circle. Notice that to find the clockwise degree of rotation, you can always just take \(360^\circ-x^\circ\).

To rotate by any angle about the origin, you do need to use some extent of trigonometry. The new formula would be \((x',y') = (x\cos\theta-y\sin\theta, x\sin\theta+y\cos\theta)\). Another common strategy is to draw a circle centered at the origin and passing through the point you wish to rotate. You can draw two radii separated by the angle of rotation, and that's the new point. Obviously, this requires a strong level of preciscion that you may not be able to perform on the AMC 10.

Rotation about a different center can be found by translating the entire system, the shape and the center of rotation, that is, such that the center of rotation lands on the origin. After performing the rotation, you perform a reverse of the translation to find your new figure.

Reflections

Reflections are transformations that flip a figure across a line of reflection. When you see your reflection in a mirror, you see a reflected version of yourself, and most importantly you'll notice that it is facing the opposite direction. Your right hand will be on the left. This is similar for reflections on the coordinate plane: reflections do not preserve the orientation of a shape. The most commonly used lines of reflection are the \(x\)-axis, the \(y\)-axis, and the line \(y=x\). For flipping over an axis, you take the coordinate of the OPPOSITE side of the axis (so if you reflect over \(x\) you look at \(y\)) and flip the sign. Thus, we have \((x,-y)\) and \((-x,y\)\) for reflections over the \(x\)-axis and \(y\)-axis, respectively. For \(y=x\), the rule is \((x',y')=(y,x)\). Note that this is the same as inverting a function.

If you want to reflect over any other line, it's still very simple. Draw a line from the point you wish to reflect to the line of reflection, ensuring it is perpendicular, and extend it by doubling it's length. Effectively, find the rise and run to a point on the line, such that it is perpendicular, and then apply those again.

Dilations

Dilations are non-rigid transformations that change the size of a shape. Note that because of this, they do not preserve congruency or distances. Angles and shape are still preserved. For dilation about the origin, for scale factor \(k\), \((x',y') = (kx, ky)\). As the shape gets enlarged, or shrunk, as do the distances from the origin. For \(0 < k < 1\), it is considered a reduction, and it makes the shape smaller. For \(k > 1\), it is considered an enlargement and makes the shape larger. Note that \(k > 0\) always. To dilate about a point other than the origin, draw lines to the point from the vertex and multiply their horizontal and vertical components by \(k\) to make them longer