11.1 Transformations
A transformation is a movement of an original object to a new position, producing an image. In a transformation, the original figure is called the object and the resulting figure is called the image.
11.2 Translation
A translation is a transformation that shifts every point on an object by a fixed distance in a specified direction without changing its shape, size, or orientation.
Vector Representation
A translation can be described using a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$ where:
- $a$ represents horizontal movement ($a > 0$ to the right, $a < 0$ to the left).
- $b$ represents vertical movement ($b > 0$ upwards, $b < 0$ downwards).
$$\text{If } P(x, y) \xrightarrow{\begin{pmatrix} a \\ b \end{pmatrix}} P'(x', y'), \text{ then } x' = x + a \text{ and } y' = y + b$$
11.3 Reflection
A reflection is a transformation that flips an object across a line called the axis of reflection. Every point on the image is at the same perpendicular distance from the axis of reflection as the corresponding point on the object.
Common Axes of Reflection
- Reflection in $x$-axis ($y = 0$): $(x, y) \rightarrow (x, -y)$
- Reflection in $y$-axis ($x = 0$): $(x, y) \rightarrow (-x, y)$
- Reflection in line $y = x$: $(x, y) \rightarrow (y, x)$
- Reflection in line $y = -x$: $(x, y) \rightarrow (-y, -x)$
11.4 Rotation
A rotation is a transformation that turns an object around a fixed point called the center of rotation through a specified angle and direction (clockwise or counterclockwise).
Common Rotations about Origin $(0, 0)$
- $90^\circ$ Clockwise / $270^\circ$ Counterclockwise: $(x, y) \rightarrow (y, -x)$
- $90^\circ$ Counterclockwise / $270^\circ$ Clockwise: $(x, y) \rightarrow (-y, x)$
- $180^\circ$ Rotation (any direction): $(x, y) \rightarrow (-x, -y)$
11.5 Isometry and Rotational Symmetry
- Isometry: A transformation in which the image has the same shape and size as the original object (congruent). Translation, reflection, and rotation are all isometries.
- Congruence: Two figures are congruent if they have identical shape and identical size.
- Rotational Symmetry: A shape has rotational symmetry if it fits onto itself more than once during a complete $360^\circ$ turn. The order of rotational symmetry is the number of times it matches itself in one full turn.