Introduction to Transformations

In geometry, a transformation is a way to move a figure on the coordinate plane. When a transformation preserves the exact size and shape of the original figure, it is called a rigid transformation. There are three basic types of rigid transformations: translations, reflections, and rotations.

Translations (Slides)

A translation "slides" a figure. Every point of the figure moves the exact same distance and in the exact same direction.

Coordinate Rule: To translate a point $a$ units horizontally and $b$ units vertically, add to the coordinates: $(x, y) \rightarrow (x + a, y + b)$

• Positive $a$ moves right, negative $a$ moves left.
• Positive $b$ moves up, negative $b$ moves down.

Example: Translate the point $(3, 2)$ four units to the right. Moving right affects the $x$-coordinate by $+4$. The $y$-coordinate stays the same. $(3 + 4, 2) \rightarrow (7, 2)$

Reflections (Flips)

A reflection "flips" a figure over a specific line, called the line of reflection, creating a mirror image.

Coordinate Rules:

• Over the $x$-axis: The $x$-coordinate stays the same, but the $y$-coordinate changes sign. $(x, y) \rightarrow (x, -y)$
• Over the $y$-axis: The $y$-coordinate stays the same, but the $x$-coordinate changes sign. $(x, y) \rightarrow (-x, y)$

Example: Reflect the point $(3, -1)$ over the $x$-axis. Keep the $x$-coordinate and change the sign of the $y$-coordinate. $(3, -(-1)) \rightarrow (3, 1)$

Rotations (Turns)

A rotation "turns" a figure around a fixed point, usually the origin $(0, 0)$. Rotations are typically measured counterclockwise.

Coordinate Rules (Counterclockwise about the origin):

• $90^\circ$ Rotation: $(x, y) \rightarrow (-y, x)$
• $180^\circ$ Rotation: $(x, y) \rightarrow (-x, -y)$
• $270^\circ$ Rotation: $(x, y) \rightarrow (y, -x)$

Example: Rotate a triangle $ABC$ by $180^\circ$ about the origin. To rotate the entire shape, you apply the $180^\circ$ rule to every single vertex. If point $A$ is at $(4, 5)$, its new location $A'$ will be $(-4, -5)$. You simply flip the signs of both coordinates for every point on the triangle.