Translations and Reflections

A translation is a transformation that slides a figure a fixed distance in a specific direction. Think of it like a chess piece moving across the board; the shape doesn't rotate or change size, it just changes position. In the coordinate plane, we describe this using a rule: $(x, y) \rightarrow (x + h, y + k)$. Here, $h$ represents the horizontal shift (left or right) and $k$ represents the vertical shift (up or down). If $h$ is positive, we move right; if negative, we move left. If $k$ is positive, we move up; if negative, we move down.

A reflection is a flip over a line called the line of reflection. Each point of the new figure is the same distance from the line as the original point, just on the opposite side.

When reflecting over the x-axis, the x-coordinate stays the same, but the y-coordinate changes sign: $(x, y) \rightarrow (x, -y)$.

When reflecting over the y-axis, the y-coordinate stays the same, but the x-coordinate changes sign: $(x, y) \rightarrow (-x, y)$.

In the real world, objects often move and flip at the same time. This is called a composition of transformations. When performing multiple steps, it is vital to follow the order strictly. For example, translating a shape and then reflecting it might result in a different position than reflecting it first and then translating it. Always label your intermediate points (like $A'$) and your final points (like $A''$) to keep track of the journey.