Translations

• Translation: A transformation that maps every point $P$ of a figure to an image point $P'$ such that the segment $PP'$ has a constant length and direction for all points.

• Pre-image and Image: The original figure is called the pre-image, while the resulting figure after the shift is known as the image.

• Rigid Motion (Isometry): Because a translation does not alter the dimensions or angles of a figure, it is classified as an isometry, meaning the pre-image and image are always congruent.

• Vector Representation: Translations are often described using a translation vector $\vec{v} = \langle a, b \rangle$, where $a$ represents the horizontal displacement and $b$ represents the vertical displacement.

Coordinate Notation

• To translate a point $(x, y)$ by $a$ units horizontally and $b$ units vertically, apply the rule:

$(x, y) \to (x + a, y + b)$

• Horizontal Shift ($a$): A positive value moves the figure to the right, while a negative value moves it to the left.
• Vertical Shift ($b$): A positive value moves the figure up, while a negative value moves it down.

Step-by-Step Mapping

• Step 1: Identify the coordinates of all vertices of the pre-image.
• Step 2: Determine the translation vector $\langle a, b \rangle$ or the verbal description of the shift.
• Step 3: Add $a$ to every $x$-coordinate and $b$ to every $y$-coordinate to find the new vertex positions.
• Step 4: Plot the new points and connect them to form the image.

• It is vital to distinguish translations from other rigid and non-rigid transformations to select the correct mathematical approach.

• Composition of Translations: When two translations are performed in sequence, the result is a single translation. If the first is $\langle a, b \rangle$ and the second is $\langle c, d \rangle$, the composite translation is $\langle a+c, b+d \rangle$.

• Verify Congruency: Always check that the image looks identical to the pre-image; if the shape appears stretched, tilted, or flipped, a calculation error has occurred.

• Check the Signs: A common mistake is moving in the wrong direction (e.g., moving left when the rule is $x+3$). Always double-check that 'plus' corresponds to right/up and 'minus' to left/down.

• Vector Consistency: Ensure that every vertex has been moved by the exact same vector. You can verify this by calculating the slope and distance between $P$ and $P'$ for at least two different vertices.

• Reverse Engineering: If an exam asks for the pre-image given the image and the rule, you must perform the inverse operation (subtract $a$ and $b$ instead of adding them).