Translations
Overview
A transformation is a function that moves or changes a figure to produce a new image. This lesson covers the translation — the transformation informally called a slide — in which every point of a figure moves the same distance in the same direction. You will describe translations with mapping rules and vectors, apply a rule to find images, work backwards to find preimages, and combine translations into compositions.
Vocabulary
- Transformation
- A function that moves or changes a figure in some way to produce a new image.
- Preimage and Image
- The preimage is the original figure; the image (or post-image) is the figure produced by the transformation.
- Prime Notation
- The image of point P is written P′ (read "P prime"). A second transformation gives P″, and so on.
- Translation
- A transformation that moves every point of a figure the same distance and in the same direction.
- Vector
- A quantity written <a, b> that describes a translation: a is the horizontal change and b is the vertical change applied to every point.
- Composition
- Two or more transformations combined and applied in a specific order.
- Rigid Motion
- Any transformation that maintains congruence — the image is congruent to the preimage.
Main Ideas
- A translation is described by a mapping rule such as (x, y) → (x + 2, y + 4), or by the vector <2, 4>: move 2 units right and 4 units up.
- To find an image, apply the rule to the coordinates of the preimage. Positive changes move right or up; negative changes move left or down.
- To find a preimage, reverse the rule: undo each change (subtract what the rule adds).
- A translation is a rigid motion: distances, angle measures, and the shape itself do not change — only the position changes.
- In a composition, transformations are applied in order, and the image of one step becomes the preimage of the next.
Lesson Notes
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Lesson Video
Watch the lesson video.
Interactive Activity
Translate figures with vectors and mapping rules, and compose two translations.
Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.