Translations - Interactive Activity

A translation is the transformation informally called a slide: every point of a figure moves the same distance and in the same direction. Because every point moves in exactly the same way, the figure does not turn, flip, or change size — only its position changes. In this lesson you will describe translations with vectors and mapping rules, apply a rule to find an image, work backwards to find a pre-image, and combine translations into compositions.

Definitions and Main Ideas

Transformation: A function that moves or changes a figure in some way to produce a new image.
Pre-image and Image: The pre-image is the figure before a transformation is applied to it. The image, also called the post-image, is the figure that results from the transformation. The terms are relative to the step being considered: in a composition, the image from one transformation becomes the pre-image for the next transformation — it is after the first step and before the second.
Prime Notation: The image of point P is written P′ (read "P prime"). A second transformation gives P″, and so on. The image produced by each step is the post-image of that step; when another transformation follows, that same figure is the pre-image for the next step.
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 pre-image, so distances, angle measures, and the shape itself do not change. A translation is a rigid motion.

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 pre-image. Positive changes move right or up; negative changes move left or down.
To find a pre-image, 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 pre-image of the next.

Translation Explorer

What you are looking at: The blue triangle is the pre-image, labeled A, B, and C. The red triangle is its image (its post-image), labeled A′, B′, and C′ — each image vertex has the same letter as the pre-image vertex it came from, with a prime mark. An arrow runs from each pre-image vertex toward its image vertex, showing how that point moved.

What to do: The pre-image stays fixed in this activity — there is nothing to drag. Use the two steppers below the diagram to change the vector <a, b>, and watch the red image move to its new position. Each stepper is explained by the line directly above it.

What to notice: Every vertex moves by the same vector, so all three arrows have the same length and direction, and the image stays congruent to the pre-image (same size and same shape), because a translation is a rigid motion. The rule and the vertex table under the diagram update with every change.

a — Horizontal change: how far every point moves right (+) or left (−).

b — Vertical change: how far every point moves up (+) or down (−).

Start over with the vector <3, 2>.

Write the Rule

What to do: Each activity shows a pre-image and its image on a grid. Work in three steps: (1) find the horizontal change a, (2) choose the complete mapping rule, and (3) apply the rule — either to find the image of another point, or to work backwards and find the pre-image of a given image point. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. When all three steps are complete, the full worked solution appears. Each choice you pick is animated on the diagram, so you can see what that choice does: a correct choice lands exactly on the true result, and a missed choice is shown as an amber ghost where that choice would land. In step 3, the extra point you are asked about is marked in purple in the diagram. Press New Activity for the next activity — the six activities cycle in order.

Land the Translation

What to do: Read the translation in the prompt, then drag the red image figure by any of its orange vertex handles. The figure snaps to grid points, and a move is refused if it would push the figure off the grid. The dashed outline is the target: the position the image must reach. Press Check when you think the figure has landed exactly on the target, or simply drag it there — an exact landing is confirmed as soon as it happens. Later challenges are compositions: two translations applied in order, and the figure must land on the target produced by both of them together. Press New Challenge for a different challenge.

Your Turn

Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.

Translations — Mr. J Geometry. All activities on this page run entirely in your browser.