Translation (geometry)

Felix Klein provided a new perspective on geometry known as transformational geometry in the nineteenth century. The majority of geometric proofs are dependent on object transformations. A graph of a function can undergo four different transformations (and translation in math is one of them). They are:

•Rotations

•Translations

•Reflections

•Dilation

We will study about translation in mathematics in this chapter by looking at translation math definitions and examples. At the bottom of the page, you can try your hand at a few translation math problems on translation geometry.

Mathematical Translation

Math Definition Translation

In arithmetic, translation shifts a shape left or right, up or down. The translated shapes appear to have the same size as the original shape, indicating that they are congruent. They’ve simply shifted in one or more directions. There is no shape change because the shape is just moved from one location to another.

The direction or path of the object’s change in location can vary, for example, the object can go left at first, then right, and so on. All points on the shape will shift by the same number of units during translating. If one point shifts two units to the right, all the other points will shift two units to the right as well.

In mathematics, one of the transformations is translation. The original shape is called the preimage when a shape is modified, and the vertices are commonly identified with uppercase letters (Example: ABCD). The vertices of the translated shape are designated with uppercase letters and a “prime” next to each (for example, A′B′C′D′, which is pronounced “A-prime, B-prime, C-prime, D-prime”).

Example of Translation in Geometry

The preimage in the diagram below is ABC, and the image is A’B’C’. A’B’C’ is formed by translating ABC in the following two ways (one after the other).

Moved 3 units up (vertically) and then 2 units right (horizontally).

Preimages and Translation Images

As we’ve seen, a shape before translations is known as “preimage,” and a shape gained after translations is known as “image.” In simple words :

The shape before translation is called “preimage.”

After translation, the shape is called “image.”

Look at the diagram below, where ABCD is the preimage and A’B’C’D’ is the ABC image.

On the Coordinate Plane, Translations

Any item in the coordinate plane can be moved vertically (up/down) or horizontally (left/right). To comprehend translations on the coordinate plane, consider the last example. As indicated in the illustration, a graph is represented in the coordinate plane. The quadrilateral is shifted 5 units horizontally to the right and 1 unit vertically upward during translation, resulting in the new translated function for the given shape being

(x, y) → (x + 5, y + 1)

You should have a good idea of how to write translations by now. The x-coordinate of a point is left/right, while the y-coordinate is affected by up/down. The following are the translation rules.

Rules of Translation

•Replace x with x – k when the form is moved to the left by k units.

•Replace x with x + k when the form is moved to the right by k units.

•Replace y with y + k when the form is shifted up by k units.

•Replace y with y – k when the form is pushed down by k units.

Steps to Make a Translation Graph

Follow the steps below to graph a shape’s translation:

•Determine the shape’s vertices.

•To find the new vertices, apply the specified transformation to each of the vertices (as shown in the above example).

•Then, on the coordinate plane, plot the new points and link them according to the supplied shape.

Important Notes on Translations

All points will change by the same number of units during translation.

After translation, the object’s shape and dimensions remain unchanged.

The domain may be influenced by horizontal translation, while the range may be affected by vertical translation.

Conclusion

In arithmetic, translation shifts a shape left or right, up or down. The translated shapes appear to have the same size as the original shape, indicating that they are congruent. They’ve simply shifted in one or more directions. There is no shape change because the shape is just moved from one location to another. The shape before translation is called “preimage.”After translation, the shape is called “image.”Any item in the coordinate plane can be moved vertically (up/down) or horizontally (left/right). To comprehend translations on the coordinate plane, consider the last example. As indicated in the illustration, a graph is represented in the coordinate plane. Determine the shape’s vertices. To find the new vertices, apply the specified transformation to each of the vertices (as shown in the above example). Then, on the coordinate plane, plot the new points and link them according to the supplied shape.