What is the Translation in Math?
Definition: In math, a translation moves a shape left, right, up, or down but does not turn. The translated shapes (or the image) appear to be the same size as the original shape, indicating that they are congruent. They’ve simply shifted in one or more directions. There is no change in the shape because the shape simply moves from one location to another.
Translations on the Coordinate Plane
Any object in the coordinate plane can be moved horizontally (left/right) or vertically (up/down) by translating it in the coordinate plane.
Let’s look at an example to see how coordinate plane translations work.
As shown in the graph, a quadrilateral has been represented in the coordinate plane. The quadrilateral is shifted 5 units horizontally to the right and 1 unit vertically upward during the translation. The resulting new translated function for the given figure being
\((x,~y)\rightarrow (x +5, ~y + 1)\).
Each of the \(x\) coordinates of the vertices have to be added by 5 units and each \(y\) coordinate will be added by 1 unit.
Translation Rule
For the following translations, let’s understand the change in the coordinates of the image formed:
Replace \(x\) with \(x -k\) when the shape is moved to the left by \(k\) units.
Replace \(x\) with \(x + k \) when the shape is moved to the right by \(k\) units.
Replace y with \(y + k\) when the shape is moved up by \(k\) units.
Replace \(y\) with \(y – k\) when the shape is moved down by \(k\) units.
Example:
When the translation \((x,y)\rightarrow (x – 2, y + 3)\) is applied to the point(2,5), what are the new coordinates?
Solution:
Consider the point to be A \((x,y) = (2, 5)\) and let the image be A’. Now, using the transformation provided apply it to this point.
\(x ~- 2 = 2~- 2 =~ 0\)
\(y~ +~ 3 = 5~ +~ 3 =~ 8\)
As a result, coordinates of the image of point A are \({A}'(0, 8)\).
Solved Translation Examples
Example 1:
Tell whether the green figure is a translation of the blue figure.
a.
Solution:
The green figure and blue figure are exactly the same in shape and size. The blue figure slides over the green figure and completely covers it. So, the green figure is a translation of the blue figure.
b.
Solution:
The blue figure turns to form the green figure.
So, the green figure is not a translation of the blue figure.
Example 2:
Translate the square and move it 4 units right and 2 units down. What will be the image’s new coordinates?
Solution:
Method 1: We use a coordinate plane to find the image. Each vertex should be moved 4 units to the right and 2 units to the bottom.
Hence, the coordinates of the image are \({A}'(2,~ 1.5), {B}'(3.5, ~0), {C}'(2, ~-1.5) \) and \({D}'(0.5, 0) \)
Method 2: Each coordinate of the vertice is added or subtracted by the required constant. Add 4 to the x-coordinates of the vertices and subtract 2 from the y-coordinates of the vertices.
\(A(-2, ~3.5)\rightarrow {A}'(-2 + 4, ~3.5 – 2)\rightarrow {A}'(2,~1.5) \)
\(B(-0.5,~ 2)\rightarrow {B}'(-0.5 + 4, 2 – 2)\rightarrow {B}'(3.5,~ 0) \)
\(C(-2, ~ 0.5)\rightarrow {C}'(-2 + 4, 0.5 -~ 2)\rightarrow {C}'(2, -1.5) \)
\(D(-3.5, ~2)\rightarrow {D}'(-3.5 + 4, ~ 2 – 2)\rightarrow {D}'(0.5, ~0) \)
Hence, the coordinates of the image are \({A}'(2,~ 1.5), {B}'(3.5, ~0), {C}'(2, ~-1.5) \) and \({D}'(0.5, ~0) \)
Example 3:
The coordinate plane is used by a garden owner to represent a garden. To represent the location of a new flower bed, he draws a triangle with vertices \(A(-2, ~1), B(2,~5) \) and \(c(1, ~2) \). The flower bed will be moved 3 units to the right and 3 units down by city officials. Find the image’s coordinates and in the coordinate plane draw the original figure and the image.
Solution:
The vertices of the flower bed are provided as coordinates. After a translation of 3 units to the right and 3 units down, you must find the coordinates and graph the original figure and its image in a coordinate plane.
Add 3 to each \(x \)-coordinate and subtract 3 from each \(y \)-coordinate to get the image’s coordinates.
\((x,y)\rightarrow (x + 3, ~y – 3) \)
\(A(-2, ~1)\rightarrow A(-2 + ~3, 1 ~- 3)\rightarrow {A}'(1, ~-2) \)
\(B(2, ~5)\rightarrow B(2 + 3, ~5-3)\rightarrow {B}'(5, ~2) \)
\(C(1, 2)\rightarrow C(1 + 3, ~2-3)\rightarrow {C}'(4,~ -1) \)
In mathematics, a translation is a transformation of a shape in a plane that preserves length, i.e., the object is transformed without losing its dimensions. It can simply be shifted to the left, right, up or down.
The changes that are done onto a figure to create an image of the figure is a transformation. Translation is an example of a transformation.
All the translated images of the figure will have the same shape and size as the original figure. Moreover, the orientation is also the same, that is the image will not turn. If these conditions are not satisfied then the image is not a translation of the figure.