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Translation is a term used in geometry to describe a function that moves an object a certain distance. The object is not altered in any other way. It is not rotated, reflected or re-sized.
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In a translation, every point of the object must be moved in the same direction and for the same distance.
When you are performing a translation, the initial object is called the pre-image, and the object after the translation is called the image. So, in the picture above, the rust-colored item is the pre-image, and the blue item is the image. We know this because the arrow tells us the direction in which the image was moved. For other images, you might be told which image is the pre-image, or you might be asked to find either the pre-image from the image, or vice versa.
When working with translation problems, the information may be presented in different ways.
You may be given a figure drawn on the coordinate plane like this:
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Then you will be asked to translate the figure. You will be given a distance and direction for the transformation.
For example, translate the figure down 7.
The way to do this is to take each vertex point individually and count down 7. So the point at (1, 5) will move to (1, -2). Notice we did not move the vertex along the x-axis, or horizontal direction. The instructions asked us to move it down only, along the y-axis.
Move the other three vertices in the same manner. The point at:
Then connect the vertices to draw the square, translated down 7.
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Another way that information might be given is like this, starting again with an image drawn on the coordinate plane:
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This time, you will not be asked to draw the translation, but instead to describe it in mathematical notation.
Describe a translation of the triangle down 2 and to the right 3.
The notation will look like this:
(x, y) → (x + 3, y - 2)
This means that for each point on the triangle (x, y), the x-coordinate is moved to the right 3 spaces, and the y-coordinate is moved down 2 spaces. Later on, anyone who sees this notation can draw the image by looking at the pre-image. You will not have to have draw it for them. When they do draw the image, it will look like this:
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Let's test your knowledge with a few additional examples. Feel free to pause the video after each question to give yourself time to find the answer.
1.) Which figure represents the translation of the yellow figure?
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The answer is Q. It is the only figure that is a translation. Figure P is reflected, so it is not facing the same direction. Figure R is larger than the original figure, therefore, it is not a translation.
2.) Write the mathematical notation for a translation that shifts up 5 and to the left 3.
To write the mathematical notation, we do not need to know anything about the figure. The notation will be the same regardless of the size or shape of the figure. For this example, the notation would look like this:
(x, y) → (x - 3, y + 5)
3.) Which figure represents the following translation. (x, y) → (x - 5, y + 2)? The maroon figure is the pre-image, and the blue one is the image.
a.)
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b.)
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c.)
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The answer is b. It is the figure that was moved to the left 5 spaces and up 2 spaces.
In geometry, a translation is the shifting of a figure from one place to another without rotating, reflecting or changing its size. This is done by moving the vertices of the figure the prescribed number of spaces on a coordinate plane and then drawing the new figure.
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