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Translations and Vectors

Graphical introduction to image translations

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Practice Translations and Vectors
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Translations

Translations are often informally called “slides”. Why is this?

Guidance

A translation is one example of a rigid transformation. A translation moves each point in a shape a specified distance in a specified direction as defined by a vector. Below, the parallelogram has been translated along vector  to create a new parallelogram (the image).

Keep in mind that the location of vector  does not matter. Vectors have a direction and a magnitude (a length), and simply tell you how to move points. Vector  essentially tells you that all points move three units to the right and one unit up

The lines that connect corresponding points will all be parallel to vector .

With the grid in the background, you can see that the slope of vector  and each line is . This should make sense because each point in the original parallelogram was moved 3 units to the right and 1 unit up to create its corresponding point in the image.

One way to think about translations is that they move points a specified distance along lines parallel to a given line.  In this case, all points were moved a distance of  (found using the Pythagorean Theorem) along lines parallel to vector .

If you are performing a translation of a shape not on a grid, the vector becomes crucial. You can no longer say “move one unit up and three units to the right” because without a grid there are no units. Below is a translation of another quadrilateral without a grid in the background.

Notice that lines parallel to vector  have been drawn through each of the original points. Vector  has been copied onto each of those lines at the points that define the original quadrilateral. The ends of the vectors define the points on the image.

Example A

Is the following transformation a translation?

Solution: One way to check if a transformation is a translation is to look at how each point moves to create its image. If all points move in the same way, then it is a translation.

 Point to Image Point Description of Motion to 4 to the right and 1 up to 4 to the right and 1 up to 4 to the right and 2 up to 4 to the right and 1 up

Because  to  is different, this is not a translation.

Example B

Describe the vector that defined the translation below.

Solution: The vector moved each point three units to the right and three units up.

Example C

Perform the translation defined by vector  on the quadrilateral below.

Solution: With the grid in the background, you can see that vector  tells you to move each point 2 units up and 1 unit to the left. Here is the translation:

Concept Problem Revisited

A translation is informally called a slide, because it essentially slides a shape to a new position. The orientation of the points does not change.

Vocabulary

A translation is a rigid transformation that moves each point in a shape a specified distance in a specified direction as defined by a vector.

Guided Practice

1. Describe the vector that defined the translation below.

2. Perform the translation defined by vector  on the triangle below.

3. Is the following transformation a translation?

1. The vector moved each point 4 units to the right and 2 units down.

2.

3. Yes, each point moves one unit to the right and two units up.

Practice

1. Is a translation a rigid transformation? Explain.

2. What role does a vector play in a translation?

3. How are parallel lines relevant to translations?

4. How can you tell if a transformation is a translation?

Describe the vector that defined each of the following translations.

5.

6.

7.

8.

Perform the translation defined by vector  on the polygons below.

9.

10.

11.

12.

Are the following transformations translations? Explain.

13.

14.

15.

Vocabulary Language: English

Image

Image

The image is the final appearance of a figure after a transformation operation.
Preimage

Preimage

The pre-image is the original appearance of a figure in a transformation operation.
Transformation

Transformation

A transformation moves a figure in some way on the coordinate plane.
Translation

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.
Rigid Transformation

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.