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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 v\begin{align*}\vec{v}\end{align*} to create a new parallelogram (the image).

Keep in mind that the location of vector v\begin{align*}\vec{v}\end{align*} does not matter. Vectors have a direction and a magnitude (a length), and simply tell you how to move points. Vector v\begin{align*}\vec{v}\end{align*} 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 v\begin{align*}\vec{v}\end{align*}.

With the grid in the background, you can see that the slope of vector v\begin{align*}\vec{v}\end{align*} and each line is 13\begin{align*}\frac{1}{3}\end{align*}. 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 10\begin{align*}\sqrt{10}\end{align*} (found using the Pythagorean Theorem) along lines parallel to vector v\begin{align*}\vec{v}\end{align*}.

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 v\begin{align*}\vec{v}\end{align*} have been drawn through each of the original points. Vector v\begin{align*}\vec{v}\end{align*} 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 A\begin{align*}A\end{align*} to A′\begin{align*}A^\prime\end{align*} 4 to the right and 1 up B\begin{align*}B\end{align*} to B′\begin{align*}B^\prime\end{align*} 4 to the right and 1 up C\begin{align*}C\end{align*} to C′\begin{align*}C^\prime\end{align*} 4 to the right and 2 up D\begin{align*}D\end{align*} to D′\begin{align*}D^\prime\end{align*} 4 to the right and 1 up

Because C\begin{align*}C\end{align*} to C\begin{align*}C^\prime\end{align*} 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 u\begin{align*}\vec{u}\end{align*} on the quadrilateral below.

Solution: With the grid in the background, you can see that vector u\begin{align*}\vec{u}\end{align*} 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 t\begin{align*}\vec{t}\end{align*} 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.

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Perform the translation defined by vector t\begin{align*}\vec{t}\end{align*} on the polygons below.

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Are the following transformations translations? Explain.

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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.