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8.9: Translation Notation

Last updated

Jun 15, 2022
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- 8.8: Translation Applications in Circle Similarity
- 8.10: Defining Rotation

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Graphical introduction to image translations. Learn to use notation to describe mapping rules,and graph images given preimage and translation

Rules for Translations

Jack describes a translation as point moving from $J(−2, 6)$ to $J′(4,9)$ . Write the mapping rule to describe this translation for Jack.

In geometry, a transformation is an operation that moves, flips, or changes a shape (called the preimage) to create a new shape (called the image). A translation is a type of transformation that moves each point in a figure the same distance in the same direction. Translations are often referred to as slides. You can describe a translation using words like "moved up 3 and over 5 to the left" or with notation. There are two types of notation to know.

One notation looks like $T_{(3, 5)}$ . This notation tells you to add 3 to the $x$ values and add 5 to the $y$ values.
The second notation is a mapping rule of the form $(x,y) \rightarrow (x−7, y+5)$ . This notation tells you that the $x$ and $y$ coordinates are translated to $x−7$ and $y+5$ .

The mapping rule notation is the most common.

Interactive Element

Let's write a mapping rule for each of the following translations:

Sarah describes a translation as point $P$ moving from $P(−2,2)$ to $P′(1,−1)$ .

In general, $P(x, y) \rightarrow P′(x+a, y+b)$ .

In this case, $P(−2, 2) \rightarrow P′(−2+a, 2+b)$ or $P(−2, 2) \rightarrow P′(1, −1)$

Therefore:

$\begin{array}{rr} -2+a=1 & \text { and } \quad 2+b=-1 \\ a=3 & \quad b=-3 \end{array}$

The rule is: $(x,y) \rightarrow (x+3, y−3)$

Mikah describes a translation as point D in a diagram moving from $D(1, −5)$ to $D′(−3,1)$ .

In general, $P(x,y) \rightarrow P′(x+a, y+b)$ .

In this case, $D(1,−5) \rightarrow D′(1+a,−5+b)$ or $D(1,−5) \rightarrow D′(−3,1)$

Therefore:

$\begin{array}{rrr} 1+a=-3 & \text { and } & -5+b=1 \\ a=-4 & & b=6 \end{array}$

The rule is: $(x,y) \rightarrow (x−4, y+6)$

The translation of the preimage $A$ to the translated image $J$ in the diagram below:

f-d_6c808fcf201befb57500106d2371b8de26e0f301099449a2781a10ce+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

First, pick a point in the diagram to use to see how it is translated.

f-d_f8b4acdccb0aaa73e968923b716dbf62d278f404425337e62a38aea2+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{2}$

$D:(−1,4) D′:(6,1)$

$D(x,y) \rightarrow D′(x+a,y+b)$

So: $D(−1,4) \rightarrow D′(−1+a,4+b)$ or $D(−1,4) \rightarrow D′(6,1)$

Therefore:

$\begin{array}{rrr} -1+a=6 & \text { and } & 4+b=1 \\ a=7 & & b=-3 \end{array}$

The rule is: $(x,y) \rightarrow (x+7,y−3)$

Example $\PageIndex{1}$

Earlier, you were told that Jack described a translation as point $J$ moving from $J(−2,6)$ to $J′(4,9)$ . What is the mapping rule that describes this translation?

Solution

$(x,y) \rightarrow (x+6, y+3)$

Example $\PageIndex{2}$

Write the mapping rule that represents the translation of the red triangle to the translated green triangle in the diagram below.

f-d_44fa5ba4bb34f45e400b74f21f39027c04e9356409d13a96742fe29f+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{3}$

Solution

$(x, y) \rightarrow (x−3, y−5)$

Example $\PageIndex{3}$

The following pattern is part of wallpaper found in a hotel lobby. Write the mapping rule that represents the translation of one blue trapezoid to a translated blue trapezoid shown in the diagram below.

f-d_f7acfb7178881d552294e3a4092d7c4b7ade8d0e4d77983e7402469b+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{4}$

Solution

If you look closely at the diagram below, there two pairs of trapezoids that are translations of each other. Therefore you can choose one blue trapezoid that is a translation of the other and pick a point to find out how much the shape has moved to get to the translated position.