8.9: Translation Notation
- Page ID
- 5999
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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](https://k12.libretexts.org/@api/deki/files/3557/f-d_6c808fcf201befb57500106d2371b8de26e0f301099449a2781a10ce%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
First, pick a point in the diagram to use to see how it is translated.
![f-d_f8b4acdccb0aaa73e968923b716dbf62d278f404425337e62a38aea2+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3558/f-d_f8b4acdccb0aaa73e968923b716dbf62d278f404425337e62a38aea2%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
\(D:(−1,4) D′:(6,1)\)
\(D(x,y) \rightarrow D′(x+a,y+b)\)
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](https://k12.libretexts.org/@api/deki/files/3559/f-d_44fa5ba4bb34f45e400b74f21f39027c04e9356409d13a96742fe29f%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/3560/f-d_f7acfb7178881d552294e3a4092d7c4b7ade8d0e4d77983e7402469b%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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.
![f-d_ec878921a3cee449546bf305c42f1693f32aad11f94564bff257b064+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3561/f-d_ec878921a3cee449546bf305c42f1693f32aad11f94564bff257b064%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
For those two trapezoids: \((x,y) \rightarrow (x+4, y−5)\)
Review
Write the mapping rule to describe the movement of the points in each of the translations below.
- \(S(1,5) \rightarrow S′(2,7)\)
- \(W(−5,−1) \rightarrow W′(−3,1)\)
- \(Q(2,−5) \rightarrow Q′(−6,3)\)
- \(M(4,3) \rightarrow M′(−2,9)\)
- \(B(−4,−2) \rightarrow B′(2,−2)\)
- \(A(2,4) \rightarrow A′(2,6)\)
- \(C(−5,−3) \rightarrow C′(−3,4)\)
- \(D(4,−1) \rightarrow D′(−4,2)\)
- \(Z(7,2) \rightarrow Z′(−3,6)\)
- \(L(−3,−2) \rightarrow L′(4,−1)\)
Write the mapping rule that represents the translation of the preimage to the image for each diagram below.
-
Figure \(\PageIndex{6}\) -
Figure \(\PageIndex{7}\) -
Figure \(\PageIndex{8}\) -
Figure \(\PageIndex{9}\) -
Figure \(\PageIndex{10}\)
Review (Answers)
To see the Review answers, open this PDF file and look for section 10.3.
Additional Resources
Interactive Element
Video: Quadrants of Coordinate Plane
Practice: Translation Notation