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8.9: Translation Notation

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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.

    1. 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.
    2. 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:

    1. 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 \(P(−2, 2) \rightarrow P′(1, −1)\)


    -2+a=1 & \text { and } \quad 2+b=-1 \\
    a=3 & \quad b=-3

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

    1. 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 \(D(1,−5) \rightarrow D′(−3,1)\)


    1+a=-3 & \text { and } & -5+b=1 \\
    a=-4 & & b=6

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

    1. The translation of the preimage \(A\) to the translated image \(J\) in the diagram below:
    Figure \(\PageIndex{1}\)

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

    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 \(D(−1,4) \rightarrow D′(6,1)\)


    -1+a=6 & \text { and } & 4+b=1 \\
    a=7 & & b=-3

    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?


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

    Example \(\PageIndex{2}\)

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

    Figure \(\PageIndex{3}\)


    \((x, y) \rightarrow (x−3, y−5)\)

    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.

    Figure \(\PageIndex{4}\)


    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.

    Figure \(\PageIndex{5}\)

    For those two trapezoids: \((x,y) \rightarrow (x+4, y−5)\)


    Write the mapping rule to describe the movement of the points in each of the translations below.

    1. \(S(1,5) \rightarrow S′(2,7)\)
    2. \(W(−5,−1) \rightarrow W′(−3,1)\)
    3. \(Q(2,−5) \rightarrow Q′(−6,3)\)
    4. \(M(4,3) \rightarrow M′(−2,9)\)
    5. \(B(−4,−2) \rightarrow B′(2,−2)\)
    6. \(A(2,4) \rightarrow A′(2,6)\)
    7. \(C(−5,−3) \rightarrow C′(−3,4)\)
    8. \(D(4,−1) \rightarrow D′(−4,2)\)
    9. \(Z(7,2) \rightarrow Z′(−3,6)\)
    10. \(L(−3,−2) \rightarrow L′(4,−1)\)

    Write the mapping rule that represents the translation of the preimage to the image for each diagram below.

    1. f-d_a6fb77effc9624ba0d43f7bb3a31fe414ad4bff193f51158e816d6a7+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{6}\)
    2. f-d_5d618b0fed9e3a8d336165dace989b1f9277f3c159db817a814810cf+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    3. f-d_0610d3e9594496dd8093d719824ed19ba6ee567bf68aced00833ba94+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    4. f-d_bb7fd49e02396021b679f5ce265a2e7c2b8e9c4ffb41a774cee55af4+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    5. f-d_a5847e235655a5497e40d7c6b1efe64f5d2b82bf10a07faed34a2a32+IMAGE_TINY+IMAGE_TINY.png
      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

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