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8.14: Rules for Reflections

  • Page ID
    6108
  • Identify and state rules describing reflections using notation

    The figure below shows a pattern of two fish. Write the mapping rule for the reflection of Image \(A\) to Image \(B\).

    f-d_d3f0f68d7a85938f8b86d4594834000c637ca44594ac0d633869b45e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A reflection is an example of a transformation that takes a shape (called the preimage) and flips it across a line (called the line of reflection) to create a new shape (called the image). By examining the coordinates of the reflected image, you can determine the line of reflection. The most common lines of reflection are the \(x\)-axis, the \(y\)-axis, or the lines \(y=x\) or\(y=−x\).

    f-d_76f4f2689cc2306b23ed3a349e87d96c65f6dc686ab52ef6e369146c+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    The preimage above has been reflected across the \(y\)-axis. This means, all of the x-coordinates have been multiplied by -1. You can describe the reflection in words, or with the following notation:

    \(r_{y-axis} (x,y)\rightarrow (−x,y)\)

    Notice that the notation tells you exactly how each \((x,y)\) point changes as a result of the transformation.

    Let's find the image of the point \((3, 2)\) that has undergone a reflection across the following lines:

    1. The \(x\)-axis,

    Reflection across the \(x\)-axis: \(r_{x-axis} (3,2)\rightarrow (3,−2)\)

    1. The \(y\)-axis

    Reflection across the \(y\)-axis: \(r_{y-axis} (3,2)\rightarrow (3,−2)\)

    1. The line \(y=x\)

    Reflection across the line \(y=x\): \(r_{y=x} (3,2)\rightarrow (2,3)\)

    1. The line\(y=−x\).

    Reflection across the line\(y=−x\): \(r_{y=−x} (3,2)\rightarrow (−2,−3)\)

    f-d_03d5091fd1931e89c0ae2c39f87b639048cc8074ec8b00b011e1afac+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Now, let's reflect Image \(A\) in the diagram below across the following lines and write the notation for each reflection:

    f-d_d82a4343980d8dbacd344b52c2370ce4233cb4ab5ec994a28e4af018+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.png
    Figure \(\PageIndex{5}\)
    1. Across the \(y\)-axis and label it \(B\).

    Reflection across the \(y\)-axis: \(r_{y-axis} A\rightarrow B=r_{y-axis} (x,y)\rightarrow (−x,y)\)

    1. Across the \(x\)-axis and label it \(O\).

    Reflection across the \(x\)-axis: \(r_{x-axis} A\rightarrow O=r_{x-axis} (x,y)\rightarrow (x,−y)\)

    1. Across the line\(y=−x\) and label it \(Z\).

    Reflection across the\(y=−x\): \(r_{y=−x} A\rightarrow Z=r_{y=−x} (x,y)\rightarrow (−y,−x) \)

    f-d_2ed28171fe940242790d77ebfc55a659482066ac8c64098e7f74b83e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Finally, let's write the notation that represents the reflection of the preimage to the image in the diagram below:

    f-d_27cd4c2be3ce9e4d41cce78d7b53662b0f588e9ed620c42c8a7353f5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    This is a reflection across the line\(y=−x\). The notation is \(r_{y=−x} (x,y)\rightarrow (−y,−x)\).

    Example \(\PageIndex{1}\)

    Earlier, you were given a problem about the figure below that shows a pattern of two fish. Write the mapping rule for the reflection of Image A\) to Image B\).

    f-d_d3f0f68d7a85938f8b86d4594834000c637ca44594ac0d633869b45e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    Solution

    To answer this question, look at the coordinate points for Image A and Image B.

    Image A \(A(−11.8,5)\) \(B(−11.8,2)\) \(C(−7.8,5)\) \(D(−4.9,2)\) \(E(−8.7,0.5)\) \(F(−10.4,3.1)\)
    Image B \(A′(−11.8,−5)\) \(B′(−11.8,−2)\) \(C′(−7.8,−5)\) \(D′(−4.9,−2)\) \(E′(−8.7,−0.5)\) \(F′(−10.4,−3.1)\)

    Notice that all of the \(y-coordinates have changed sign. Therefore Image A has reflected across the \(x\)-axis. To write a rule for this reflection you would write: \(r_{x-axis} (x,y)\rightarrow (x,−y)\).

    Example \(\PageIndex{2}\)

    Thomas describes a reflection as point \(J moving from \(J(−2,6)\) to \(J′(−2,−6)\). Write the notation to describe this reflection for Thomas.

    \(J:(−2,6)\qquad J′:(−2,−6)\)

    Solution

    Since the y-coordinate is multiplied by -1 and the \(x-coordinate remains the same, this is a reflection in the \(x\)-axis. The notation is: \(r_{x-axis} J\rightarrow J′=r_{x-axis} (−2,6)\rightarrow (−2,6)\)

    Example \(\PageIndex{3}\)

    Write the notation that represents the reflection of the yellow diamond to the reflected green diamond in the diagram below.

    f-d_88eaac2581b654d7d7b6f887c8891d16bf6254041398c05db6ac5643+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)

    Solution

    In order to write the notation to describe the reflection, choose one point on the preimage (the yellow diamond) and then the reflected point on the green diamond to see how the point has moved. Notice that point \(E\) is shown in the diagram:

    \(E(−1,3)\rightarrow E′(3,−1)\)

    Since both \(x\)- and \(y\)-coordinates are reversed numbers, the reflection is in the line \(y=x\). The notation for this reflection would be: \(r_{y=x} (x,y)\rightarrow (y,x)\).

    Example \(\PageIndex{4}\)

    Karen was playing around with a drawing program on her computer. She created the following diagrams and then wanted to determine the transformations. Write the notation rule that represents the transformation of the purple and blue diagram to the orange and blue diagram.

    f-d_c04044d8d1ee4981c3d7fc1d247c7313499bc57c936e157b90635782+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)

    Solution

    In order to write the notation to describe the transformation, choose one point on the preimage (purple and blue diagram) and then the transformed point on the orange and blue diagram to see how the point has moved. Notice that point A is shown in the diagram:

    \(C(7,0)\rightarrow C′(−7,0)\)

    Since both x-coordinates only are multiplied by -1, the transformation is a reflection is in \(y\)-axis. The notation for this reflection would be: \(r_{y-axis} (x,y)\rightarrow (−x,y)\).

    f-d_257c5dc571c7df70b2251ce94c4edba06403ccca9055d95c1895068e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{11}\)

    Review

    Write the notation to describe the movement of the points in each of the reflections below.

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

    Write the notation that represents the reflection of the preimage image for each diagram below.

    1. f-d_0cc75f8a43579eb56c0e3c0392d06097439e8668e0a392b141a5dd12+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    2. f-d_c87f8229d4b1311f1a97d6326e7b17e6e1a42344eaaad30473504990+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)
    3. f-d_a38ccbd1f9331700b60ce7a5bed73cdce2c892ba0ecb988c602e9017+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    4. f-d_dddcb4d26c2afa175e4fe13ed1490919dd115dc8dd8494490cf979ca+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 10.6.

    Additional Resources

    Interactive Element

    Practice: Rules for Reflections