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Exam questions · Maths · Transformations and Similarity

Reflections and Translations

  • 6 exam questions
  • 18 marks
  • 9 quick checks
  1. 1 Write down [2 marks]

    Write down the coordinates of the image of the point \((3, 4)\) after a reflection in the \(y\)-axis.

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    Model answer

    A reflection in the \(y\)-axis changes the sign of the \(x\)-coordinate, so the image is \((-3, 4)\).

    Mark scheme

    • The \(x\)-coordinate changes sign — M1
    • \((-3, 4)\) — A1
  2. 2 Reflect [4 marks]

    Triangle \(P\) is drawn on the grid. (a) Reflect triangle \(P\) in the line \(x = 4\). Label the image \(Q\). (2 marks) (b) Translate triangle \(P\) by the vector \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\). Label the image \(R\). (2 marks)

    Triangle P on a grid with the mirror line x equals 4.
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    Model answer

    (a) The vertices of \(P\) are \((1, 1)\), \((3, 1)\) and \((1, 4)\), which are 3, 1 and 3 units to the left of \(x = 4\). So \(Q\) has vertices \((7, 1)\), \((5, 1)\) and \((7, 4)\). (b) Adding 2 to each \(x\) and 3 to each \(y\) gives \(R\) with vertices \((3, 4)\), \((5, 4)\) and \((3, 7)\).

    Mark scheme

    • (a) Reflects at least two vertices correctly — M1
    • (a) \(Q\) with vertices \((7, 1)\), \((5, 1)\) and \((7, 4)\) — A1
    • (b) Translates at least two vertices correctly — M1
    • (b) \(R\) with vertices \((3, 4)\), \((5, 4)\) and \((3, 7)\) — A1
  3. 3 Describe [3 marks]

    Triangle \(B\) is the image of triangle \(A\) after a single transformation. (a) Describe fully the single transformation that maps \(A\) onto \(B\). (2 marks) (b) Triangle \(A\) has an area of 3 square units. Write down the area of triangle \(B\). (1 mark)

    Triangle A and its image triangle B on a grid.
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    Model answer

    (a) Corresponding points are the same distance from the line \(y = 2\) on opposite sides, so it is a reflection in the line \(y = 2\). (b) A reflection does not change the size, so the area of \(B\) is also 3 square units.

    Mark scheme

    • (a) Reflection — B1
    • (a) In the line \(y = 2\) — B1
    • (b) 3 — B1
  4. 4 Write down [3 marks]

    \(A\) is the point \((2, 3)\). \(A\) is translated by the vector \(\begin{pmatrix} -4 \\ 1 \end{pmatrix}\) to give the point \(B\). \(B\) is reflected in the \(x\)-axis to give the point \(C\). Write down the coordinates of (a) \(B\), (b) \(C\). (3 marks)

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    Model answer

    (a) \(B = (2 - 4, 3 + 1) = (-2, 4)\). (b) A reflection in the \(x\)-axis changes the sign of the \(y\)-coordinate, so \(C = (-2, -4)\).

    Mark scheme

    • (a) \((-2, 4)\) — B1
    • (b) Changes the sign of the \(y\)-coordinate of their \(B\) — M1
    • (b) \((-2, -4)\) — A1 (follow through from (a))
  5. 5 Work out [3 marks]

    A translation maps the point \((-1, 4)\) onto the point \((5, -2)\). (a) Write down the column vector of the translation. (2 marks) (b) The same translation maps \((3, 3)\) onto the point \(Q\). Write down the coordinates of \(Q\). (1 mark)

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    Model answer

    (a) The change in \(x\) is \(5 - (-1) = 6\) and the change in \(y\) is \(-2 - 4 = -6\), so the vector is \(\begin{pmatrix} 6 \\ -6 \end{pmatrix}\). (b) \(Q = (3 + 6, 3 - 6) = (9, -3)\).

    Mark scheme

    • (a) \(5 - (-1)\) or \(-2 - 4\) — M1
    • (a) \(\begin{pmatrix} 6 \\ -6 \end{pmatrix}\) — A1
    • (b) \((9, -3)\) — B1 (follow through from (a))
  6. 6 Work out [3 marks]

    The point \(P\) is \((3, 1)\). \(P\) is reflected in the line \(y = x\) to give \(Q\). \(Q\) is reflected in the \(y\)-axis to give \(R\). Write down the coordinates of \(R\).

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    Model answer

    Reflecting in \(y = x\) swaps the coordinates, so \(Q = (1, 3)\). Reflecting in the \(y\)-axis changes the sign of \(x\), so \(R = (-1, 3)\).

    Mark scheme

    • \(Q = (1, 3)\) — B1
    • Changes the sign of the \(x\)-coordinate of their \(Q\) — M1
    • \((-1, 3)\) — A1

Quick check

  1. 1

    What is the image of the point \((3, 5)\) in the \(x\)-axis?

    1. A\((-3, 5)\)
    2. B\((3, -5)\)
    3. C\((5, 3)\)
    4. D\((-3, -5)\)
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    B: \((3, -5)\)

    A reflection in the \(x\)-axis changes the sign of the \(y\)-coordinate.

  2. 2

    What is the image of the point \((2, 3)\) in the \(y\)-axis?

    1. A\((-2, 3)\)
    2. B\((2, -3)\)
    3. C\((3, 2)\)
    4. D\((-2, -3)\)
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    A: \((-2, 3)\)

    A reflection in the \(y\)-axis changes the sign of the \(x\)-coordinate.

  3. 3

    What is the image of \((4, 1)\) in the line \(y = x\)?

    1. A\((-1, -4)\)
    2. B\((4, -1)\)
    3. C\((-4, 1)\)
    4. D\((1, 4)\)
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    D: \((1, 4)\)

    In the line \(y = x\) the coordinates swap.

  4. 4

    The point \((5, 2)\) is reflected in the line \(x = 3\). What is the image?

    1. A\((-5, 2)\)
    2. B\((3, 2)\)
    3. C\((1, 2)\)
    4. D\((1, -2)\)
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    C: \((1, 2)\)

    \((5, 2)\) is 2 right of the line, so the image is 2 left of it, at \((1, 2)\).

  5. 5

    What does the column vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

    1. A2 right and 5 up
    2. B2 left and 5 up
    3. C5 left and 2 up
    4. D2 left and 5 down
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    B: 2 left and 5 up

    The top number is left or right, and the bottom number is up or down.

  6. 6

    The point \((4, 1)\) is translated by \(\begin{pmatrix} -3 \\ 2 \end{pmatrix}\). What is the image?

    1. A\((1, 3)\)
    2. B\((7, 3)\)
    3. C\((1, -1)\)
    4. D\((7, -1)\)
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    A: \((1, 3)\)

    \((4 - 3, 1 + 2) = (1, 3)\).

  7. 7

    A translation takes \((2, 5)\) to \((7, 3)\). What is the column vector?

    1. A\(\begin{pmatrix} -5 \\ 2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 9 \\ 8 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\)
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    D: \(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\)

    The change in \(x\) is \(7 - 2 = 5\) and the change in \(y\) is \(3 - 5 = -2\).

  8. 8

    What must you give to describe a reflection fully?

    1. AThe scale factor
    2. BThe centre and the angle
    3. CThe equation of the mirror line
    4. DA column vector
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    C: The equation of the mirror line

    A reflection is described by its mirror line, such as \(x = 1\).

  9. 9

    What is the image of \((4, 1)\) in the line \(y = -x\)?

    1. A\((1, 4)\)
    2. B\((-1, -4)\)
    3. C\((-4, -1)\)
    4. D\((-1, 4)\)
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    B: \((-1, -4)\)

    In the line \(y = -x\) the coordinates swap and both change sign.