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Exam questions · Maths · Functions, Sequences and Rates of Change

Transformations of Graphs

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

    The graph of \(y = f(x)\) is shown. The turning point is \((3, -4)\). (a) Write down the coordinates of the turning point of the graph of \(y = f(x) - 2\). [1 mark] (b) Write down the coordinates of the turning point of the graph of \(y = f(x - 2)\). [1 mark]

    The graph of y equals f of x with its turning point marked.
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    Model answer

    (a) \((3, -6)\). (b) \((5, -4)\).

    Mark scheme

    • (a) \((3, -6)\) — B1
    • (b) \((5, -4)\) — B1
  2. 2 Write down [2 marks]

    The graph of \(y = g(x)\) has a minimum point at \((-2, -5)\). (a) Write down the coordinates of the maximum point of \(y = -g(x)\). [1 mark] (b) Write down the coordinates of the minimum point of \(y = g(-x)\). [1 mark]

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    Model answer

    (a) \((-2, 5)\). (b) \((2, -5)\).

    Mark scheme

    • (a) \((-2, 5)\) — B1
    • (b) \((2, -5)\) — B1
  3. 3 Write down [3 marks]

    The graph of \(y = x^2\) is transformed. Write down the equation of the new graph after (a) a translation of 4 units to the left, [1 mark] (b) a translation of 3 units down, [1 mark] (c) a translation by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\). [1 mark]

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    Model answer

    (a) \(y = (x + 4)^2\). (b) \(y = x^2 - 3\). (c) \(y = (x - 2)^2 - 3\).

    Mark scheme

    • (a) \(y = (x + 4)^2\) — B1
    • (b) \(y = x^2 - 3\) — B1
    • (c) \(y = (x - 2)^2 - 3\) — B1
  4. 4 Write down [4 marks]

    The graph of \(y = f(x)\) has a maximum point at \((-2, 6)\). Write down the coordinates of the maximum point of the graph of (a) \(y = f(x + 3)\) [1 mark] (b) \(y = f(x) + 1\) [1 mark] (c) \(y = -f(x)\) [1 mark] (d) \(y = f(-x)\) [1 mark]

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    Model answer

    (a) \((-5, 6)\). (b) \((-2, 7)\). (c) \((-2, -6)\), which is now a minimum. (d) \((2, 6)\).

    Mark scheme

    • (a) \((-5, 6)\) — B1
    • (b) \((-2, 7)\) — B1
    • (c) \((-2, -6)\) — B1
    • (d) \((2, 6)\) — B1
  5. 5 Write down [3 marks]

    The diagram shows the graph of \(y = \sin x\) and a transformation of it, for \(0^\circ \le x \le 360^\circ\). (a) Write down the equation of the transformed graph. [1 mark] (b) Describe fully the single transformation. [2 marks]

    The graph of y equals sin x and a transformed curve, a sine curve moved 90 degrees to the right.
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    Model answer

    (a) \(y = \sin(x - 90^\circ)\). (b) A translation by the vector \(\begin{pmatrix} 90 \\ 0 \end{pmatrix}\), which is \(90^\circ\) to the right.

    Mark scheme

    • (a) \(y = \sin(x - 90^\circ)\) — B1
    • (b) Translation — B1
    • (b) Vector \(\begin{pmatrix} 90 \\ 0 \end{pmatrix}\), or \(90^\circ\) to the right — B1
  6. 6 Work out [4 marks]

    \(f(x) = x^2 - 6x + 5\). Solve \(f(x - 2) = 0\). [4 marks]

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    Model answer

    \(f(x) = (x - 1)(x - 5)\), so \(f(x - 2) = (x - 3)(x - 7)\). So \(x = 3\) or \(x = 7\).

    Mark scheme

    • \(f(x) = (x - 1)(x - 5)\) — B1
    • \(f(x - 2) = (x - 2 - 1)(x - 2 - 5)\) or \((x - 3)(x - 7)\) — M1
    • \(x = 3\) — A1
    • \(x = 7\) — A1

Quick check

  1. 1

    What does \(y = f(x) + 3\) do to the graph of \(y = f(x)\)?

    1. AMoves it right 3
    2. BMoves it up 3
    3. CMoves it left 3
    4. DMoves it down 3
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    B: Moves it up 3

    Adding to the function moves the graph up.

  2. 2

    What does \(y = f(x + 2)\) do to the graph of \(y = f(x)\)?

    1. AMoves it left 2
    2. BMoves it right 2
    3. CMoves it up 2
    4. DMoves it down 2
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    A: Moves it left 2

    A plus inside the bracket moves the graph left.

  3. 3

    What does \(y = -f(x)\) do to the graph of \(y = f(x)\)?

    1. AReflects it in the \(y\)-axis
    2. BMoves it down
    3. CTurns it by \(180^\circ\) about the origin
    4. DReflects it in the \(x\)-axis
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    D: Reflects it in the \(x\)-axis

    A minus outside changes the \(y\)-values.

  4. 4

    What does \(y = f(-x)\) do to the graph of \(y = f(x)\)?

    1. AReflects it in the \(x\)-axis
    2. BMoves it left
    3. CReflects it in the \(y\)-axis
    4. DMoves it down
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    C: Reflects it in the \(y\)-axis

    A minus inside changes the \(x\)-values.

  5. 5

    The maximum of \(y = f(x)\) is at \((3, 5)\). Where is the maximum of \(y = f(x - 2)\)?

    1. A\((1, 5)\)
    2. B\((5, 5)\)
    3. C\((3, 7)\)
    4. D\((3, 3)\)
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    B: \((5, 5)\)

    The graph moves right 2.

  6. 6

    The maximum of \(y = f(x)\) is at \((3, 5)\). Where is the turning point of \(y = -f(x)\)?

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

    The \(y\)-coordinate changes sign.

  7. 7

    What is the equation of \(y = x^2\) after a translation of 3 units to the right?

    1. A\(y = (x + 3)^2\)
    2. B\(y = x^2 + 3\)
    3. C\(y = x^2 - 3\)
    4. D\(y = (x - 3)^2\)
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    D: \(y = (x - 3)^2\)

    Moving right replaces \(x\) with \(x - 3\).

  8. 8

    What is the maximum value of \(y = \sin x + 1\)?

    1. A1
    2. B0
    3. C2
    4. D\(-1\)
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    C: 2

    The sine graph is moved up by 1, so its maximum is \(1 + 1 = 2\).

  9. 9

    Which equation gives the same graph as \(y = \cos x\)?

    1. A\(y = \sin(x - 90^\circ)\)
    2. B\(y = \sin(x + 90^\circ)\)
    3. C\(y = -\sin x\)
    4. D\(y = \sin x + 90\)
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    B: \(y = \sin(x + 90^\circ)\)

    Moving the sine graph left by \(90^\circ\) gives the cosine graph.