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Exam questions · Maths · Graphs

Cubic, Reciprocal and Other Graphs

  • 6 exam questions
  • 16 marks
  • 9 quick checks
  1. 1 Match [3 marks]

    Here are three graphs, \(A\), \(B\) and \(C\). The three equations are \(y = \dfrac{1}{x}\), \(y = x^3\) and \(y = x^2 - 1\). Match each equation to the correct graph.

    Three graphs: an S-shaped curve, a curve in two branches and a U-shaped curve.
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    Model answer

    Graph \(A\) is an S-shaped curve through the origin, so it is \(y = x^3\). Graph \(B\) has two branches, so it is \(y = \dfrac{1}{x}\). Graph \(C\) is a U shape with its lowest point at \((0, -1)\), so it is \(y = x^2 - 1\).

    Mark scheme

    • \(A\) is \(y = x^3\) — B1
    • \(B\) is \(y = \dfrac{1}{x}\) — B1
    • \(C\) is \(y = x^2 - 1\) — B1
  2. 2 Complete [2 marks]

    Complete the table of values for \(y = x^3\). \(x = -2, -1, 0, 1, 2\)

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    Model answer

    The values are \(-8, -1, 0, 1, 8\).

    Mark scheme

    • At least three correct values — M1
    • \(-8, -1, 0, 1, 8\) — A1
  3. 3 Complete [3 marks]

    (a) Complete the table of values for \(y = \dfrac{8}{x}\). \(x = 1, 2, 4, 8, -2, -4\) (2 marks) (b) Explain why there is no point on the graph with \(x = 0\). (1 mark)

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    Model answer

    (a) The values are \(8, 4, 2, 1, -4, -2\). (b) You cannot divide by zero, so \(\dfrac{8}{0}\) has no value.

    Mark scheme

    • (a) At least four correct values — M1
    • (a) \(8, 4, 2, 1, -4, -2\) — A1
    • (b) You cannot divide by zero — C1
  4. 4 Complete [3 marks]

    (a) Complete the table of values for \(y = 2^x\). \(x = -1, 0, 1, 2, 3\) (2 marks) (b) Write down the coordinates of the point where the graph of \(y = 2^x\) crosses the \(y\)-axis. (1 mark)

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    Model answer

    (a) The values are \(\dfrac{1}{2}, 1, 2, 4, 8\). (b) When \(x = 0\), \(y = 2^0 = 1\), so the point is \((0, 1)\).

    Mark scheme

    • (a) At least three correct values — M1
    • (a) \(\dfrac{1}{2}, 1, 2, 4, 8\) — A1
    • (b) \((0, 1)\) — B1
  5. 5 Work out [2 marks]

    Work out the value of \(y\) when \(x = -3\) for \(y = x^3 - x\).

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    Model answer

    \((-3)^3 - (-3) = -27 + 3 = -24\).

    Mark scheme

    • \((-3)^3 = -27\) — M1
    • \(-24\) — A1
  6. 6 Show that [3 marks]

    (a) Write down the equation of the circle with centre \((0, 0)\) and radius 6. (1 mark) (b) Show that the point \((4, 5)\) lies on the circle \(x^2 + y^2 = 41\). (2 marks)

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    Model answer

    (a) \(x^2 + y^2 = 36\). (b) \(4^2 + 5^2 = 16 + 25 = 41\), so the point lies on the circle.

    Mark scheme

    • (a) \(x^2 + y^2 = 36\) — B1
    • (b) \(4^2 + 5^2\) or \(16 + 25\) — M1
    • (b) 41 with a conclusion — C1

Quick check

  1. 1

    What shape is the graph of \(y = x^3\)?

    1. AAn S-shaped curve through the origin
    2. BA U-shaped curve
    3. CTwo separate branches
    4. DA straight line
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    A: An S-shaped curve through the origin

    Cubic graphs have an S shape.

  2. 2

    What is special about the graph of \(y = \dfrac{1}{x}\)?

    1. AIt passes through the origin
    2. BIt is a straight line
    3. CIt is a closed curve
    4. DIt has two branches and never touches either axis
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    D: It has two branches and never touches either axis

    You cannot divide by 0, and \(\dfrac{1}{x}\) is never 0.

  3. 3

    Where does the graph of \(y = 3^x\) cross the \(y\)-axis?

    1. A\((0, 3)\)
    2. B\((0, 0)\)
    3. C\((0, 1)\)
    4. D\((1, 0)\)
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    C: \((0, 1)\)

    \(3^0 = 1\).

  4. 4

    What is the value of \(x^3\) when \(x = -3\)?

    1. A\(27\)
    2. B\(-27\)
    3. C\(-9\)
    4. D\(9\)
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    B: \(-27\)

    \((-3) \times (-3) \times (-3) = -27\).

  5. 5

    Which of these equations gives a cubic graph?

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

    A cubic has \(x^3\) as its highest power.

  6. 6

    Work out \(y\) when \(x = 2\) on \(y = x^3 - 3x\).

    1. A\(14\)
    2. B\(-2\)
    3. C\(6\)
    4. D\(2\)
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    D: \(2\)

    \(8 - 6 = 2\).

  7. 7

    What is \(2^3\)?

    1. A\(6\)
    2. B\(9\)
    3. C\(8\)
    4. D\(5\)
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    C: \(8\)

    \(2 \times 2 \times 2 = 8\).

  8. 8

    What shape is the graph of \(y = -x^2\)?

    1. AA U shape
    2. BAn upside-down U
    3. CAn S shape
    4. DA straight line
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    B: An upside-down U

    A negative \(x^2\) term turns the parabola upside down.

  9. 9

    What is the radius of the circle \(x^2 + y^2 = 36\)?

    1. A\(6\)
    2. B\(36\)
    3. C\(18\)
    4. D\(72\)
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    A: \(6\)

    The radius is \(\sqrt{36} = 6\).