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Exam questions · Maths · Graphs

Cubic, Reciprocal and Other Graphs

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

    The diagram shows three graphs, \(A\), \(B\) and \(C\). The three equations are \(y = x^2\), \(y = 2x - 1\) and \(y = x^3 - 3x\). Match each equation to the correct graph. [3 marks]

    Three graphs: an S-shaped curve with two turning points, a U-shaped curve touching the origin and an upward-sloping line.
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    Model answer

    Graph \(A\) is an S shape with a hill and a valley, so \(y = x^3 - 3x\). Graph \(B\) is a U shape touching the origin, so \(y = x^2\). Graph \(C\) is a straight line, so \(y = 2x - 1\).

    Mark scheme

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

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

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    Model answer

    The values are \(-6, 0, 0, 0, 6\).

    Mark scheme

    • At least three correct values — M1
    • \(-6, 0, 0, 0, 6\) — A1
  3. 3 Complete [3 marks]

    (a) Complete the table of values for \(y = \dfrac{12}{x}\). \(x = 1, 2, 3, 4, 6, 12\) [2 marks] (b) For which value of \(x\) is the graph of \(y = \dfrac{12}{x}\) not defined? [1 mark]

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    Model answer

    (a) The values are \(12, 6, 4, 3, 2, 1\). (b) It is not defined at \(x = 0\), because you cannot divide by zero.

    Mark scheme

    • (a) At least four correct values — M1
    • (a) \(12, 6, 4, 3, 2, 1\) — A1
    • (b) \(x = 0\) — B1
  4. 4 Calculate [3 marks]

    A population of bacteria is 100 at the start. It doubles every hour, so after \(t\) hours it is \(P = 100 \times 2^t\). (a) Calculate \(P\) when \(t = 3\). [1 mark] (b) Calculate \(P\) when \(t = 5\). [1 mark] (c) Explain why the graph of \(P\) against \(t\) never touches the \(t\)-axis. [1 mark]

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    Model answer

    (a) \(100 \times 8 = 800\). (b) \(100 \times 32 = 3200\). (c) \(2^t\) is always greater than 0, so \(P\) is never 0.

    Mark scheme

    • (a) 800 — B1
    • (b) 3200 — B1
    • (c) \(2^t\) is never zero, so the population is never 0 — C1
  5. 5 Calculate [3 marks]

    A curve has equation \(y = x^2 - 9\). (a) Calculate the coordinates of the points where the curve crosses the \(x\)-axis. [2 marks] (b) Write down the \(y\)-intercept. [1 mark]

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    Model answer

    (a) When \(y = 0\), \(x^2 = 9\), so \(x = 3\) or \(x = -3\). The points are \((3, 0)\) and \((-3, 0)\). (b) The \(y\)-intercept is \(-9\).

    Mark scheme

    • (a) \(x^2 = 9\) — M1
    • (a) \((3, 0)\) and \((-3, 0)\) — A1
    • (b) \(-9\) — B1
  6. 6 Show that [3 marks]

    A circle has equation \(x^2 + y^2 = 13\). (a) Write down the radius of the circle in surd form. [1 mark] (b) Show that the point \((2, 3)\) lies on the circle. [2 marks]

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    Model answer

    (a) The radius is \(\sqrt{13}\). (b) \(2^2 + 3^2 = 4 + 9 = 13\), so the point lies on the circle.

    Mark scheme

    • (a) \(\sqrt{13}\) — B1
    • (b) \(2^2 + 3^2\) or \(4 + 9\) — M1
    • (b) 13 with a conclusion — A1

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 \(\sqrt{36} = 6\).