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Exam questions · Maths · Graphs

Quadratic Graphs

  • 6 exam questions
  • 17 marks
  • 9 quick checks
  1. 1 Complete [2 marks]

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

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    Model answer

    The values are \(3, 0, -1, 0, 3\).

    Mark scheme

    • At least three correct values — M1
    • \(3, 0, -1, 0, 3\) — A1
  2. 2 Write down [4 marks]

    The diagram shows the graph of \(y = 4x - x^2\). (a) Write down the coordinates of the maximum point. [1 mark] (b) Use the graph to solve \(4x - x^2 = 0\). [2 marks] (c) Write down the equation of the line of symmetry. [1 mark]

    The graph of the curve y equals 4x minus x squared.
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    Model answer

    (a) The highest point is \((2, 4)\). (b) The curve crosses the \(x\)-axis at \(x = 0\) and \(x = 4\). (c) The line of symmetry is \(x = 2\).

    Mark scheme

    • (a) \((2, 4)\) — B1
    • (b) One of \(x = 0\) or \(x = 4\) — M1
    • (b) \(x = 0\) and \(x = 4\) — A1
    • (c) \(x = 2\) — B1
  3. 3 Find [3 marks]

    A curve has equation \(y = x^2 + 2x - 3\). (a) Write down the coordinates of the point where the curve crosses the \(y\)-axis. [1 mark] (b) Solve \(x^2 + 2x - 3 = 0\) to find where the curve crosses the \(x\)-axis. [2 marks]

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    Model answer

    (a) \((0, -3)\). (b) \(x^2 + 2x - 3 = (x + 3)(x - 1) = 0\), so \(x = -3\) and \(x = 1\).

    Mark scheme

    • (a) \((0, -3)\) — B1
    • (b) \((x + 3)(x - 1)\) — M1
    • (b) \(x = -3\) and \(x = 1\) — A1
  4. 4 Find [3 marks]

    The curve \(y = x^2 - 4x + k\) has its turning point at \((2, -1)\). Find the value of \(k\).

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    Model answer

    At the turning point \(x = 2\) and \(y = -1\), so \(-1 = 2^2 - 4 \times 2 + k = -4 + k\) and \(k = 3\).

    Mark scheme

    • \(x = 2\) and \(y = -1\) substituted — M1
    • \(-1 = 4 - 8 + k\) — M1
    • 3 — A1
  5. 5 Calculate [3 marks]

    (a) Write \(x^2 - 10x + 7\) in the form \((x - a)^2 + b\). [2 marks] (b) Write down the coordinates of the turning point of the graph of \(y = x^2 - 10x + 7\). [1 mark]

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    Model answer

    (a) \((x - 5)^2 - 25 + 7 = (x - 5)^2 - 18\). (b) The turning point is \((5, -18)\).

    Mark scheme

    • (a) \((x - 5)^2\) seen — M1
    • (a) \((x - 5)^2 - 18\) — A1
    • (b) \((5, -18)\) — B1
  6. 6 Find [2 marks]

    The graph of \(y = x^2 - 4x + 3\) crosses the \(x\)-axis at \(x = 1\) and \(x = 3\). Write down the values of \(x\) for which \(y\) is negative.

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    Model answer

    The graph is a U shape, so it is below the \(x\)-axis between the roots: \(1 < x < 3\).

    Mark scheme

    • Between the roots 1 and 3 identified — M1
    • \(1 < x < 3\) — A1

Quick check

  1. 1

    What is the shape of the graph of a quadratic equation?

    1. AA straight line
    2. BAn S-shaped curve
    3. CTwo separate branches
    4. DA parabola, a smooth U or upside-down U
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    D: A parabola, a smooth U or upside-down U

    Quadratic graphs are parabolas.

  2. 2

    What are the roots of a graph?

    1. AThe \(y\)-values where the curve crosses the \(y\)-axis
    2. BThe highest point of the curve
    3. CThe \(x\)-values where the curve crosses the \(x\)-axis
    4. DThe line of symmetry
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    C: The \(x\)-values where the curve crosses the \(x\)-axis

    At the roots \(y = 0\), so the curve meets the \(x\)-axis.

  3. 3

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

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

    \((-2)^2 - 2 \times (-2) - 3 = 4 + 4 - 3 = 5\).

  4. 4

    What is the \(y\)-intercept of \(y = x^2 + 4x - 7\)?

    1. A\(-7\)
    2. B\(7\)
    3. C\(4\)
    4. D\(0\)
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    A: \(-7\)

    Put \(x = 0\): \(y = -7\).

  5. 5

    The graph of \(y = x^2 - 2x - 3\) crosses the \(x\)-axis at \(-1\) and 3. What are the solutions of \(x^2 - 2x - 3 = 0\)?

    1. A\(x = 1\) and \(x = -3\)
    2. B\(x = -1\) and \(x = -3\)
    3. C\(x = 0\) and \(x = 2\)
    4. D\(x = -1\) and \(x = 3\)
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    D: \(x = -1\) and \(x = 3\)

    The solutions are the \(x\)-values where \(y = 0\).

  6. 6

    A parabola crosses the \(x\)-axis at \(x = 1\) and \(x = 5\). What is its line of symmetry?

    1. A\(x = 2.5\)
    2. B\(x = 5\)
    3. C\(x = 3\)
    4. D\(x = -3\)
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    C: \(x = 3\)

    The line of symmetry is halfway between the roots: \(\dfrac{1 + 5}{2} = 3\).

  7. 7

    The curve \(y = x^2 - 4x + 3\) crosses the \(x\)-axis at 1 and 3. What are the coordinates of its turning point?

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

    \(x = 2\) is halfway between the roots. Then \(y = 4 - 8 + 3 = -1\).

  8. 8

    Which line do you draw on the graph of \(y = x^2 - 2x - 3\) to solve \(x^2 - 2x - 3 = 2\)?

    1. A\(y = 2\)
    2. B\(x = 2\)
    3. C\(y = -3\)
    4. D\(y = 0\)
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    A: \(y = 2\)

    Solutions are where the curve meets the horizontal line \(y = 2\).

  9. 9

    What are the coordinates of the turning point of \(y = (x - 3)^2 - 4\)?

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

    In \(y = (x - a)^2 + b\) the turning point is \((a, b)\).