Quadratic Graphs

1. 1 Complete [3 marks]

   (a) Complete the table of values for \(y = x^2 - 3x\). \(x = -1, 0, 1, 2, 3, 4\) (2 marks) (b) Write down the equation of the line of symmetry of the graph. (1 mark)

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   Model answer

   (a) The values are \(4, 0, -2, -2, 0, 4\). (b) The line of symmetry is halfway between \(x = 0\) and \(x = 3\), so \(x = 1.5\).

   Mark scheme

   • (a) At least three correct values — M1
   • (a) \(4, 0, -2, -2, 0, 4\) — A1
   • (b) \(x = 1.5\) — B1
2. 2 Write down [4 marks]

   The diagram shows the graph of \(y = x^2 - 4x + 3\). (a) Write down the coordinates of the turning point. (1 mark) (b) Use the graph to solve \(x^2 - 4x + 3 = 0\). (2 marks) (c) Write down the equation of the line of symmetry of the graph. (1 mark)

   

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   Model answer

   (a) The lowest point is \((2, -1)\). (b) The curve crosses the \(x\)-axis at \(x = 1\) and \(x = 3\). (c) The line of symmetry goes through the turning point, so \(x = 2\).

   Mark scheme

   • (a) \((2, -1)\) — B1
   • (b) One of \(x = 1\) or \(x = 3\) — M1
   • (b) \(x = 1\) and \(x = 3\) — A1
   • (c) \(x = 2\) — B1
3. 3 Work out [3 marks]

   (a) Write down the \(y\)-intercept of the graph of \(y = x^2 + 5x - 6\). (1 mark) (b) Solve \(x^2 + 5x - 6 = 0\) to find where the graph crosses the \(x\)-axis. (2 marks)

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   Model answer

   (a) Put \(x = 0\): \(y = -6\). (b) \(x^2 + 5x - 6 = (x + 6)(x - 1) = 0\), so \(x = -6\) and \(x = 1\).

   Mark scheme

   • (a) \(-6\) — B1
   • (b) \((x + 6)(x - 1)\) — M1
   • (b) \(x = -6\) and \(x = 1\) — A1
4. 4 Work out [3 marks]

   A curve has equation \(y = x^2 - 6x + 5\). (a) Work out the coordinates of the points where the curve crosses the \(x\)-axis. (2 marks) (b) Write down the equation of the line of symmetry of the curve. (1 mark)

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   Model answer

   (a) \(x^2 - 6x + 5 = (x - 1)(x - 5) = 0\), so the points are \((1, 0)\) and \((5, 0)\). (b) The line of symmetry is halfway between them: \(x = 3\).

   Mark scheme

   • (a) \((x - 1)(x - 5)\) — M1
   • (a) \((1, 0)\) and \((5, 0)\) — A1
   • (b) \(x = 3\) — B1
5. 5 Work out [4 marks]

   (a) Write \(x^2 - 8x + 7\) in the form \((x - a)^2 - b\). (2 marks) (b) Write down the coordinates of the turning point of the graph of \(y = x^2 - 8x + 7\). (1 mark) (c) Write down the equation of its line of symmetry. (1 mark)

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   Model answer

   (a) \(x^2 - 8x + 7 = (x - 4)^2 - 16 + 7 = (x - 4)^2 - 9\). (b) The turning point is \((4, -9)\). (c) The line of symmetry is \(x = 4\).

   Mark scheme

   • (a) \((x - 4)^2\) seen — M1
   • (a) \((x - 4)^2 - 9\) — A1
   • (b) \((4, -9)\) — B1
   • (c) \(x = 4\) — B1
6. 6 Work out [3 marks]

   The equation of a curve is \(y = 4x - x^2\). Work out the coordinates of the maximum point of the curve.

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   Model answer

   The curve crosses the \(x\)-axis where \(x(4 - x) = 0\), at \(x = 0\) and \(x = 4\). The maximum is halfway between, at \(x = 2\), and \(y = 8 - 4 = 4\). The point is \((2, 4)\).

   Mark scheme

   • Roots 0 and 4 found, or \(x = 2\) seen — M1
   • \(y = 4 \times 2 - 2^2\) — M1
   • \((2, 4)\) — 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 \((a, b)\).