16N.1.SL.TZ0.T_9

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The graph of the quadratic function $f(x) = c + bx - {x^2}$ intersects the $x$ -axis at the point ${\text{A}}( - 1,{\text{ }}0)$ and has its vertex at the point ${\text{B}}(3,{\text{ }}16)$ .

N16/5/MATSD/SP1/ENG/TZ0/09

Write down the equation of the axis of symmetry for this graph.

[2]

Find the value of $b$ .

[2]

Write down the range of $f(x)$ .

[2]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$x = 3$ (A1)(A1) (C2)

Note: Award (A1) for $x =$ constant, (A1) for the constant being 3.

The answer must be an equation.

[2 marks]

$\frac{{ - b}}{{2( - 1)}} = 3$ (M1)

Note: Award (M1) for correct substitution into axis of symmetry formula.

$b - 2x = 0$ (M1)

Note: Award (M1) for correctly differentiating and equating to zero.

$c + b( - 1) - {( - 1)^2} = 0$ (or equivalent)

$c + b(3) - {(3)^2} = 16$ (or equivalent) (M1)

Note: Award (M1) for correct substitution of $( - 1,{\text{ }}0)$ and $(3,{\text{ }}16)$ in the original quadratic function.

$(b = ){\text{ }}6$ (A1)(ft) (C2)

Note: Follow through from part (a).

[2 marks]

$( - \infty ,{\text{ 16]}}$ OR $] - \infty ,{\text{ }}16]$ (A1)(A1)

Note: Award (A1) for two correct interval endpoints, (A1) for left endpoint excluded and right endpoint included.

[2 marks]

Examiners’ report

[N/A]