21M.1.SL.TZ2.12

pestleMathematicsAISLPaper 121M· sl-2-5-modelling-functionssource ↗

Consider the function $f (x) = a x^{2} + b x + c$ . The graph of $y = f (x)$ is shown in the diagram. The vertex of the graph has coordinates $(0.5, - 12.5)$ . The graph intersects the $x$ -axis at two points, $(- 2, 0)$ and $(p, 0)$ .

Find the value of $p$ .

[1]

Find the value of

(i) $a$ .

(ii) $b$ .

(iii) $c$ .

[5]

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

[1]

Markscheme / solution

$3$ A1

Note: Accept $(3, 0)$ seen.

[1 mark]

METHOD 1

$0 = 4 a - 2 b + c, 0 = 9 a + 3 b + c, - \frac{25}{2} = \frac{1}{4} a + \frac{1}{2} b + c$ (M1)(A1)

(i) $2$ A1

(ii) $- 2$ A1

(iii) $- 12$ A1

Note: Award the (M1)(A1) if at least one correct value is seen. Do not apply FT form part (a) if workings are not shown.

METHOD 2

$- 12.5 = a (0.5 + 2) (0.5 - 3)$ (M1)

(i) $a = 2$ A1

$0 = 2 \times {(3)}^{2} + 3 b + c$

$0 = 2 \times {(- 2)}^{2} + (- 2) b + c$ (M1)

(ii) $b = - 2$ A1

(iii) $c = - 12$ A1

[5 marks]

$x = 0.5$ A1

Note: Do not FT from their part (b), this is a contradiction with the diagram.

[1 mark]

Examiners’ report

[N/A]