18M.1.SL.TZ1.T_12

pestleMathematicsAISLPaper 118M· sl-2-5-modelling-functionssource ↗

Consider the quadratic function $f\left( x \right) = a{x^2} + bx + 22$ .

The equation of the line of symmetry of the graph $y = f\left( x \right){\text{ is }}x = 1.75$ .

The graph intersects the x-axis at the point (−2 , 0).

Using only this information, write down an equation in terms of a and b.

[1]

Using this information, write down a second equation in terms of a and b.

[1]

Hence find the value of a and of b.

[2]

The graph intersects the x-axis at a second point, P.

Find the x-coordinate of P.

[2]

Markscheme / solution

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

$1.75 = \frac{{ - b}}{{2a}}$ (or equivalent) (A1) (C1)

Note: Award (A1) for $f\left( x \right) = {\left( {1.75} \right)^2}a + 1.75b$ or for $y = {\left( {1.75} \right)^2}a + 1.75b + 22$ or for $f\left( {1.75} \right) = {\left( {1.75} \right)^2}a + 1.75b + 22$ .

[1 mark]

${\left( { - 2} \right)^2} \times a + \left( { - 2} \right) \times b + 22 = 0$ (or equivalent) (A1) (C1)

Note: Award (A1) for ${\left( { - 2} \right)^2} \times a + \left( { - 2} \right) \times b + 22 = 0$ seen.

Award (A0) for $y = {\left( { - 2} \right)^2} \times a + \left( { - 2} \right) \times b + 22$ .

[1 mark]

a = −2, b = 7 (A1)(ft)(A1)(ft) (C2)

Note: Follow through from parts (a) and (b).
Accept answers(s) embedded as a coordinate pair.

[2 marks]

−2x² + 7x + 22 = 0 (M1)

Note: Award (M1) for correct substitution of a and b into equation and setting to zero. Follow through from part (c).

(x =) 5.5 (A1)(ft) (C2)

Note: Follow through from parts (a) and (b).

x-coordinate = 1.75 + (1.75 − (−2)) (M1)

Note: Award (M1) for correct use of axis of symmetry and given intercept.

(x =) 5.5 (A1) (C2)

[2 marks]

Examiners’ report

[N/A]