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Date	May 2010	Marks available	1	Reference code	10M.1.sl.TZ2.1
Level	SL only	Paper	1	Time zone	TZ2
Command term	Write down	Question number	1	Adapted from	N/A

Question

Let $f(x) = p(x - q)(x - r)$ . Part of the graph of f is shown below.

The graph passes through the points (−2, 0), (0, − 4) and (4, 0) .

Write down the value of q and of r.

[2]

Write down the equation of the axis of symmetry.

[1]

Find the value of p.

[3]

Markscheme

$q = - 2$ , $r = 4$ or $q = 4$ , $r = - 2$ A1A1 N2

[2 marks]

$x = 1$ (must be an equation) A1 N1

[1 mark]

substituting $(0{\text{, }} - 4)$ into the equation (M1)

e.g. $- 4 = p(0 - ( - 2))(0 - 4)$ , $- 4 = p( - 4)(2)$

correct working towards solution (A1)

e.g. $- 4 = - 8p$

$p = \frac{4}{8}$ $\left( { = \frac{1}{2}} \right)$ A1 N2

[3 marks]

Examiners report

The majority of candidates were successful on some or all parts of this question, with some candidates using a mix of algebra and graphical reasoning and others ignoring the graph and working only algebraically. Some did not recognize that p and q are the roots of the quadratic function and hence gave the answers as 2 and $- 4$ .

A common error in part (b) was the absence of an equation. Some candidates wrote down the equation $x = \frac{{ - b}}{{2a}}$ but were not able to substitute correctly. Those students did not realize that the axis of symmetry is always halfway between the x-intercepts.

More candidates had trouble with part (c) with erroneous substitutions and simplification mistakes commonplace.

Syllabus sections

Topic 2 - Functions and equations » 2.4 » The quadratic function

$x \mapsto a{x^2} + bx + c$ : its graph,

$y$ -intercept

$(0, c)$ . Axis of symmetry.

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