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Date May 2010 Marks available 3 Reference code 10M.1.sl.TZ2.1
Level SL only Paper 1 Time zone TZ2
Command term Find 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]
a.

Write down the equation of the axis of symmetry.

[1]
b.

Find the value of p.

[3]
c.

Markscheme

\(q = - 2\) , \(r = 4\) or \(q = 4\) , \(r = - 2\)     A1A1     N2

[2 marks]

a.

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

[1 mark]

b.

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]

c.

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.

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.

b.

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

c.

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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