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Date May 2010 Marks available 6 Reference code 10M.1.sl.TZ1.8
Level SL only Paper 1 Time zone TZ1
Command term Write down Question number 8 Adapted from N/A

Question

Let \(f(x) = \frac{1}{2}{x^3} - {x^2} - 3x\) . Part of the graph of f is shown below.


There is a maximum point at A and a minimum point at B(3, − 9) .

Find the coordinates of A.

[8]
a.

Write down the coordinates of

(i)     the image of B after reflection in the y-axis;

(ii)    the image of B after translation by the vector \(\left( {\begin{array}{*{20}{c}}
{ - 2}\\
5
\end{array}} \right)\) ;

(iii)   the image of B after reflection in the x-axis followed by a horizontal stretch with scale factor \(\frac{1}{2}\) .

[6]
b(i), (ii) and (iii).

Markscheme

\(f(x) = {x^2} - 2x - 3\)     A1A1A1

evidence of solving \(f'(x) = 0\)     (M1)

e.g. \({x^2} - 2x - 3 = 0\)

evidence of correct working     A1

e.g. \((x + 1)(x - 3)\) ,  \(\frac{{2 \pm \sqrt {16} }}{2}\)

\(x =  - 1\) (ignore \(x = 3\) )     (A1)

evidence of substituting their negative x-value into \(f(x)\)     (M1)

e.g. \(\frac{1}{3}{( - 1)^3} - {( - 1)^2} - 3( - 1)\) , \( - \frac{1}{3} - 1 + 3\)

\(y = \frac{5}{3}\)     A1

coordinates are \(\left( { - 1,\frac{5}{3}} \right)\)     N3

[8 marks]

a.

(i) \(( - 3{\text{, }} - 9)\)     A1     N1

(ii) \((1{\text{, }} - 4)\)     A1A1    N2

(iii) reflection gives \((3{\text{, }}9)\)     (A1)

stretch gives \(\left( {\frac{3}{2}{\text{, }}9} \right)\)     A1A1     N3

[6 marks]

b(i), (ii) and (iii).

Examiners report

A majority of candidates answered part (a) completely.

a.

Candidates were generally successful in finding images after single transformations in part (b). Common incorrect answers for (biii) included \(\left( {\frac{3}{2},\frac{9}{2}} \right)\) , (6, 9) and (6, 18) , demonstrating difficulty with images from horizontal stretches.

b(i), (ii) and (iii).

Syllabus sections

Topic 2 - Functions and equations » 2.3 » Transformations of graphs.
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