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Date May 2011 Marks available 2 Reference code 11M.1.hl.TZ2.1
Level HL only Paper 1 Time zone TZ2
Command term Determine Question number 1 Adapted from N/A

Question

The quadratic function \(f(x) = p + qx - {x^2}\) has a maximum value of 5 when x = 3.

Find the value of p and the value of q .

[4]
a.

The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.

[2]
b.

Markscheme

METHOD 1

\(f'(x) = q - 2x = 0\)     M1

\(f'(3) = q - 6 = 0\)

q = 6     A1

f(3) = p + 18 − 9 = 5     M1

p = −4     A1 

METHOD 2

\(f(x) = - {(x - 3)^2} + 5\)     M1A1

\( = - {x^2} + 6x - 4\)

q = 6, p = −4     A1A1

[4 marks]

a.

\(g(x) = - 4 + 6(x - 3) - {(x - 3)^2}{\text{ }}( = - 31 + 12x - {x^2})\)     M1A1

Note: Accept any alternative form which is correct.

Award M1A0 for a substitution of (x + 3) .

 

[2 marks]

b.

Examiners report

In general candidates handled this question well although a number equated the derivative to the function value rather than zero. Most recognised the shift in the second part although a number shifted only the squared value and not both x values.

a.

In general candidates handled this question well although a number equated the derivative to the function value rather than zero. Most recognised the shift in the second part although a number shifted only the squared value and not both x values.

b.

Syllabus sections

Topic 2 - Core: Functions and equations » 2.3 » Transformations of graphs: translations; stretches; reflections in the axes.

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