Express \(4{x^2} - 4x + 5\) in the form \(a{(x - h)^2} + k\) where a, h, \(k \in \mathbb{Q}\).

The graph of \(y = {x^2}\) is transformed onto the graph of \(y = 4{x^2} - 4x + 5\). Describe a sequence of transformations that does this, making the order of transformations clear.

Sketch the graph of \(y = f(x)\).

Find the range of f.

By using a suitable substitution show that \(\int {f(x){\text{d}}x = \frac{1}{4}\int {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \).

Prove that \(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{\pi }{{16}}} \).

\(4{(x - 0.5)^2} + 4\)     A1A1

Note: A1 for two correct parameters, A2 for all three correct.

[2 marks]

translation \(\left( {\begin{array}{*{20}{c}}
  {0.5} \\
  0
\end{array}} \right)\) (allow “0.5 to the right”)     A1

stretch parallel to y-axis, scale factor 4 (allow vertical stretch or similar)     A1

translation \(\left( {\begin{array}{*{20}{c}}
  0 \\
  4
\end{array}} \right)\) (allow “4 up”)     A1

Note: All transformations must state magnitude and direction.

Note: First two transformations can be in either order.

It could be a stretch followed by a single translation of \(\left( {\begin{array}{*{20}{c}}
  {0.5} \\
  4
\end{array}} \right)\). If the vertical translation is before the stretch it is \(\left( {\begin{array}{*{20}{c}}
  0 \\
  1
\end{array}} \right)\).

[3 marks]

general shape (including asymptote and single maximum in first quadrant),     A1

intercept \(\left( {0,\frac{1}{5}} \right)\) or maximum \(\left( {\frac{1}{2},\frac{1}{4}} \right)\) shown     A1

[2 marks]

\(0 < f(x) \leqslant \frac{1}{4}\)     A1A1

Note: A1 for \( \leqslant \frac{1}{4}\), A1 for \(0 < \).

[2 marks]

let \(u = x - \frac{1}{2}\)     A1

\(\frac{{{\text{d}}u}}{{{\text{d}}x}} = 1\,\,\,\,\,{\text{(or d}}u = {\text{d}}x)\)     A1

\(\int {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \int {\frac{1}{{4{{\left( {x - \frac{1}{2}} \right)}^2} + 4}}{\text{d}}x} } \)     A1

\(\int {\frac{1}{{4{u^2} + 4}}{\text{d}}u = \frac{1}{4}\int {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \)     AG

Note: If following through an incorrect answer to part (a), do not award final A1 mark.

[3 marks]

\(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{1}{4}\int_{0.5}^3 {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \)     A1

Note: A1 for correct change of limits. Award also if they do not change limits but go back to x values when substituting the limit (even if there is an error in the integral).

\(\frac{1}{4}\left[ {\arctan (u)} \right]_{0.5}^3\)     (M1)

\(\frac{1}{4}\left( {\arctan (3) - \arctan \left( {\frac{1}{2}} \right)} \right)\)     A1

let the integral = I

\(\tan 4I = \tan \left( {\arctan (3) - \arctan \left( {\frac{1}{2}} \right)} \right)\)     M1

\(\frac{{3 - 0.5}}{{1 + 3 \times 0.5}} = \frac{{2.5}}{{2.5}} = 1\)     (M1)A1

\(4I = \frac{\pi }{4} \Rightarrow I = \frac{\pi }{{16}}\)     A1AG

[7 marks]

This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e).

This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.

(b) Exam technique would have helped those candidates who could not get part (a) correct as any solution of the form given in the question could have led to full marks in part (b). Several candidates obtained expressions which were not of this form in (a) and so were unable to receive any marks in (b) Many missed the fact that if a vertical translation is performed before the vertical stretch it has a different magnitude to if it is done afterwards. Though on this occasion the markscheme was fairly flexible in the words it allowed to be used by candidates to describe the transformations it would be less risky to use the correct expressions.

This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.

(c) Generally the sketches were poor. The general rule for all sketch questions should be that any asymptotes or intercepts should be clearly labelled. Sketches do not need to be done on graph paper, but a ruler should be used, particularly when asymptotes are involved.

This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e).

This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.

(e) and (f) were well done up to the final part of (f), in which candidates did not realise they needed to use the compound angle formula.

This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.

(e) and (f) were well done up to the final part of (f), in which candidates did not realise they needed to use the compound angle formula.