Factorise the expression \({x^2} - kx\) .

Hence solve the equation \({x^2} - kx = 0\) .

The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of \(k\).

Write down the value of \(k\) for this function.

The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of \(k\).

Find the minimum value of the function \(y = f(x)\) .

\(x(x - k)\)     (A1)     (C1)

[1 mark]

\(x = 0\) or \(x = k\)     (A1)     (C1)

Note: Both correct answers only.

[1 mark]

\(k = 3\)     (A1)     (C1)

[1 mark]

\({\text{Vertex at }}x = \frac{{ - ( - 3)}}{{2(1)}}\)     (M1)

Note: (M1) for correct substitution in formula.

\(x = 1.5\)     (A1)(ft)

\(y = - 2.25\)     (A1)(ft)

OR

\(f'(x) = 2x - 3\)     (M1)

Note: (M1) for correct differentiation.

\(x = 1.5\)     (A1)(ft)
\(y = - 2.25\)     (A1)(ft)

OR

for finding the midpoint of their 0 and 3     (M1)
\(x = 1.5\)     (A1)(ft)
\(y = - 2.25\)     (A1)(ft)

Note: If final answer is given as \((1.5{\text{, }}{- 2.25})\) award a maximum of (M1)(A1)(A0)

[3 marks]

This question was poorly answered by all but the best candidates. The links between the parts were not made. The idea of the line of symmetry for the graph was seldom investigated. The “minimum value of the function” was often incorrectly given as a coordinate pair.

This question was poorly answered by all but the best candidates. The links between the parts were not made. The idea of the line of symmetry for the graph was seldom investigated. The “minimum value of the function” was often incorrectly given as a coordinate pair.

This question was poorly answered by all but the best candidates. The links between the parts were not made. The idea of the line of symmetry for the graph was seldom investigated. The “minimum value of the function” was often incorrectly given as a coordinate pair.

This question was poorly answered by all but the best candidates. The links between the parts were not made. The idea of the line of symmetry for the graph was seldom investigated. The “minimum value of the function” was often incorrectly given as a coordinate pair.