Sketch the graph of the function \(f(x)\), for \( - 10 \leqslant x \leqslant 10\) . Indicating clearly the axis intercepts and any asymptotes.

Write down the equation of the vertical asymptote.

On the same diagram as part (a) sketch the graph of \(g(x) = x + 0.5\) .

Using your graphical display calculator write down the coordinates of one of the points of intersection on the graphs of \(f\) and \(g\), giving your answer correct to five decimal places.

Write down the gradient of the line \(g(x) = x + 0.5\) .

The line \(L\) passes through the point with coordinates \(( - 2{\text{, }} - 3)\) and is perpendicular to the line \(g(x)\) . Find the equation of \(L\).

     (A6)

Notes: (A1) for labels and some idea of scale.
(A1) for \(x\)-intercept seen, (A1) for \(y\)-intercept seen in roughly the correct places (coordinates not required).
(A1) for vertical asymptote seen, (A1) for horizontal asymptote seen in roughly the correct places (equations of the lines not required).
(A1) for correct general shape.

[6 marks]

\(x = - 4\)     (A1)(A1)(ft)

Note: (A1) for \(x =\), (A1)(ft) for \( - 4\).

[2 marks]

     (A1)(A1)

Note: (A1) for correct axis intercepts, (A1) for straight line

[2 marks]

\(( - 2.85078{\text{, }} - 2.35078)\) OR \((0.35078{\text{, }}0.85078)\)     (G1)(G1)(A1)(ft)

Notes: (A1) for \(x\)-coordinate, (A1) for \(y\)-coordinate, (A1)(ft) for correct accuracy. Brackets required. If brackets not used award (G1)(G0)(A1)(ft).
Accept \(x = - 2.85078\), \(y = - 2.35078\) or \(x = 0.35078\), \(y = 0.85078\).

[3 marks]

\({\text{gradient}} = 1\)     (A1)

[1 mark]

\({\text{gradient of perpendicular}} = - 1\)     (A1)(ft)

(can be implied in the next step)

\(y = mx + c\)

\( - 3 = - 1 \times - 2 + c\)     (M1)

\(c = - 5\)

\(y = - x - 5\)     (A1)(ft)(G2)

OR

\(y + 3 = - (x + 2)\)     (M1)(A1)(ft)(G2)

Note: Award (G2) for correct answer with no working at all but (A1)(G1) if the gradient is mentioned as \( - 1\) then correct answer with no further working.

[3 marks]

This was not very well done. The graph was often correct but was so small that it was difficult to check if axes intercepts were correct or not. Often the vertical asymptote looked as if it were joined to the rest of the graph. Very few of the candidates put a scale and/or labels on their axes.

Reasonably well done. Some put \(y = - 4\) while others omitted the minus sign.

Fairly well done – but once again too small to check the axes intercepts properly. Also, many candidates did not appear to have a ruler to draw the straight line.

Well done.

Most could find the gradient of the line.

Many forgot to find the gradient of the perpendicular line. Others had problems with the equation of a line in general.