Consider the curve \(y = 1 + \frac{1}{{2x}},\,\,x \ne 0.\)

For this curve, write down

i)     the value of the \(x\)-intercept;

ii)    the equation of the vertical asymptote.

Sketch the curve for \( - 2 \leqslant x \leqslant 4\) on the axes below.

i)     \(\left( {x = } \right) - 0.5\,\,\left( { - \frac{1}{2}} \right)\)       (A1)

ii)    \(x = 0\)        (A1)(A1)    (C3)

Note: Award (A1) for “\(x = \)” and (A1) for “\(0\)” seen as part of an equation.

(A1)(ft)(A1)(ft)(A1)    (C3)

Note: Award (A1)(ft) for correct \(x\)-intercept, (A1)(ft) for asymptotic behaviour at \(y\)-axis, (A1) for approximately correct shape (cannot intersect the horizontal asymptote of \(y = 1\)). Follow through from part (a).

Question 8: Rational function.

Few candidates could find the \(x\)-intercept of the rational function. Many candidates did appreciate that the curve does not cross the asymptote. Often the candidates wrote down the equation of the horizontal asymptote rather than the equation of the vertical asymptote. The most frequent incorrect sketch was that of \(y = \frac{1}{2}x + 1\) suggesting that the candidate did not understand that the curve \(y = 1 + \frac{1}{{2x}}\) is not linear and had taken insufficient care in entering the function into the calculator. Some candidates that appreciated the shape of the curve did not earn marks on account of the poor quality of their sketches, which either crossed, or veered away from, the asymptotes.

Question 8: Rational function.

Few candidates could find the \(x\)-intercept of the rational function. Many candidates did appreciate that the curve does not cross the asymptote. Often the candidates wrote down the equation of the horizontal asymptote rather than the equation of the vertical asymptote. The most frequent incorrect sketch was that of \(y = \frac{1}{2}x + 1\) suggesting that the candidate did not understand that the curve \(y = 1 + \frac{1}{{2x}}\) is not linear and had taken insufficient care in entering the function into the calculator. Some candidates that appreciated the shape of the curve did not earn marks on account of the poor quality of their sketches, which either crossed, or veered away from, the asymptotes.