Graphs: including reciprocal graphs

The idea

A reciprocal graph comes from an equation y = k/x, where k is a non-zero constant. Because division by zero is undefined, x cannot be zero, so the graph splits into two separate branches. For positive k, branches lie in the first and third quadrants; for negative k, they lie in the second and fourth. Each branch is a smooth curve that approaches, but never touches, the x-axis and y-axis, which act as asymptotes. As x increases, y approaches zero; as x approaches zero, the magnitude of y increases. The graph has no intercepts. To sketch it, plot points such as (1, k), (2, k/2), (0.5, 2k) and their negatives, then join them with smooth curves that flatten towards the axes without crossing them.

What earns the marks

• Recognise that a reciprocal graph has equation y = k/x with k ≠ 0 and that x = 0 is excluded from the domain.
• Identify the two separate branches and state the quadrant pair: first and third for k > 0, second and fourth for k < 0.
• State that the x-axis and y-axis are asymptotes that the curve approaches but never meets.
• Explain that the graph has no intercepts with either axis and that y is undefined at x = 0.
• Sketch the correct shape by plotting points such as (1, k), (2, k/2), (0.5, 2k) and their negative counterparts, then joining with smooth curves.

Common mistakes and how to put them right

• Joining the two branches with a single continuous curve.

  Put it right: The branches are separate because x = 0 is undefined, so they must be drawn as two disconnected curves.

• Drawing the curve crossing an axis.

  Put it right: A reciprocal graph never touches or crosses either axis, because k/x can never equal zero and x can never equal zero.

• Treating the graph as a straight line through the origin.

  Put it right: y = k/x is a curve with asymptotes, not a linear graph, so it must bend towards the axes.

• Assuming the graph exists at x = 0.

  Put it right: Substituting x = 0 gives division by zero, so there is no point on the graph there.

Exam tips

• When asked to sketch y = k/x, mark at least one point on each branch, such as (1, k) and (−1, −k), to show the correct quadrants.
• Label the asymptotes or state clearly that the axes are asymptotes, because this is a key feature that distinguishes reciprocal from other graphs.
• Check the sign of k before drawing: a positive k gives branches in the first and third quadrants, while a negative k gives branches in the second and fourth.
• Use a smooth curve for each branch and do not join them across the y-axis; examiners look for two separate branches.