Reciprocal graphs

Reciprocal graphs are the graphs of equations like y = 1/x or y = k/x, where x is on the bottom of a fraction. They have two separate curves, one in each of two opposite quarters of the grid, and the curves get closer and closer to the axes without ever touching them. Those lines they approach are called asymptotes. A reciprocal graph never crosses x = 0, because you can't divide by zero.

The graph of y = 1/x

Here x and y are the usual coordinates. For positive x the curve is in the top right, and for negative x it is in the bottom left, because a number and its reciprocal always have the same sign. AQA's GCSE Maths specification asks students to recognise, sketch and interpret the reciprocal function y = 1/x, noting that x ≠ 0. More on reciprocals themselves is in the reciprocal guide.

Features of a reciprocal graph

• Two branches. The curve breaks into two pieces either side of x = 0.
• Asymptotes. y = 1/x gets closer to the x-axis (y = 0) as x gets large, and closer to the y-axis (x = 0) as x gets near zero, but never meets either.
• No intercepts. It never crosses either axis.
• Symmetry. It is symmetrical about the lines y = x and y = −x.

The graph of y = k/x

Changing the number on top stretches the curve, and changing its sign moves it to the other pair of quarters.

Here x and y are the coordinates again. With k = 6, every point has x × y = 6: (1, 6), (2, 3), (3, 2) and (6, 1). Bigger values of k push the curve further from the corner.

When k is negative, as in y = −4/x, the branches move to the top left and bottom right: a positive x gives a negative y.

How to plot a reciprocal graph

Make a table of values, but leave out x = 0. For y = 6/x:

1. Plot each point carefully.
2. Join the points on each side with a smooth curve, not straight lines.
3. Never join the two branches across x = 0. There is a gap there, because 6/0 has no value.
4. Let each branch get close to the axes without touching them.

Transformations of reciprocal graphs

Shifting the graph moves the asymptotes with it. For y = 1/(x − a) + b:

• The vertical asymptote moves to x = a, where the bottom of the fraction is zero.
• The horizontal asymptote moves to y = b.

Here x and y are the coordinates. The whole curve has moved 2 right and 3 up, so the asymptotes are x = 2 and y = 3, and the centre of symmetry is (2, 3). Try one, then press Mark to check it in the course.

The graph of y = 1/x²

At A-level you also meet y = k/x². Squaring makes every value positive, so for positive k both branches sit above the x-axis, like a volcano with a gap down the middle. It is symmetrical about the y-axis, and it falls away faster than y = 1/x as x grows.

Reciprocal graphs and inverse proportion

If y is inversely proportional to x, then y = k/x, and its graph is a reciprocal graph. For example, the time t hours to drive 120 km at a steady speed v km/h is t = 120/v. At 60 km/h it takes 2 hours, at 40 km/h 3 hours, and at 30 km/h 4 hours: halve the speed and the time doubles. Only the positive branch makes sense here, because speeds and times aren't negative.

Common mistakes

• Joining the branches. There is always a gap at the vertical asymptote.
• Letting the curve touch an axis. It gets close but never meets it.
• Putting a negative k graph in the wrong quarters. y = −4/x is top left and bottom right.
• Using straight lines between points. Draw a smooth curve.
• Forgetting the asymptote moves in y = 1/(x − a) + b.

To practise graph sketching under time pressure, try today's GCSE Maths Higher short timed paper. The Graphing Functions course has an asymptote explorer to try shifting the curve yourself.

Read next

Maths · NumberReciprocal: what it is, how to find it, with examplesThe reciprocal of a number is 1 divided by it. Find the reciprocal of whole numbers, fractions, decimals and negatives, and see the graph of y = 1/x.