Reciprocal Graphs

The graph of $y=\frac{6}{x}$y = 6/x

In $y=\frac{6}{x}$y = 6/x, x is the denominator: y is 6 divided by x. Work out some points. At x = 1, y = 6. At x = 2, y = 3. At x = 3, y = 2. At x = 6, y = 1. At x = 12, y = 0.5.

In every pair, x × y = 6. So when x gets bigger, y must get smaller, and doubling x halves y. The points do not lie on a straight line: plotted and joined, they make a curve that falls steeply at first and then levels out.

The graph of $y=\frac{6}{x}$y = 6/x through (1, 6), (2, 3), (3, 2) and (6, 1). It falls toward the x-axis and rises toward the y-axis.

Why it never touches either axis

As x grows, $y=\frac{6}{x}$y = 6/x gets smaller and smaller: at x = 60 it is 0.1, and at x = 600 it is 0.01. But it never reaches 0. For $\frac{6}{x}$6/x to equal 0, some number x would have to give x × 0 = 6, and no number does. So the curve comes closer and closer to the x-axis without ever touching it.

As x gets closer to 0, y gets larger and larger: at x = 0.1, y = 60, and at x = 0.01, y = 600. At x = 0 there is no value at all, because you cannot divide by zero. So the curve climbs closer and closer to the y-axis without ever touching it.

A line that a curve comes closer and closer to, without ever reaching, is called an asymptote. The graph of $y=\frac{6}{x}$y = 6/x has two: the x-axis and the y-axis.

Negative values of x

The rule works for negative x too. At x = −2, $y=6/\left(-2\right)=-3$y = 6 / (−2) = −3. At x = −6, y = −1. A negative x always gives a negative y, so this part of the curve lies in the bottom-left of the grid.

The full graph of $y=\frac{6}{x}$y = 6/x is two separate pieces, called branches, one on each side of the y-axis. They are never joined, because there is no point at x = 0.

The graph of $y=\frac{6}{x}$y = 6/x has two branches. (2, 3) is on one, and (−2, −3) is on the other.

A bigger numerator

Compare $y=\frac{12}{x}$y = 12/x with $y=\frac{6}{x}$y = 6/x. For every x, $\frac{12}{x}$12/x is twice $\frac{6}{x}$6/x: at x = 2 it is 6 instead of 3, and at x = 4 it is 3 instead of 1.5. So the whole curve sits further from the axes.

It still never touches them, for the same reason: $\frac{12}{x}$12/x is never 0, and there is no value at x = 0. The graph of $y=\frac{k}{x}$y = k/x, for any positive number k, has this shape, and its points all satisfy x × y = k.

The gold curve is $y=\frac{12}{x}$y = 12/x, through (2, 6) and (4, 3). The white curve is $y=\frac{6}{x}$y = 6/x, closer to the axes.

Inverse proportion on a graph

When x × y stays the same, y is inversely proportional to x, and $y=\frac{k}{x}$y = k/x is the equation of inverse proportion. Its graph is a reciprocal curve: doubling x halves y, and no value of x, however large, makes y reach 0.

Drag x from 2 to 4. On the left, y = 2x doubles. On the right, $y=\frac{8}{x}$y = 8/x halves, and the rectangle under the point keeps its area, x × y = 8. Drag x further right and y keeps shrinking but never reaches 0.

The usual mistakes

Multiplying instead of dividing. In $y=\frac{12}{x}$y = 12/x, x is the denominator: at x = 4, $y=12/4=3$y = 12 / 4 = 3, not 48.

Drawing the curve touching an axis. However far the curve is drawn, it only comes closer to the axes.

Joining the two branches across the y-axis. There is no point at x = 0, so the two branches stay apart.

Drawing a straight line. Equal steps in x do not give equal steps in y: from x = 1 to 2, $\frac{6}{x}$6/x falls by 3, but from x = 2 to 3 it falls by only 1.