A reciprocal graph has the equation y = k/x, where k is a non-zero constant. Its distinctive double-curve shape — called a hyperbola — never touches the axes, because x can never be zero. At KS3 you learn to plot these curves, read off values, and link them to inverse proportion.
What does "reciprocal" mean?
The reciprocal of any number is 1 divided by that number. The reciprocal of 4 is 1/4; the reciprocal of x is 1/x. So y = 1/x gives the reciprocal of every x-value as the y-value.
More generally, y = k/x scales all the y-values by the constant k. If k = 4, then at x = 2 the y-value is 4/2 = 2. At x = 8 the y-value is 4/8 = 0.5. As x grows larger, y gets smaller — the relationship is inverse.
What are the key features of a reciprocal graph?
The graph of y = k/x (for k > 0) has these properties:
| Feature | Detail |
|---|---|
| Shape | Two smooth curves — a hyperbola |
| Quadrants | Curve in 1st quadrant (x > 0, y > 0) and 3rd quadrant (x < 0, y < 0) |
| x-axis | Asymptote: y never equals zero |
| y-axis | Asymptote: x can never equal zero |
| Symmetry | Rotational symmetry of 180° about the origin |
| Intersection with axes | None — it never crosses either axis |
If k < 0, the two branches move to the 2nd and 4th quadrants instead.
How do you plot a reciprocal graph?
Worked example: Plot y = 6/x for −6 ≤ x ≤ 6 (x ≠ 0).
Step 1: Build a table of values (avoid x = 0).
| x | −6 | −3 | −2 | −1 | −0.5 | 0.5 | 1 | 2 | 3 | 6 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | −1 | −2 | −3 | −6 | −12 | 12 | 6 | 3 | 2 | 1 |
Step 2: Plot these points on a pair of axes, keeping note that x = 0 and y = 0 are both asymptotes.
Step 3: Draw two smooth curves — one curving through the positive-x points (1st quadrant) and one through the negative-x points (3rd quadrant). Do not join the two curves: there is a gap where x = 0.
Step 4: Draw dashed lines along x = 0 (the y-axis) and y = 0 (the x-axis) to indicate the asymptotes.
A common mistake: drawing the curve touching or crossing the axes. The curve gets very close but never actually reaches them.
What is an asymptote?
An asymptote is a line that a curve approaches but never reaches. For y = k/x there are two:
- x = 0 (the y-axis): as x → 0 from the positive side, y → +∞. As x → 0 from the negative side, y → −∞. The curve shoots off towards infinity on both sides of the y-axis.
- y = 0 (the x-axis): as x → +∞, y → 0. As x → −∞, y → 0. The curve flattens towards the x-axis but never touches it.
You show asymptotes on your sketch using dashed lines. In an exam, failing to show asymptotes or drawing the curve crossing the axis are the most penalised errors.
How does a reciprocal graph connect to inverse proportion?
A graph of y against x is a straight line through the origin when y ∝ x (direct proportion). If instead the graph shows a hyperbola shape — two curves in opposite quadrants, never touching the axes — then y ∝ 1/x (inverse proportion).
This gives you a visual test: look at the graph shape to decide the type of proportionality. If the product x × y is constant for all points on the curve, the graph is a reciprocal graph.
From the table above: 6 × 1 = 6, 3 × 2 = 6, 2 × 3 = 6, (−1) × (−6) = 6. The product is always 6 = k. ✓
How does a reciprocal graph differ from a quadratic or linear graph?
| Graph type | Equation example | Shape | Crosses x-axis? |
|---|---|---|---|
| Linear | y = 2x + 1 | Straight line | Yes (once) |
| Quadratic | y = x² | U-shaped parabola | Yes (zero, one, or twice) |
| Cubic | y = x³ | S-shaped curve | Yes (once or three times) |
| Reciprocal | y = 6/x | Hyperbola (two branches) | No |
The reciprocal is the only one of these that never crosses the x-axis (for k ≠ 0) and splits into two disconnected branches.
Frequently asked questions
Why can't x = 0 in y = k/x?
Dividing any number by zero is undefined — it produces no numerical answer. So x = 0 is simply not part of the domain of a reciprocal function. This is why the graph has a gap at the y-axis and the y-axis acts as an asymptote.
What happens to the graph if k is larger?
A larger value of k "pushes" the hyperbola away from the origin. The curves are further from the axes. For example, y = 12/x has branches in the same quadrants as y = 6/x but further out — at x = 1, y = 12 instead of 6. The shape is otherwise identical. All reciprocal graphs (with the same sign of k) look the same; only their scale changes.
How is the reciprocal graph different from a straight-line inverse proportion graph?
If you plot y against x, the reciprocal graph is a hyperbola. But if you plot y against 1/x (i.e. put 1/x on the horizontal axis), you get a straight line through the origin with gradient k. This plotting trick — called a linearisation — is useful at GCSE for confirming an inverse proportion relationship from data.
Is y = k/x the only type of reciprocal graph I need to know at KS3?
At KS3, y = k/x is the focus. At GCSE Higher, you may also meet y = k/x² (inverse square proportion), which has a different shape: both branches are in the upper half of the coordinate plane (since x² is always positive, y is always positive for k > 0). The asymptotes remain the same, but the curve falls more steeply.
Let Professor Pi help you plot and understand reciprocal graphs step by step — add the AI Tutors connector at aitutors.me.