If y is inversely proportional to x, plotting y against 1/x gives a straight line through the origin. Its gradient is the constant of proportionality in y = k/x. A straight line with a non-zero intercept on these axes describes a different relationship.
The 2027 SEC G3 Mathematics K310 syllabus includes inverse proportion. This worked example shows how to check the relationship from a table and read the constant from a graph.
- Calculate the reciprocal of each x-value.
- Plot y vertically against 1/x horizontally.
- Check whether the line goes through the origin.
- Find the gradient to obtain k.
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Choose the horizontal axis
Suppose x takes the values 2, 3 and 6 while y takes the values 12, 8 and 4. Plotting y against x gives a curve for the model y = 24/x. Plotting y against 1/x instead gives the points (1/2, 12), (1/3, 8) and (1/6, 4).
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Worked example | Read the constant from the line
Take the points (1/2, 12) and (1/6, 4). The vertical change is 12 − 4 = 8. The horizontal change is 1/2 − 1/6 = 1/3. The gradient is 8 ÷ 1/3 = 24. The line passes through the origin, so the model is y = 24 × (1/x).
Use the original equation to check the result. For x = 3, it gives y = 24/3 = 8, which agrees with the table. If x = 12, the model predicts y = 2.
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Check a graph with an intercept
Suppose each y-value were increased by 2. The transformed graph would still have a gradient of 24 but would cross the vertical axis at 2. The new relationship would be y = 24/x + 2. Multiplying by x now gives xy = 24 + 2x, so xy would vary with x.
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Frequently asked questions
Can I use the table without drawing?
Yes. For exact values, check whether xy is constant. The graph is another way to see the relationship.
Does the origin mean I can substitute x = 0?
The transformed horizontal axis represents 1/x. Its zero mark does not make x = 0 a valid input to y = k/x.