Recognising direct and inverse proportion, setting up and using proportion equations with a constant, and interpreting their graphs.

What this dot point is asking

AQA wants you to recognise whether a relationship is direct or inverse proportion, set up the proportion equation with a constant of proportionality $k$, use it to find unknown values, and interpret the graphs of each type. At Higher tier this extends to proportion involving squares, cubes and roots. The skill is finding $k$ from a given pair, then using the resulting formula.

Recognising the type

Direct proportion: doubling one quantity doubles the other, and their ratio is constant. Examples are cost and quantity, or distance and time at constant speed. Inverse proportion: doubling one quantity halves the other, and their product is constant. Examples are speed and time for a fixed journey, or the number of workers and the time to finish a job. Reading the words carefully decides which model applies.

Setting up the proportion equation

The method is always the same three steps: write the proportion equation with $k$, substitute a known pair to find $k$, then use the completed formula. The symbol $\propto$ means "is proportional to", so "$y$ is proportional to $x^3$" becomes $y = kx^3$.

Interpreting the graphs

The graph of $y = kx$ is a straight line through the origin with gradient $k$; a straight line not through the origin is not direct proportion. The graph of $y = \dfrac{k}{x}$ is a reciprocal curve with two branches that approach but never touch the axes. A square-law relationship $y = kx^2$ gives a parabola through the origin. Recognising the shape from a graph, or matching a graph to its equation, is a common short question.

Worded problems

Many proportion questions are dressed as recipes, currency, fuel use or physics. Decide direct or inverse from the wording, find $k$ from the data, and answer with the formula. If a quantity is shared inversely, more parts means a smaller share each, so the product of share and number stays fixed.

The constant of proportionality has meaning

The constant $k$ is not just an algebraic device; it often has a real interpretation. In $\text{cost} = k \times \text{quantity}$, the constant $k$ is the price per item. In $\text{distance} = k \times \text{time}$, the constant is the speed. When you find $k$ from given data, it pays to say what it represents, because a question may ask you to interpret it. A larger $k$ in a direct relationship means a steeper line, so the dependent quantity grows faster for each unit of the other.

Combining proportion with graphs

The graph of a relationship reveals the model at a glance, which examiners exploit. A straight line through the origin is direct proportion, and its gradient is $k$. A curve that drops steeply then levels toward the axes is inverse proportion. A parabola through the origin signals a square law, $y = kx^2$. Being asked to match a sketch to one of these models, or to explain why a given graph cannot represent direct proportion (because it does not pass through the origin), is a recurring style of question that rewards knowing each shape.