Direct and Inverse Proportion Graphs Revision Notes | GCSE AQA Higher Maths

1. Why Proportion Graphs Matter

We will also emphasize additional teaching elements to deepen understanding of key topics related to direct and inverse proportions as outlined in specification point 3.14.1.6.

Think about speed and journey time. Drive faster and the time drops. That is not direct proportion — it is inverse. But drive for longer at a fixed speed and the distance increases steadily. That is direct proportion. The graph shape is the giveaway.

Higher-tier questions will show you a graph and ask you to identify or interpret the proportion relationship. You need to know the two key shapes cold — and understand what each feature of the graph is telling you about the relationship between the variables. Let's apply this by identifying the relationship from a typical exam question.

2. Direct Proportion Graphs

When $y$ is directly proportional to $x$, we write $y \propto x$. This means $y = kx$ for some constant $k$. Every time $x$ doubles, $y$ doubles. Every time $x$ triples, $y$ triples. The ratio $\frac{y}{x}$ is always the same.

The graph of a direct proportion relationship has two non-negotiable features: it is a straight line, and it passes through the origin. Both conditions must hold. A straight line that does not pass through the origin represents a linear relationship, but not direct proportion. For practice, consider how this applies to various graphical representations in A-level exams.

The gradient of the line is the constant of proportionality $k$. A steeper line means a larger $k$, so $y$ grows faster relative to $x$. A shallower line means a smaller $k$. The gradient is always positive for standard direct proportion (both variables increase together).

You can find $k$ from a graph by picking any point on the line (other than the origin) and calculating $k = \frac{y}{x}$. For example, if the line passes through $(4, 10)$, then $k = \frac{10}{4} = 2.5$, so the relationship is $y = 2.5x$. Try a similar question: if the point is $(6, 12)$, what is $k$?

The graph above shows two direct proportion relationships: $y = 2x$ (steeper, larger $k$) and $y = 0.5x$ (shallower, smaller $k$). Both pass through the origin. Notice that neither line has a $y$-intercept other than zero — that is the defining visual signature of direct proportion.

This is where students often slip up. Can you identify the direct proportion graph from a set of options?

3. Inverse Proportion Graphs

Now for the other key shape. When $y$ is inversely proportional to $x$, we write $y \propto \frac{1}{x}$. This means $y = \frac{k}{x}$. As $x$ increases, $y$ decreases — but not in a straight-line way. The decrease slows down as $x$ gets larger.

The graph of an inverse proportion relationship is a smooth curve that sweeps from top-left to bottom-right (for positive $k$). It gets very close to both the $x$-axis and the $y$-axis, but never actually reaches either. Those axes are called asymptotes.

To find $k$ from an inverse proportion graph, pick any point on the curve and multiply the coordinates together: $k = x \times y$. For example, if the curve passes through $(2, 6)$, then $k = 2 \times 6 = 12$, so the relationship is $y = \frac{12}{x}$. This product stays constant at every point on the curve — that is the key check.

Think about the asymptotic behaviour carefully. As $x$ gets very large, $y$ gets very small — but $y$ never actually reaches zero. This makes physical sense: if you are sharing a fixed amount of money between more and more people, each person gets less and less, but you can never reach a point where each person gets exactly nothing (unless there is no money at all).

Here is a check that catches students out. Can you apply the product rule to identify an inverse proportion relationship?

Fill in the blanks: In direct proportion, the relationship can be expressed as $y = kx$, where $k$ is the constant of proportionality. In inverse proportion, the relationship can be expressed as $y = \frac{k}{x}$.

4. Reading and Interpreting Proportion Graphs

Recognising the shape is step one. Interpreting what the graph tells you about the relationship is step two — and that is where the exam marks live. GCSE Higher exam questions often give you a graph and ask you to find $k$, write the equation, or explain what a feature of the graph represents. Additionally, consistent practice with different question types will enhance your understanding and retention of concepts.

Worked example: Finding $k$ from a Direct Proportion Graph

A graph shows a straight line through the origin passing through the point $(5, 20)$. Write the equation connecting $y$ and $x$. [2 marks]

1. Straight line through the origin $\Rightarrow$ direct proportion: $y = kx$ — Both conditions confirmed: straight line AND passes through $(0,0)$. [M1 for identifying direct proportion]
2. $20 = k \times 5$ — Use the point $(5, 20)$ — substitute $x = 5$ and $y = 20$ into $y = kx$.
3. $k = \frac{20}{5} = 4$ — Divide both sides by 5. Alternatively, $k = \frac{y}{x} = \frac{20}{5} = 4$.
4. $y = 4x$ — Substitute $k = 4$ back into $y = kx$. [A1 for correct equation]

Answer: $y = 4x$

Worked example: Finding $k$ from an Inverse Proportion Graph

A graph shows a curve that passes through the point $(3, 8)$. The curve approaches both axes but never crosses them. Write the equation connecting $y$ and $x$. [2 marks]

1. Curve approaching both axes $\Rightarrow$ inverse proportion: $y = \frac{k}{x}$ — The hyperbola shape with asymptotes along both axes is the signature of inverse proportion. [M1 for identifying inverse proportion]
2. $8 = \frac{k}{3}$ — Use the point $(3, 8)$ — substitute $x = 3$ and $y = 8$ into $y = \frac{k}{x}$.
3. $k = 8 \times 3 = 24$ — Multiply both sides by 3. Equivalently, $k = x \times y = 3 \times 8 = 24$.
4. $y = \frac{24}{x}$ — Substitute $k = 24$ back into $y = \frac{k}{x}$. [A1 for correct equation]

Answer: $y = \frac{24}{x}$

Earlier we established that the gradient of a direct proportion graph equals $k$. Now you can see the same logic applied to inverse proportion: instead of reading the gradient, you multiply coordinates. Two different graph shapes, two different methods — but the same goal of finding $k$.

Worked example: Interpreting a Proportion Graph from a Table

The table below shows values of $x$ and $y$. Determine whether $y$ is directly or inversely proportional to $x$, and find the equation. [3 marks] $x$: 2, 4, 8 $y$: 15, 7.5, 3.75

1. $\frac{15}{2} = 7.5$, $\quad \frac{7.5}{4} = 1.875$, $\quad \frac{3.75}{8} = 0.46875$ — The ratios are not equal, so $y$ is NOT directly proportional to $x$.
2. $2 \times 15 = 30$, $\quad 4 \times 7.5 = 30$, $\quad 8 \times 3.75 = 30$ — The product is constant at 30 for all pairs. This confirms inverse proportion. [M1 for correct check method]
3. $y \propto \frac{1}{x}$ with $k = 30$ — Since $xy = 30$ is constant, $y = \frac{k}{x}$ with $k = 30$. [M1 for identifying inverse proportion]
4. $y = \frac{30}{x}$ — [A1 for correct equation]

Answer: $y = \frac{30}{x}$ (inverse proportion with $k = 30$)

Notice how checking the product $xy$ is the quickest way to test for inverse proportion from a table of values. If $xy$ is constant, you have inverse proportion. If $\frac{y}{x}$ is constant, you have direct proportion. These two checks are your diagnostic tools.

5. Deliberate Practice

The best way to lock in these graph shapes is to practise identifying them and extracting $k$ under exam conditions. Work through these problems before checking the solutions.

6. Exam Traps and Common Mistakes

This topic has a small number of very predictable traps. examiners know exactly where students go wrong, and they design questions to probe those weak spots. Here is what to watch for.

The most common error is confusing a linear graph with a direct proportion graph. A straight line is not automatically direct proportion — it must pass through the origin. If the line has a non-zero $y$-intercept, the relationship is $y = kx + c$, which is linear but not proportional. Always check the origin.

The second trap is misreading the asymptotic behaviour of an inverse proportion graph. Students sometimes say the curve 'reaches zero' or 'touches the axis'. It does not. The curve gets arbitrarily close to both axes, but $y$ is never zero and $x$ is never zero in the domain of the relationship. If an exam question asks you to describe the graph, use the word 'asymptote' or say 'approaches but never reaches'.

A third trap appears when students are given a graph and asked to find $k$. For direct proportion, they correctly compute $\frac{y}{x}$. But then they apply the same method to an inverse proportion graph — dividing instead of multiplying. Remember: direct proportion uses $k = \frac{y}{x}$; inverse proportion uses $k = x \times y$.

Before the tips block, try this question — it is designed to catch the most common trap in this topic.

Try this question — it is designed to catch the most common trap in this topic: If $y$ is directly proportional to $x$ and $y = 12$ when $x = 3$, what is the value of $y$ when $x = 9? [2 marks]