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Proportional Graphs

FoundationHigherAQAEdexcelOCR

Practise proportional graphs with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through recognising proportion from graphs, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Direct proportion gives a straight line through the origin.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

A proportional graph shows the relationship between two quantities visually, and the shape tells you which kind of proportion is at work.

Direct proportion gives a straight line through the origin. The line must pass through the origin: a straight line crossing the axis elsewhere shows a linear relationship but not a proportional one.

Inverse proportion gives a reciprocal curve with two branches, approaching but never touching the axes. Recognising which graph is shown lets you pick the right formula before doing any calculation.

Revision notes

Direct proportion graphs

A straight line through the origin, with gradient equal to the constant \(k\).

So a line through \((0,0)\) and \((4, 12)\) has \(k = 3\), giving \(y = 3x\).

Inverse proportion graphs

A curve falling steeply then flattening, never meeting either axis.

The product of the coordinates at any point on the curve gives \(k\), so a point at \((4, 5)\) means \(k = 20\).

Telling them apart

Check whether the graph is straight or curved, and whether it passes through the origin.

Straight through the origin means direct; a two-branch curve means inverse; a straight line not through the origin is neither.

Key points

  • Direct proportion gives a straight line through the origin.
  • The gradient equals the constant \(k\).
  • Inverse proportion gives a reciprocal curve.
  • The curve never touches the axes.
  • Multiply coordinates on an inverse curve to find \(k\).
  • A line not through the origin is not proportional.

Worked examples

Example 1

A direct proportion graph passes through \((5, 15)\). Find \(k\).

Working

\[k = \frac{y}{x}\]the gradient gives the constant
\[\frac{15}{5} = 3\]divide to find k

Example 2

An inverse proportion curve passes through \((6, 4)\). Find \(k\).

Working

\[k = xy\]the product is constant
\[6 \times 4 = 24\]multiply the coordinates

Example 3

Explain why \(y = 2x + 5\) is not a direct proportion.

Working

\[\text{The line does not pass through the origin}\]at x = 0, y = 5 not 0
\[\text{So it is linear but not proportional}\]state the conclusion

Common mistakes

  • Calling any straight line direct proportion.

    It must pass through the origin. y = 2x + 5 does not.

  • Dividing coordinates on an inverse curve.

    For inverse proportion you multiply the coordinates to find k.

  • Expecting an inverse curve to touch an axis.

    The axes are asymptotes, so the curve approaches but never meets them.

  • Reading the gradient from a curve.

    Gradient applies to straight lines. For a curve, use the product of coordinates.

Exam tips

  • Check first whether the graph is straight or curved.
  • Confirm a straight line passes through the origin before calling it proportional.
  • Divide coordinates for direct, multiply for inverse.
  • Use any clear point on the graph to find k.

Key terms

Direct proportion
A straight-line relationship through the origin.
Inverse proportion
A reciprocal relationship with constant product.
Asymptote
A line a curve approaches but never touches.
Gradient
The steepness of a straight line, equal to \(k\) here.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.