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Real-life Linear Graphs

FoundationHigherAQAEdexcelOCR

Practise real-life linear graphs with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through plotting straight-line graphs and finding gradients, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. The gradient often represents a rate, such as cost per unit.

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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

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Real-life linear graphs use straight lines to model situations such as taxi fares, phone tariffs and currency conversion. The mathematics is the same as any straight line, but the gradient and intercept now carry meaning.

The intercept is usually a fixed charge that applies regardless of usage, such as a standing charge or a call-out fee. The gradient is the rate, such as the cost per mile or per minute.

Interpreting those two numbers in context is what most questions are really testing. An answer of two is worth little; an answer of two pounds per mile shows you understand what the graph is describing.

Revision notes

Interpreting the intercept

Where the line crosses the vertical axis is the value when the horizontal quantity is zero.

A taxi graph crossing at \(£3\) means a \(£3\) fixed charge before any distance is travelled. Always name the units.

Interpreting the gradient

The gradient is the rate of change, found as the change in the vertical quantity divided by the change in the horizontal one.

If the fare rises by \(£6\) over \(4\) miles, the gradient is \(£1.50\) per mile. State the units as part of the answer.

Reading values and comparing options

Read from the graph by drawing lines across and down, showing your construction lines.

When two tariffs are drawn on the same axes, the point where the lines cross is where the costs are equal. Before that point one option is cheaper; after it the other is.

Key points

  • The intercept is the fixed charge or starting value.
  • The gradient is the rate of change.
  • Always state units in your interpretation.
  • Show construction lines when reading values.
  • Where two lines cross, the two options cost the same.
  • The steeper line has the higher rate.

Worked examples

Example 1

A taxi graph crosses the vertical axis at \(£4\) and the fare rises to \(£16\) after \(6\) miles. Find the cost per mile.

Working

\[16 - 4 = 12\]find the change in fare above the fixed charge
\[12 \div 6\]divide by the number of miles
\[= £2 \text{ per mile}\]state the rate with units

Example 2

Interpret the intercept of \(£4\) in that graph.

Working

\[\text{The fare when distance is zero}\]the intercept is the value at x = 0
\[\text{A £4 fixed charge}\]explain what it means in context

Example 3

Two tariffs cross at \(5\) hours. Explain what this means.

Working

\[\text{Both cost the same at 5 hours}\]the point of intersection gives equal cost
\[\text{One is cheaper before, the other after}\]compare the options either side of the crossing point

Common mistakes

  • Giving a number without units.

    A gradient of 2 means little; £2 per mile answers the question.

  • Reading the intercept as the total cost.

    It is the cost before any usage, not the final amount.

  • Not showing construction lines.

    Marks are awarded for the lines drawn across and down to the axes.

  • Ignoring different axis scales.

    A steeper-looking line is not always the higher rate if the scales differ.

Exam tips

  • Always interpret the gradient and intercept in the words of the question.
  • Include units in every numerical answer.
  • Draw and leave your construction lines on the graph.
  • At an intersection, say which option is better either side of it.

Key terms

Fixed charge
A cost applied regardless of usage, shown by the intercept.
Rate
The cost per unit, shown by the gradient.
Intersection
The point where two lines cross.
Interpret
To explain what a value means in context.

Related topics

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