Real-Life Graphs

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The Big Idea

Real-life graphs show relationships between two quantities
The gradient (steepness) tells you the rate of change
The shape of a graph tells you what's happening in real time
You can extract exact information by reading values carefully and calculating areas or gradients

Conversion Graphs

Conversion graphs are straight-line graphs that let you convert (change) between two different units or quantities. Think of them as a translator between two measurement systems.

Common examples:

Temperature: °C ↔ °F
Currency: Pounds ↔ Dollars
Volume: Litres ↔ Gallons
Prices: Cost vs. quantity (e.g. £ vs. kg)
Taxi fares: Cost vs. distance travelled

The Gradient of a Conversion Graph Gradient = Change in y / Change in x = Rise / Run

This tells you the rate of change. For example, if y-axis is cost (£) and x-axis is distance (miles), a gradient of 5 means the cost increases by £5 for every 1 mile travelled.

How to Read a Conversion Graph

The process is always the same — draw perpendicular lines to the axes:

To find the y-value for a given x-value:

Start at your x-value on the horizontal axis
Draw a vertical line up to the graph line
From that point, draw a horizontal line across to the y-axis
Read off the value

To find the x-value for a given y-value:

Start at your y-value on the vertical axis
Draw a horizontal line across to the graph line
From that point, draw a vertical line down to the x-axis
Read off the value

Tip: Using Proportion If the graph starts at the origin (0,0), you can use proportion to find values not on the axes. For example, if 20kg costs £12, then 120kg (which is 6 × 20kg) will cost 6 × £12 = £72.

When the Graph Doesn't Start at the Origin

Some conversion graphs have a y-intercept (where they cross the y-axis) that isn't zero. This is important — it often represents a fixed cost.

Example: A taxi company might charge £5 just to turn up (fixed fee), then £2 per mile. If you look at the graph where distance = 0, you'll see the cost is £5, not £0. That's the y-intercept.

Key Rule: You can only use proportion if the graph starts at the origin. If there's a fixed cost (non-zero y-intercept), you must find the equation of the line or read values directly from the graph.

Worked Example: Plumber's Charges

A plumber charges for the time spent on a job. The graph shows price (£) on the y-axis and hours on the x-axis.

Question (a): Estimate the price for a 3-hour job

Solution: At 3 hours, draw a vertical line up to the graph, then a horizontal line to the y-axis. The value reads approximately £225 (answers between £220–£230 are acceptable).

Question (b): A job costs £320. How long did it take (to nearest 0.5 hour)?

4.5 hours

Question (c): What is the fixed callout fee?

y-intercept £45

Practice Question

Distance-Time Graphs

Distance-time graphs

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Also in the full note

Speed-Time Graphs
Rates-of-Change Graphs
Key Terms to Memorise
Concepts Checklist
Common Mistakes & Exam Traps
Quick Reference: What Does Gradient Mean?
Quick Reference: What Does the Shape Mean?

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Real-Life Graphs

Real-Life Graphs

The Big Idea

Conversion Graphs

Worked Example: Plumber's Charges

Practice Question

Distance-Time Graphs

More Mathematics topics