🚗 Real-life Graphs

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

Interpret distance–time graphs (speed as gradient)
Read and draw travel graphs (journeys with stops)
Interpret conversion graphs and cost graphs
Read values and rates from real-life graphs
Sketch graphs from written descriptions

📖 Learn

1. Distance–Time Graphs

A distance–time graph shows how far something has travelled over time.

Speed = gradient = distance ÷ time

Steep line: fast speed (large gradient)

Shallow line: slow speed (small gradient)

Horizontal line: stopped (gradient = 0, speed = 0)

Line going back down: returning to start

💡 Distance must always be ≥ 0. The line can never go below the x-axis.

⚠️ Vertical sections of a d–t graph are impossible — you can't be in two places at the same time!

2. Speed–Time Graphs

A speed–time graph shows how speed changes over time.

Acceleration = gradient Distance = area under graph

Rising line: accelerating (speeding up)

Horizontal line: constant speed

Falling line: decelerating (slowing down)

Area under graph: = distance travelled (rectangle + triangle areas)

💡 Area of a triangle = ½ × base × height. Area of rectangle = length × width.

3. Conversion Graphs

A conversion graph converts between two units (e.g. miles and km, £ and $).

Always a straight line through the origin (0 km = 0 miles)

To convert: draw a vertical line from the x-axis, then horizontal to the y-axis

Gradient: gives the conversion rate (e.g. 1.6 km per mile)

Example: Miles to km: y = 1.6x, so 50 miles = 80 km

💡 The conversion rate IS the gradient of the graph.

4. Cost Graphs

A cost graph shows total cost vs. quantity (items, time, etc.).

y-intercept: fixed cost (cost before any units are bought)

Gradient: cost per unit (variable cost)

Example: Phone plan: £10/month + £0.05/minute. y = 0.05x + 10

Comparing plans: plot both lines — cheaper plan depends on usage

💡 Intersection of two cost lines = the break-even point (both plans cost the same).

5. Sketching from Descriptions

Sketch graphs to represent written descriptions — no numbers needed, just the shape.

Increasing: line going up

Decreasing: line going down

Constant: horizontal line

Fast then slow: steep section then shallow section

Example: "Water fills a bath at a constant rate" → straight line going up

Example: "A car accelerates, then stops suddenly" → rising line then drops vertically to 0 on speed–time graph

✏️ Worked Examples

Example 1 – Reading a Distance–Time Graph

A cyclist travels 30 km in 1 hour, rests for 30 minutes, then returns 30 km in 2 hours. Find: (a) speed in first section (b) speed returning.

(a) Speed out: gradient = 30 km ÷ 1 h = 30 km/h

(b) Speed back: gradient = 30 km ÷ 2 h = 15 km/h

Note: During the rest, the graph is horizontal (speed = 0).

Example 2 – Speed–Time Graph Area

A train accelerates from 0 to 60 m/s over 30 s, then travels at 60 m/s for 120 s, then decelerates to rest in 20 s. Find total distance.

Triangle 1 (0–30 s): ½ × 30 × 60 = 900 m

Rectangle (30–150 s): 120 × 60 = 7200 m

Triangle 2 (150–170 s): ½ × 20 × 60 = 600 m

Total: 900 + 7200 + 600 = 8700 m

Example 3 – Conversion Graph

1 mile ≈ 1.6 km. Convert: (a) 25 miles to km (b) 56 km to miles.

(a) 25 × 1.6 = 40 km

(b) 56 ÷ 1.6 = 35 miles

💡 On the graph: find 25 on the miles axis, go up to the line, read across to km axis.

Example 4 – Two Cost Plans

Plan A: £5 + £2/hour. Plan B: £3/hour. When are they equal? Which is cheaper for 4 hours?

Plan A: C = 2h + 5 Plan B: C = 3h

Equal when: 2h + 5 = 3h → h = 5 hours (intersection)

At h = 4: A = 2(4)+5 = £13. B = 3(4) = £12 → Plan B cheaper for 4 hours

💡 Below 5 hours: Plan A is cheaper. Above 5 hours: Plan B is cheaper.

🎨 Visualizer

🚗 Travel Graph Simulator

Choose a journey scenario to visualise a distance–time graph.

🔄 Conversion Graph

Enter a conversion rate (y per x) and a value to convert.

Rate (y per x): Convert x =

💰 Cost Plan Comparison

Plan A: fixed + rate per hour. Plan B: rate per hour only. Find the break-even.

Plan A fixed: £ Rate: £/hr

Plan B rate: £/hr

Exercise 1 – Distance–Time Graph Calculations

A car travels: 60 km in the first 1 hour, stops for 30 minutes, then drives 40 km in the next hour, then returns home in 2 hours.

Exercise 2 – Speed, Distance, Time

Use Speed = Distance ÷ Time. Distance = Speed × Time. Time = Distance ÷ Speed.