Travel Graphs and Curves

On a distance–time graph the gradient is the speed. On a speed–time graph the gradient is the acceleration and the area underneath is the distance. Learn the shapes of quadratic, cubic, reciprocal and exponential graphs so you can recognise them at a glance.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Read speed from the gradient of a distance–time graph.
Find acceleration (gradient) and distance (area) from a speed–time graph.
Recognise the shapes of $y=x^2$ , $y=x^3$ , $y=\dfrac ax$ and $y=a^x$ .
Find roots, turning points and intersections of curves.

Pause and think

A distance–time graph has a horizontal section. What is happening during that time?

Section 2 of 14

Core idea

Start hereA walk in a picture

A graph of distance from home against time tells a whole story: walking out, stopping at a shop, coming back slower. The steepness tells you speed, and flat bits mean you've stopped.

The big idea

Distance–time: gradient = speed. Speed–time: gradient = acceleration, area = distance. Recognise $x^2$ , $x^3$ , $\frac1x$ , $a^x$ shapes.

1. Real-life graphs

A walk to a shop and back

Steeper is faster. Flat means stopped. Sloping down means coming back.

Worked exampleA walk

600 m out in 10 min, 5 min stop, back in 15 min.

Goal: Speeds.
Strategy: Gradient = distance ÷ time.

$600\div10=60\text{ m/min}$
Out.
$600\div15=40\text{ m/min}$
Back (steeper going out).

Answer

60

and

40

m/min.

✓ CheckTotal time 30 min ✓.

Worked exampleSpeed–time

0 to 12 m/s in 6 s, then steady for 10 s.

Goal: Acceleration; distance.
Strategy: Gradient; area (triangle + rectangle).

$12\div6=2\text{ m/s}^2$
Gradient.
$\tfrac12(6)(12)+10(12)=36+120=156$
Area.

Answer

2

m/s²;

156

✓ CheckAverage speed

156/16=9.75

, between 6 and 12 ✓.

2. Curve shapes

Curve shapes to recognise

Quadratic: U. Cubic: snake. Reciprocal: two separate curves near the axes. Exponential: flat then shooting up.

$y=ax^2+\ldots$ : U (∩ if $a<0$ ).
$y=ax^3+\ldots$ : S-shape.
$y=\frac ax$ : two branches, asymptotes on the axes.
$y=a^x$ : exponential.

Common misconceptionDistance vs speed graphs A horizontal line means "stopped" on distance–time but "constant speed" on speed–time.

Pause and think

Describe $y=5-x^2$ .

Summary

Gradients and areas carry meaning.
Know the basic shapes.

Asymptote: Line approached but not reached.

HistoryThe first line graphs. William Playfair, a Scottish engineer, published the first widely used line graphs and bar charts in his Commercial and Political Atlas in 1786, plotting England's trade over time. Before that, almost nobody drew numbers as pictures; now it is how we read everything from journeys to share prices.

Fun factEvery fountain draws a parabola. Ignoring air resistance, a jet of water follows a quadratic curve. Next time you see a fountain or a basketball shot, you are watching

y=ax^2+bx+c

with

a<0

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