Gradients of Curves and Area under Graphs — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers two ways of extracting meaning from a graph: the gradient, which measures how fast one quantity changes with another, and the area between the graph and the horizontal axis. By the end of this guide you should be able to estimate the gradient of a curve at a point using a tangent, find an average rate of change using a chord, and estimate an area by splitting it into strips.

You should also be able to apply both skills to motion graphs, where the gradient and the area each carry a physical meaning, and to say what those meanings are on a distance–time graph and on a speed–time graph.

The organising idea is that the axes tell you what the gradient and the area mean. A gradient is always the y-quantity divided by the x-quantity, and an area is always the y-quantity multiplied by the x-quantity. So on a graph of distance against time, the gradient is distance ÷ time, which is speed. On a graph of speed against time, the area is speed × time, which is distance. You never need to memorise a list of cases — you read the units off the axes and the meaning follows.

Key terms and definitions

Gradient — the steepness of a graph: the change in the vertical quantity divided by the change in the horizontal one.

Tangent — a straight line touching a curve at exactly one point, used to estimate the gradient at that instant.

Chord — a straight line joining two points on a curve, whose gradient gives the average rate of change between them.

Rate of change — how fast one quantity changes with respect to another.

Area under a graph — the region between the graph and the horizontal axis.

Kinematics — the mathematics of motion: distance, speed, velocity and acceleration.

Acceleration — the rate at which speed changes, measured in m/s².

Core concepts

Why a curve needs a tangent

On a straight-line graph the gradient is the same everywhere, so one calculation describes the whole line. On a curve the steepness changes continuously, so there is no single gradient — only a gradient at a particular point.

To estimate it, draw a straight line that touches the curve at that point without crossing it, and find that line's gradient instead. The tangent matches the curve's steepness at the point of contact, so its gradient is the best available estimate.

Because the tangent is drawn by eye, the answer is an estimate, and exam questions say so. Two reasonable attempts can give slightly different values and both earn the marks.

Draw the tangent long. A short tangent makes the two readings close together, so a small error in either produces a large error in the gradient.

Calculating a gradient

Pick two points that are far apart on the tangent, ideally where it crosses grid lines so the readings are exact.

Gradient = change in the vertical quantity ÷ change in the horizontal quantity.

For a tangent through (1, 2) and (5, 14): the vertical change is 14 − 2 = 12 and the horizontal change is 5 − 1 = 4, so the gradient is 12 ÷ 4 = 3.

Take the changes in the same order both times. Subtracting the coordinates in opposite orders produces a gradient with the wrong sign.

A downward-sloping graph has a negative gradient, which on a speed–time graph means the object is slowing down.

Chords and average rates

A chord joins two points on the curve, and its gradient gives the average rate of change over that interval rather than the rate at an instant.

On a distance–time graph, a tangent gives the speed at one moment and a chord gives the average speed over a stretch of the journey. These are different numbers, and a question asking for "the speed at 4 seconds" wants a tangent while one asking for "the average speed over the first 10 seconds" wants a chord.

Area under a graph

The area is estimated by splitting the region into shapes whose areas you can calculate — rectangles, triangles and trapeziums — and adding them.

A rectangle is base × height. A triangle is ½ × base × height. A trapezium is ½ × (the two parallel sides added) × the width, which is the same as averaging the two heights and multiplying by the width.

Where the top of the region is curved, the trapezium rule works well: divide the region into vertical strips of equal width and treat the top of each as a straight line. More strips give a better estimate.

A trapezium whose top slopes upwards sits slightly below the curve it approximates when the curve bends upwards, so the estimate is a slight underestimate — and a slight overestimate when the curve bends the other way. Questions do ask which it is.

Motion graphs: what the gradient means

On a distance–time graph, the gradient is distance ÷ time, so it is the speed. A steeper line means a faster journey, and a horizontal line means the distance is not changing, so the object is stationary.

On a speed–time (or velocity–time) graph, the gradient is speed ÷ time, so it is the acceleration. A horizontal line here means constant speed, not a stationary object — which is the commonest confusion between the two graph types. A negative gradient means the object is slowing down.

Motion graphs: what the area means

On a speed–time graph, the area is speed × time, so it is the distance travelled. This is the reason these questions are worth so many marks: a shape you can measure gives a quantity you cannot read directly off either axis.

For a speed rising steadily from 0 to 12 m/s over 6 seconds, the region is a triangle, so the distance is ½ × 6 × 12 = 36 m.

For a constant speed of 8 m/s over 5 seconds, the region is a rectangle, so the distance is 8 × 5 = 40 m.

On a distance–time graph the area has no useful meaning, since distance × time is not a quantity anyone measures. Only the gradient matters there.

Units

The units follow the same rule as the meanings. A gradient's units are the vertical unit per the horizontal unit, so metres per second on a distance–time graph, and metres per second per second — m/s² — on a speed–time graph.

An area's units are the two multiplied, so m/s × s gives metres.

Worked examples

Example 1: Estimating a gradient from a tangent

A tangent drawn to a distance–time curve passes through the points (2, 5) and (8, 29), with distance in metres and time in seconds. Estimate the speed at the point of contact.

Gradient = (29 − 5) ÷ (8 − 2) = 24 ÷ 6 = 4.

The graph plots distance against time, so the gradient is distance ÷ time, which is speed.

The speed is 4 m/s.

Example 2: Distance from a speed–time graph

A cyclist accelerates steadily from rest to 10 m/s over 8 seconds, then holds 10 m/s for a further 12 seconds. Find the total distance travelled.

The first stage forms a triangle: ½ × 8 × 10 = 40 m.

The second forms a rectangle: 10 × 12 = 120 m.

Total distance = 40 + 120 = 160 m.

Splitting the region at the point where the motion changes is the whole technique. Trying to treat the entire journey as one shape gives the wrong answer.

Example 3: Acceleration

A car's speed increases from 4 m/s to 24 m/s in 5 seconds. Find its acceleration.

On a speed–time graph the gradient is the acceleration.

Gradient = (24 − 4) ÷ 5 = 20 ÷ 5 = 4.

The acceleration is 4 m/s². The units come from speed divided by time: m/s divided by s.

Common mistakes and how to avoid them

Confusing the two graph types. A horizontal line on a distance–time graph means stationary; on a speed–time graph it means constant speed. Read the vertical axis label first.

Taking the area under a distance–time graph. It has no meaning. Only speed–time areas give distance.

Drawing the tangent too short. Extend it well past the point of contact so the two readings are far apart.

Reading coordinates in opposite orders. Subtract the two vertical values in the same order as the two horizontal values, or the sign will be wrong.

Treating a multi-stage journey as one shape. Split the region wherever the graph changes direction and add the pieces.

Forgetting that a tangent gives an estimate. Say so, and do not present the value as exact.

Dropping the units, or using the wrong ones. Gradient units are vertical per horizontal; area units are the two multiplied.

Exam technique for "Gradients of Curves and Area under Graphs"

Read both axis labels before you do anything. They tell you what the gradient and the area mean, which removes the need to remember which case you are in.

Draw the tangent with a ruler and mark the two points you read from. Marks are awarded for a correctly drawn tangent even when the arithmetic that follows is wrong.

Show the subtraction as a fraction, with the vertical change over the horizontal change. The examiner can award a method mark for the correct fraction alone.

For an area, show the shapes you split the region into and give each one's area separately before adding. A single unexplained number earns much less.

State the units with every answer, and say whether an estimate is above or below the true value when the question asks.

Use a tangent for a rate at an instant and a chord for an average rate. The wording of the question tells you which is wanted.

Quick revision summary

The axes tell you the meaning. A gradient is the vertical quantity divided by the horizontal one; an area is the two multiplied.

On a distance–time graph, the gradient is speed; a horizontal line means stationary; the area means nothing.

On a speed–time graph, the gradient is acceleration and the area is the distance travelled; a horizontal line means constant speed, and a negative gradient means slowing down.

To find the gradient at a point on a curve, draw a long tangent and calculate its gradient from two widely separated points. The result is an estimate.

To find an average rate between two points, use a chord instead.

To estimate an area, split it into rectangles, triangles and trapeziums, work out each, then add. More and narrower strips give a better estimate.

Units follow the operation: m ÷ s gives m/s, m/s ÷ s gives m/s², and m/s × s gives m.