What you'll learn
This topic covers finding gradients of curves and areas under graphs — including their use in kinematics (motion). In this guide you will learn how to estimate the gradient of a curve using a tangent, how to estimate the area under a curve, and how these relate to speed, acceleration and distance. These are higher-tier interpretation skills that link algebra and real-life graphs.
Key terms and definitions
Gradient — the steepness of a graph, representing a rate of change.
Tangent — a straight line that just touches a curve at one point.
Chord — a line joining two points on a curve, giving an average rate.
Area under a graph — the region between the curve and the horizontal axis.
Kinematics — the study of motion (distance, speed and acceleration).
Core concepts
Gradient of a curve
On a curve the gradient changes, so to find the gradient at a point you draw a tangent at that point and find its gradient (change in y ÷ change in x). This gives the instantaneous rate of change there. The longer and more accurate the tangent, the better the estimate.
Average rate using a chord
To find the average rate of change between two points, draw a chord joining them and find its gradient. On a distance–time graph this gives the average speed over that interval, as opposed to the instantaneous speed from a tangent.
Area under a graph
The area under a graph often has a physical meaning. To estimate it, split the region into strips — rectangles, triangles and trapeziums — find each area and add them. The trapezium rule (averaging the parallel sides) is a common way to handle curved tops.
Kinematics: gradients
In motion graphs, gradients give rates. On a distance–time graph the gradient is speed; on a speed–time graph the gradient is acceleration. Use a tangent for the value at an instant on a curved graph.
Kinematics: areas
On a speed–time graph, the area under the graph is the distance travelled. Estimate it by splitting into shapes. This connects the area skill directly to a real quantity, a popular higher-mark question.
Worked examples
Example 1: Gradient from a tangent
A tangent to a curve passes through (2, 3) and (6, 11). Find the gradient.
Gradient = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.
Example 2: Area as a triangle
A speed–time graph rises straight from 0 to 12 m/s over 6 s. Find the distance.
Area of triangle = ½ × 6 × 12 = 36 m.
Example 3: Trapezium strip
Estimate the area of a strip with parallel heights 4 and 6 and width 2.
Area = ½ (4 + 6) × 2 = 10.
Common mistakes and how to avoid them
Reading the gradient off the curve directly. You must draw a tangent first.
Using a short tangent. A longer tangent gives a more accurate gradient.
Forgetting area = distance on a speed–time graph.
Miscounting strip widths. Keep the strips equal and add carefully.
Confusing average and instantaneous rates. Chord = average, tangent = instant.
Exam technique for Gradients and Areas under Graphs
Draw a tangent for the gradient at a point on a curve.
Use a chord for an average rate between two points.
Split the area into rectangles, triangles and trapeziums.
Interpret physically — gradient as speed/acceleration, area as distance.
State units appropriate to the axes.
Quick revision summary
On a curve the gradient changes, so to find the gradient at a point draw a tangent and find its gradient — this gives the instantaneous rate of change. A chord between two points gives the average rate over that interval. The area under a graph is estimated by splitting it into rectangles, triangles and trapeziums (the trapezium rule handles curved tops) and adding the parts. In kinematics, gradients give rates — distance–time gradient is speed, speed–time gradient is acceleration — and the area under a speed–time graph is the distance travelled. The common errors are reading a gradient straight off a curve instead of drawing a tangent, using too short a tangent, forgetting that area gives distance, and confusing average (chord) with instantaneous (tangent) rates. Draw tangents for instant gradients, chords for averages, split areas into simple shapes, and always interpret the result physically with correct units.