A graph of taxi cost against distance is a straight line from to , with distance in km and cost in pounds.
- (a)Write down the fixed charge.[1]
- (b)Work out the cost per kilometre.[2]
Loading...
A graph of a journey tells a story: when the car sped up, when it stopped, how far it went. Two measurements unlock that story — the , which gives a rate such as speed or acceleration, and, for velocity–time graphs, the under the graph, which gives the distance travelled.
The big picture
Gradient always means "how fast the vertical quantity changes per unit of the horizontal one". On a distance–time graph that is speed; on a velocity–time graph it is acceleration; on a graph of water depth against time it is the rate of filling. For curves, the gradient changes, so it is estimated by drawing a tangent.
Area under a velocity–time graph is distance because velocity multiplied by time is distance. Edexcel Higher questions ask you to estimate the area under a curve using trapezia and to say whether the estimate is too big or too small.
What you'll learn
The gradient of a distance–time graph is the . Steeper means faster; a horizontal section means the object is stationary.
A line sloping back down to the time axis means the object is returning to its starting point.
Average speed for the whole journey is total distance divided by total time — not the average of the individual speeds.
A car travels 90 km in 1 hour 30 minutes. What is its average speed in km/h?
Tip — Convert minutes to hours before calculating km/h: 40 minutes is hour.
The gradient of a velocity–time graph is the . A negative gradient is deceleration; a horizontal line means constant velocity.
The between the graph and the time axis is the distance travelled. Split it into rectangles, triangles and trapezia.
A cyclist accelerates uniformly from rest to 15 m/s in 5 seconds. What is the acceleration, in m/s²?
When a graph is curved, the rate of change is different at every point. To estimate it at a point, draw a — a straight line touching the curve at that point — and find the tangent’s gradient.
The rate of change between two points is the gradient of the chord joining them.
Tangent gradients are estimates, so a range of answers is accepted if the tangent is drawn carefully.
The difference between the chord and the tangent is the difference between average speed over a journey and the speedometer reading at an instant. At A-Level, differentiation calculates the tangent gradient exactly.
To estimate the area under a curve, split it into vertical strips of equal width and treat each as a trapezium.
If the curve bends so that the straight tops of the trapezia lie below it, the estimate is an underestimate; if they lie above, it is an overestimate.
A velocity–time graph is a straight line from to , with time in seconds and velocity in m/s. How far does the object travel?
Tip — Sketch the trapezia on the curve to decide over- or underestimate: if the curve is above the trapezium tops, the trapezia miss some area.
Think like an examiner
Rates and areas
Watch out for these
Stretch yourself
A lift starts from rest, accelerates uniformly for 3 s to 2 m/s, travels at 2 m/s for seconds, then decelerates uniformly to rest in 2 s. It rises 25 m in total. Find .
Hint — Write the area under the velocity–time graph in terms of and set it equal to 25.
Questions students ask
Key takeaways
Exam-style questions
3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.
A graph of taxi cost against distance is a straight line from to , with distance in km and cost in pounds.
Unlimited practice
A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.
Generating a question…
Test yourself
Ready to practise Real-Life Graphs & Rates of Change? Pick a mode and earn XP & Dobloons.
Real past-paper questions on Real-Life Graphs & Rates of Change, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.