Ch 6 · 2.9 Graphs in Practical Situations · O Level Mathematics 4024
Topic 6 of 8 in Algebra and Graphs — O Level Mathematics 4024 — 46 lessons.
Introduction
A graph tells a story: a family's car journey with a stop for lunch, the cost of electricity as units are used, a sprinter speeding up then slowing down. This lesson reads those stories, using the gradient as a rate of change and the area under a speed–time graph as a distance. (Syllabus section 2.9.)
Key ideas — Conversion and other practical graphs
A straight-line conversion graph (e.g. dollars to euros) lets you read across and down. The gradient of a straight-line graph is the rate of change: dollars per euro, litres per minute, cost per unit. A line not through the origin includes a fixed charge.
🧑🏫 Teacher's explanation
Imagine you are at a laundromat. You pay a fixed fee of $5 just to use the machine, plus $2 for every kilogram of clothes you wash. If you graph the total cost against the weight, you don't start at zero because of that initial $5 fee. This is a conversion graph with a non-zero intercept.
📖 Definitions
🔍 Worked example — Easy
A taxi company charges a fixed fee of £3 plus £1.50 per kilometer traveled. Write an equation for the total cost C in terms of distance d, and find the cost for a 10 ext km journey.
🔍 Worked example — Standard
A car travels at a constant speed of 20 ext m/s for 10 ext seconds, then accelerates uniformly to 30 ext m/s over the next 5 ext seconds. Calculate the total distance traveled.
🔍 Worked example — Exam-style
A graph shows the relationship between temperature T (in ^circC) and time t (in minutes). The graph is a straight line passing through (2, 10) and (5, 25). Find the rate of change of temperature per minute.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Distance–time graphs
🧑🏫 Teacher's explanation
Imagine you are tracking a marathon runner using a GPS. A distance-time graph doesn't show how fast they are going directly, but it shows how much ground they have covered at any given moment. If the line is steep, they are sprinting; if it is flat, they have stopped to tie their laces.
📖 Definitions
🔍 Worked example — Easy
A cyclist travels 15 km in 45 minutes. Calculate the average speed in km/h.
🔍 Worked example — Standard
A car's journey is plotted on a distance-time graph. From t=0 to t=2 hours, the distance increases from 0 to 100 km. From t=2 to t=3 hours, the distance remains at 100 km. Find the speed during the first 2 hours and describe the motion during the third hour.
🔍 Worked example — Exam-style
A particle moves such that its distance d (in meters) from a point O is given by d = 5t for the first 4 seconds, where t is time in seconds. At t=4, it changes direction and moves back towards O at a constant speed of 3 m/s for 2 seconds. Find the total distance travelled.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Calculate the speed of the car during the first 2 hours. [2] (b) Calculate the average speed for the entire 5-hour journey. [3] [5 marks]
Key ideas — Speed–time graphs
🧑🏫 Teacher's explanation
Imagine you are driving a car. A speed-time graph doesn't show you where you are, but how fast your speedometer is moving. If the line is flat, your speedometer is stuck on one number—you are moving at a constant speed. If the line goes up, you are pressing the accelerator; if it goes down, you are braking.
📖 Definitions
🔍 Worked example — Easy
A cyclist travels at a constant speed of 5 m/s for 10 seconds. Sketch the speed-time graph and calculate the distance traveled.
🔍 Worked example — Exam-style
A train's speed is given by a graph where it accelerates from 0 m/s to 30 m/s in 10 seconds, then immediately decelerates to 0 m/s in 5 seconds. Find the total distance traveled.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Curved graphs and tangents
For a curve, the rate of change at a point is the gradient of the tangent there: draw a ruled line touching the curve at that point, pick two points far apart on it, and work out rise ÷ run. The answer is an estimate.
🧑🏫 Teacher's explanation
Imagine you are driving a car. If you look at your speedometer, it tells you your speed at that exact moment. However, if you look at a graph of your speed over time, and the line is curved, your speed is constantly changing. A straight line on a graph means a constant rate, but a curve means the rate is shifting every millisecond.
📖 Definitions
🔍 Worked example — Easy
A speed-time graph shows a car accelerating from 0 m/s to 10 m/s in 5 seconds following a curve. Estimate the acceleration at t = 2 s if a tangent drawn at that point passes through (2, 4) and (5, 10).
🔍 Worked example — Exam-style
A speed-time graph for a journey consists of a triangle with vertices at (0,0), (5, 10), and (12, 0). Calculate the total distance travelled.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
In the graph, a car speeds up from rest to 8 m/s in 4 s, travels at 8 m/s until 10 s, then slows to rest at 12 s.
🔍 Worked example — Easy
A car starts from rest and accelerates uniformly to a speed of 8 m/s in 4 s. It then travels at this constant speed for 6 s before decelerating uniformly to rest in 2 s. Calculate the total distance travelled.
🔍 Worked example — Standard
A cyclist rides 12 km in 40 minutes, rests for 20 minutes, and then returns home in 30 minutes. Calculate the average speed for the entire journey in km/h.
🔍 Worked example — Exam-style
A speed-time graph shows a particle accelerating from 0 m/s to 20 m/s in 10 s. It then maintains this speed for 5 s. Calculate the acceleration during the first 10 s and the total distance covered in the first 15 s.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) gradient 1.5 (\$ per km); intercept 3 (the fixed charge). (2) 1.5 m/s²; tfrac12 × 20 × 30 = 300 m. (3) tfrac12(30 + 15) × 10 = 225 m. (4) 3 ÷ 0.75 = 4 km/h.