Mathematics • Year 8 • Unit 2 • Lesson 12
Reading Real-World Graphs
Use the two-step method (read c, calculate m) on real-world data: rainwater tanks, parking fees, taxi fares, candle burning and bushwalk altitude.
1. Word problems
For each scenario, find the equation of the straight line in y = mx + c form, then state what m and c stand for in the real context.
1.1 — Rainwater tank. A graph of water volume V (litres) against time t (hours) shows the line passes through (0, 200) and (4, 600).
(a) Find the equation V = mt + c.
(b) State in words what m and c mean for this tank. 3 marks
1.2 — Parking fee. A car park graph shows cost C ($) against hours h. The line passes through (0, 3) and (2, 11).
(a) Find C = mh + c.
(b) Predict the cost for 5 hours of parking using your equation. 3 marks
1.3 — Taxi fare. Charlie's taxi receipt graph plots fare F ($) against distance d (km). The line crosses the F-axis at $4 and rises by $2.50 for each extra km.
(a) Write the equation F = md + c.
(b) Use it to find the fare for a 6 km trip. 3 marks
1.4 — Candle burning. A graph of candle height H (cm) against time t (minutes) shows the line passes through (0, 15) and (30, 9).
(a) Find H = mt + c.
(b) Why is m negative? What does its value mean physically? 3 marks
1.5 — Bushwalk altitude. Reuben's altitude graph shows his elevation A (metres) against time t (hours). The line passes through (0, 100) and (3, 700).
(a) Find A = mt + c.
(b) How fast (m/hour) is he gaining altitude, and how high was he at the start of the walk? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate sees a downhill straight line on a temperature-vs-time graph crossing the y-axis at (0, 30) and passing through (5, 10). They write the equation as y = 4x + 30. In your own words explain (i) what mistake they made when calculating m, (ii) what the correct gradient and equation should be, and (iii) what the correct gradient means about how the temperature is changing. Use the phrase "rise over run" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Rainwater tank
(a) c = 200. m = (600 − 200)/(4 − 0) = 400/4 = 100. V = 100t + 200.
(b) m = 100 means the tank is filling at 100 L per hour. c = 200 means there were already 200 L in the tank at the start (t = 0).
1.2 — Parking fee
(a) c = 3. m = (11 − 3)/(2 − 0) = 8/2 = 4. C = 4h + 3.
(b) At h = 5: C = 4(5) + 3 = 20 + 3 = $23.
1.3 — Taxi fare
(a) c = 4 (flag-fall), m = 2.50 (per km). F = 2.50d + 4.
(b) F = 2.50(6) + 4 = 15 + 4 = $19.
1.4 — Candle burning
(a) c = 15. m = (9 − 15)/(30 − 0) = −6/30 = −0.2. H = −0.2t + 15.
(b) m is negative because the candle is getting shorter — height decreases as time increases. The value −0.2 cm/min means it shortens by 0.2 cm every minute (i.e. 6 cm in 30 minutes ✓).
1.5 — Bushwalk altitude
(a) c = 100. m = (700 − 100)/(3 − 0) = 600/3 = 200. A = 200t + 100.
(b) Reuben is gaining altitude at 200 m per hour, and he started the walk at 100 m.
2.1 — Explain your thinking (sample response)
The classmate calculated the gradient as if the line went uphill. Using rise over run between (0, 30) and (5, 10), the rise is 10 − 30 = −20 (not +20), and the run is 5 − 0 = 5, so m = −20/5 = −4, not +4. The correct equation is y = −4x + 30. A negative gradient means the temperature is FALLING — by 4°C every hour — not rising. The classmate forgot to flip the sign when the line slopes downhill.
Marking: 1 mark for spotting the sign error in m; 1 mark for the correct gradient −4; 1 mark for the correct equation y = −4x + 30; 1 mark for a clear full-sentence explanation that uses "rise over run".