Mathematics • Year 10 • Unit 2 • Lesson 17
Linear Graphs in the Real World
Graph and interpret real-world linear relationships — taxi fares, swimming pace, account balance, plant growth and movie download. Each problem asks for intercepts, gradient, and a meaning statement.
1. Word problems
For each: identify the equation, the gradient and y-intercept, what each one means in context, and use the equation to answer the asked question.
1.1 — Taxi fare. A taxi charges a $4 flag fall plus $2.50 per km. Let d = distance in km, C = cost in $.
(a) Write the linear equation C = ____.
(b) State the gradient and what it means in dollars per km.
(c) State the y-intercept and what it means.
(d) Find C for d = 8 km. 4 marks
1.2 — Swimming pace. Mai swims at a constant pace. She's at the 100 m mark after 2 minutes and at the 250 m mark after 5 minutes. Let t = time in minutes, d = distance in metres.
(a) Find the gradient (speed in m/min).
(b) Find the equation of the line in y = mx + c form using one of the points.
(c) When is she at the 400 m mark? 4 marks
1.3 — Account balance. Marco's bank balance B (in $) after t weeks is B = −25t + 500 (he is withdrawing $25 a week from $500).
(a) State the gradient and y-intercept and explain each in context.
(b) When does the balance reach zero? (Find the x-intercept.)
(c) Make a small table for t = 0, 4, 8, 12, 16 weeks. 4 marks
1.4 — Plant growth. A seedling is 3 cm tall when planted (day 0) and grows at 1.5 cm per day. Let h = height in cm, d = days after planting.
(a) Write the linear equation h = ____.
(b) Make a table for d = 0, 2, 4, 6, 8 days.
(c) On what day does the plant first exceed 20 cm? 3 marks
1.5 — Movie download. A movie download progress is P = 4t (megabytes downloaded after t minutes). The file is 80 MB.
(a) State the gradient and what it means.
(b) Find both intercepts (note: the y-intercept here is the starting state at t = 0).
(c) When is the download complete (P = 80)? 3 marks
2. Explain your thinking
Communication, not just numbers. 4 marks
2.1 A classmate looks at the taxi fare C = 2.50d + 4 and says: "The y-intercept here is 4, but the line never actually starts at (0, 4) because no one takes a 0 km taxi." Using the words real-world domain, y-intercept and extrapolation, explain (i) why the y-intercept of 4 is still meaningful (what it represents), (ii) why the graph's "physically reasonable" portion only starts at some positive d (state a sensible minimum), and (iii) when reading the graph beyond a "reasonable" d we are doing extrapolation — give an example where extrapolation might fail.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Taxi fare
(a) C = 2.50d + 4. (b) Gradient m = 2.50: cost rises by $2.50 per km. (c) y-intercept 4: the $4 flag fall charged before driving. (d) C = 2.50(8) + 4 = 20 + 4 = $24.
1.2 — Swimming pace
(a) m = (250 − 100)/(5 − 2) = 150/3 = 50 m/min. (b) Using (2, 100): d − 100 = 50(t − 2) → d = 50t (clean form because she started at 0 m at t = 0). (c) 400 = 50t → t = 8 minutes.
1.3 — Account balance
(a) Gradient m = −25 ($/week withdrawal); y-int 500 (starting balance at t = 0). (b) 0 = −25t + 500 → t = 20 weeks. (c) Table: t=0→500; t=4→400; t=8→300; t=12→200; t=16→100.
1.4 — Plant growth
(a) h = 1.5d + 3. (b) Table: d=0→3; d=2→6; d=4→9; d=6→12; d=8→15. (c) 20 = 1.5d + 3 → 1.5d = 17 → d ≈ 11.33, so first exceeds 20 cm on day 12.
1.5 — Movie download
(a) m = 4 MB/min (download speed). (b) y-intercept: (0, 0) — no MB at the start. x-intercept: (0, 0) (same point — line through origin). (c) 80 = 4t → t = 20 minutes.
2.1 — Explain (sample response)
(i) The y-intercept of 4 represents the flag fall — the fee a passenger is charged the moment they sit in the taxi, before any movement. It is meaningful even though no one books a 0 km ride. (ii) The real-world domain only kicks in at some small positive d (say d ≥ 0.5 km — the shortest fare most taxis will accept). For d below that, no passenger is being charged anything because no ride happens. (iii) Extrapolation happens when you read the graph beyond a reasonable range. Example: at d = 1000 km the model gives C = 2.50(1000) + 4 = $2504. In real life a 1000 km taxi trip might switch to a flat-rate or hourly model, or simply not be offered — so the linear model fails.
Marking: 1 for explaining the y-intercept meaning; 1 for stating a sensible minimum d; 1 for using all three required words; 1 for a plausible extrapolation-failure example.