Mathematics • Year 8 • Unit 2 • Lesson 13
Intercepts in the Real World
Use x- and y-intercepts to answer real questions: when does the lemonade sell out, when does the rainwater tank empty, when do two delivery riders break even. Real-world intercepts have units and meanings.
1. Word problems
For each scenario the y-intercept = the starting value, and the x-intercept = the moment something hits zero.
1.1 — Rainwater tank emptying. A tank holds water modelled by V = −20t + 600, where V is litres and t is hours.
(a) Find the V-intercept (set t = 0). What does it tell you?
(b) Find the t-intercept (set V = 0). What does it tell you in plain English? 3 marks
1.2 — Phone credit. Riley's prepaid credit C ($) after sending n texts is C = −0.25n + 20.
(a) Find the C-intercept and explain what it means.
(b) Find the n-intercept and explain when the credit runs out. 3 marks
1.3 — Lemonade stand. Maya runs a stand. Her stock S (cups) over hours h follows the equation 4h + S = 60 (4 cups sold per hour, starting stock 60 cups).
(a) Find both intercepts.
(b) Interpret each intercept in the lemonade context. 3 marks
1.4 — Sketch from intercepts. A line has x-intercept (4, 0) and y-intercept (0, 3).
(a) Sketch the line in the space below using the intercept method (plot both intercepts and join).
(b) Find the gradient using the two intercepts. 3 marks
1.5 — Race start positions. In a race, two runners' positions are modelled by R₁: y = 6t and R₂: y = 4t + 5 (y in metres, t in seconds).
(a) What is the y-intercept of each line? What does each intercept mean about how the race starts?
(b) Without solving, who is faster? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked for the x-intercept of 2x + 5y = 20. They write "x-intercept: set x = 0 → 5y = 20 → y = 4 → (0, 4)". In your own words explain (i) which step is wrong, (ii) what to set to zero to find the x-intercept, and (iii) what the correct x-intercept is. Use the phrase "x-axis means y = 0" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Rainwater tank
(a) V-intercept (t = 0): V = 600 → (0, 600). The tank starts with 600 L.
(b) t-intercept (V = 0): 0 = −20t + 600 → 20t = 600 → t = 30 → (30, 0). The tank is empty after 30 hours.
1.2 — Phone credit
(a) C-intercept (n = 0): C = 20 → (0, 20). Riley starts with $20 of credit.
(b) n-intercept (C = 0): 0 = −0.25n + 20 → 0.25n = 20 → n = 80 → (80, 0). Riley's credit runs out after sending 80 texts.
1.3 — Lemonade stand
(a) S-intercept (h = 0): S = 60 → (0, 60). h-intercept (S = 0): 4h = 60 → h = 15 → (15, 0).
(b) She starts with 60 cups and sells out after 15 hours (at 4 cups per hour).
1.4 — Sketch from intercepts
(a) Plot (0, 3) on the y-axis and (4, 0) on the x-axis, then draw a straight line through both points. (Mark each axis with a scale before plotting.)
(b) Gradient m = (0 − 3)/(4 − 0) = −3/4. (The line goes down 3 units for every 4 units across — i.e. the equation is y = −¾x + 3.)
1.5 — Race start positions
(a) R₁ y-intercept = (0, 0): Runner 1 starts at the starting line (0 m). R₂ y-intercept = (0, 5): Runner 2 gets a 5 m head start.
(b) Runner 1 is faster (gradient 6 m/s vs 4 m/s). Whether and when R₁ catches R₂ depends on the gradients — that's a later lesson on simultaneous equations.
2.1 — Explain your thinking (sample response)
The classmate's mistake is in step (i): they set x = 0 to find the x-intercept. That's the trap! Because the x-axis means y = 0, you set y = 0 to find an x-intercept — the opposite letter. The correct working is 2x + 5(0) = 20 → 2x = 20 → x = 10, giving the x-intercept (10, 0). What the classmate actually computed, (0, 4), is the y-intercept of the line.
Marking: 1 mark for spotting the wrong variable was set to 0; 1 mark for the right rule (y = 0 to find x-intercept); 1 mark for the correct answer (10, 0); 1 mark for a clear full-sentence explanation using "x-axis means y = 0".