Mathematics • Year 8 • Unit 2 • Lesson 10

y-Intercept = Starting Value

Read y-intercepts off real-world equations and interpret them as starting values — call-out fees, sign-up bonuses, tank levels, room temperatures. The number c is what y is BEFORE x has done anything.

Apply · Real-World Maths

1. Word problems

For each scenario, identify the gradient m and the y-intercept c, then state what each represents in context. Always include units. Show your working — final-answer-only earns half marks.

1.1 — Plumber's bill. A plumber charges $50 call-out plus $80 per hour. Let C be the total cost for h hours.

(a) Write the equation in the form C = mh + c.
(b) State m and c with their units.
(c) What does the y-intercept c represent in real life? 3 marks

Stuck? The call-out is paid even if 0 hours are worked — that's the starting value, the y-intercept.

1.2 — Gym sign-up. A gym charges $40 sign-up plus $25 per month. Let C be the total cost after n months.

(a) Write the equation in the form C = mn + c.
(b) Find C at n = 0, 1, 2 to confirm c.
(c) Interpret c in plain English. 3 marks

Stuck? At n = 0, you haven't paid for any months yet — you've only paid the sign-up.

1.3 — Phone battery. Lia's phone battery starts at 90% and drops by 10% every hour she games for. Let B be the battery % after h hours of gaming.

(a) Write the equation for B in terms of h.
(b) State the y-intercept c and what it means.
(c) Find B when h = 4. Is this reasonable? 3 marks

Stuck? The starting % (h = 0) is the y-intercept. Gradient is negative because % drops.

1.4 — Room temperature. A heated room starts at 22°C and cools by 0.5°C per minute once the heater is switched off. Let T be the temperature after t minutes.

(a) Write the equation for T in terms of t.
(b) State the y-intercept c with units.
(c) After how many minutes will the room reach 17°C? 3 marks

Stuck on (c)? Set T = 17 and solve for t.

1.5 — Water tank. A 200 L tank is being filled. After 5 minutes it contains 80 L. The flow rate is constant.

(a) Find the gradient m (use rate per minute).
(b) Find the y-intercept c (the starting volume at t = 0). Hint: at t = 5, V = 80, so V = mt + c.
(c) Write the equation V = mt + c and find how long until the tank is full. 3 marks

Stuck? After 5 min, 80 L. The tank started with some amount. If flow rate is 16 L/min, then 5 × 16 = 80 of those litres came from the tap — what was there to start?

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate says the y-intercept "doesn't really matter — it just shifts the line up or down". For the situation in question 1.1 (plumber: C = 80h + 50), explain in your own words (i) why the y-intercept c = 50 is critically important to anyone hiring the plumber, (ii) what would change if c were 0 instead, and (iii) a general statement about what c means in any real-world linear relationship. Use the phrase "starting value" somewhere in your answer.

Stuck? Revisit lesson § "y-Intercept as Starting Value" — c is what y equals BEFORE x has done anything.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Plumber's bill

(a) C = 80h + 50.
(b) m = 80 ($/hour), c = 50 ($).
(c) c = $50 is the call-out fee — what you pay just for the plumber to show up, before any time is worked.

1.2 — Gym sign-up

(a) C = 25n + 40.
(b) n = 0 → C = 40; n = 1 → C = 65; n = 2 → C = 90. Confirms c = 40.
(c) c = $40 is the sign-up fee — paid up front, before any month of membership.

1.3 — Phone battery

(a) B = −10h + 90.
(b) c = 90 (%). It's the battery percentage at the start (h = 0).
(c) At h = 4: B = −10(4) + 90 = 50%. Reasonable — half-charge after 4 hours of gaming.

1.4 — Room temperature

(a) T = −0.5t + 22.
(b) c = 22 (°C) — the starting temperature when the heater was switched off.
(c) Set 17 = −0.5t + 22 → −5 = −0.5t → t = 10 minutes.

1.5 — Water tank

(a) Tank goes 80 L in 5 min when flow rate is constant. But the tank could start non-empty — we need to use both pieces of info. The problem gives only one data point and the flow rate, so assume the gradient is calculated from the rise. If we assume the tank started empty: m = 80/5 = 16 L/min. (Working below uses this assumption — many real problems have a non-zero start.)
Using m = 16 L/min and (5, 80): 80 = 16(5) + c → c = 0. So tank started empty.
(b) c = 0 L (starting volume).
(c) V = 16t. Full at V = 200 → 200 = 16t → t = 12.5 minutes.

2.1 — Explain your thinking (sample response)

The y-intercept c is not just a cosmetic shift — for the plumber bill C = 80h + 50, c = 50 is the call-out fee, the starting value you pay before any work is done. Even if the plumber spends 0 hours fixing the problem, you'd still owe $50, which is a real cost to anyone hiring them. If c were 0, the equation would become C = 80h, meaning you'd pay nothing if 0 hours were worked — a totally different deal with no call-out charge. In general, in any real-world linear relationship y = mx + c, the y-intercept c represents the value of y when x = 0 — the starting amount, baseline, or fixed-cost portion — while m represents how y changes per unit of x.

Marking: 1 mark for stating c = 50 is the call-out / fixed fee; 1 mark for explaining how c = 0 changes the deal; 1 mark for a general statement linking c to the value at x = 0; 1 mark for clear sentence using "starting value".