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.
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
1.2, Gym sign-up. A gym charges $40 sign-up plus $25 per month. Let C be the total cost after n months.
(a) Write the equation in the form C = mn + c.
(b) Find C at n = 0, 1, 2 to confirm c.
(c) Interpret c in plain English. 3 marks
1.3, Phone battery. Lia's phone battery starts at 90% and drops by 10% every hour she games for. Let B be the battery % after h hours of gaming.
(a) Write the equation for B in terms of h.
(b) State the y-intercept c and what it means.
(c) Find B when h = 4. Is this reasonable? 3 marks
1.4, Room temperature. A heated room starts at 22°C and cools by 0.5°C per minute once the heater is switched off. Let T be the temperature after t minutes.
(a) Write the equation for T in terms of t.
(b) State the y-intercept c with units.
(c) After how many minutes will the room reach 17°C? 3 marks
1.5, Water tank. A 200 L tank is being filled at a constant rate of 16 L per minute. When timing starts (t = 0) the tank already contains some water, and after 5 minutes it contains 80 L.
(a) State the gradient m (the flow rate in L/min).
(b) Find the y-intercept c (the starting volume at t = 0). Use V = mt + c with the data point t = 5, V = 80.
(c) Write the equation V = mt + c and find how long until the tank is full (200 L). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says the y-intercept "doesn't really matter, it just shifts the line up or down". For the situation in question 1.1 (plumber: C = 80h + 50), explain in your own words (i) why the y-intercept c = 50 is critically important to anyone hiring the plumber, (ii) what would change if c were 0 instead, and (iii) a general statement about what c means in any real-world linear relationship. Use the phrase "starting value" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1, Plumber's bill
(a) C = 80h + 50.
(b) m = 80 ($/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) Flow rate = m = 16 L/min (given).
(b) Use V = 16t + c with t = 5, V = 80: 80 = 16(5) + c → 80 = 80 + c → c = 0 L. The tank was empty when timing started.
(c) V = 16t. Full at V = 200 → 200 = 16t → t = 12.5 → 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".