Mathematics • Year 8 • Unit 2 • Lesson 14

Sketching for Real Decisions

Use linear sketches to make decisions: which streaming plan is cheaper, which delivery company suits short trips, when two cars meet. Sketch a quick line and read off the answer.

Apply · Real-World Maths

1. Word problems

For each scenario, identify the equation, decide which sketching method is fastest, find two key points, and answer the question.

1.1 — Streaming choice. StreamPlus costs $7 per month plus a one-off $5 signup. The total cost is C = 7m + 5 over m months.

(a) Which sketching method fits this equation form best?
(b) Find C at m = 0 and m = 6, then sketch the line in your own axes.
(c) Predict the cost after 12 months from your line. 3 marks

Stuck? Equation is y = mx + c, so gradient-intercept is fastest. m = 7 (per month); c = 5 (signup).

1.2 — Delivery company budget. A delivery firm charges according to 5x + 2y = 40, where x is parcels delivered and y is leftover budget ($).

(a) Find both intercepts and state what each means in this context.
(b) Sketch the line, then read off the leftover budget when 4 parcels have been delivered. 3 marks

Stuck? Standard form ax + by = c → use the intercept method.

1.3 — Hourly wage. Yasmin earns $20 per hour with no base pay. Her earnings E follow E = 20h.

(a) State the y-intercept and explain what it means.
(b) Pick two points (h = 0 and h = 5) and sketch the line. Mark the point (5, 100) on it. 3 marks

Stuck? No base pay → c = 0 → line passes through the origin. Use two-point method with (0, 0) and one other point.

1.4 — Falling temperature. The afternoon temperature in a Sydney suburb is modelled by T = −2h + 30, where T (°C) is the temperature h hours after 12 noon.

(a) State the T-intercept and the h-intercept.
(b) Sketch the line, then describe in one sentence what happens at the h-intercept. 3 marks

Stuck? T-intercept (h = 0): T = 30°C (start at noon). h-intercept (T = 0): when does the temperature hit 0°C?

1.5 — Two delivery riders. Rider A's distance from base: dₐ = 12t. Rider B's: d_B = 8t + 10 (km and minutes).

(a) Sketch BOTH lines on the same axes between t = 0 and t = 5.
(b) Roughly when (in minutes) does Rider A overtake Rider B? Show the meeting point on your sketch. 3 marks

Stuck? Sketch each line through its y-intercept using its gradient. They meet where the two lines cross.

2. Explain your thinking

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

2.1 A classmate is asked to sketch the equation 3x − y = 6. They write "It's in y = mx + c form, so I'll plot c = 6 on the y-axis and use gradient 3." In your own words explain (i) why their identification of the equation's form is wrong, (ii) what method they should use, and (iii) give the correct y-intercept and x-intercept for 3x − y = 6. Use the phrase "rearrange to y = mx + c" somewhere in your answer.

Stuck? Revisit lesson § Card 5 (Choosing the Best Method) — 3x − y = 6 is in standard form, not y = mx + c.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — StreamPlus

(a) Gradient-intercept method — equation is y = mx + c with m = 7, c = 5.
(b) C(0) = 5; C(6) = 7(6) + 5 = 47. Sketch passes through (0, 5) and (6, 47).
(c) C(12) = 7(12) + 5 = $89.

1.2 — Delivery budget

(a) y-int (x = 0): 2y = 40 → y = 20 → (0, 20) means $20 budget left when 0 parcels delivered (i.e. starting budget is $20). x-int (y = 0): 5x = 40 → x = 8 → (8, 0) means the budget runs out after 8 parcels.
(b) At x = 4: 5(4) + 2y = 40 → 2y = 20 → y = $10 leftover budget.

1.3 — Hourly wage

(a) y-intercept (0, 0) — Yasmin earns nothing if she works 0 hours (no base pay).
(b) At h = 0: E = $0. At h = 5: E = $100 → point (5, 100). Line through origin and (5, 100).

1.4 — Falling temperature

(a) T-intercept: T(0) = 30 → (0, 30). h-intercept: 0 = −2h + 30 → h = 15 → (15, 0).
(b) Line falls from (0, 30) to (15, 0). At the h-intercept the modelled temperature would reach 0°C, which is 15 hours after noon (3 a.m. the next day) — well outside the realistic afternoon range, but that's what the line predicts.

1.5 — Two delivery riders

(a) Rider A: through (0, 0) with gradient 12. Rider B: through (0, 10) with gradient 8. Both lines on same axes, t from 0 to 5.
(b) Set equal: 12t = 8t + 10 → 4t = 10 → t = 2.5 minutes. At that time both riders are at d = 12(2.5) = 30 km. They meet at the point (2.5, 30) on the sketch.

2.1 — Explain your thinking (sample response)

The classmate misread the form. The equation 3x − y = 6 is in standard form (ax + by = c), not gradient-intercept form, so c = 6 is NOT the y-intercept and 3 is NOT the gradient. The fastest method here is the intercept method: set x = 0 to get −y = 6 → y = −6 (y-intercept (0, −6)); set y = 0 to get 3x = 6 → x = 2 (x-intercept (2, 0)). If you really want gradient-intercept form, you must first rearrange to y = mx + c: y = 3x − 6, giving m = 3 and the correct y-intercept c = −6 (not +6).

Marking: 1 mark for noticing the form is standard, not y = mx + c; 1 mark for naming the intercept method; 1 mark for the correct intercepts (0, −6) and (2, 0); 1 mark for a clear full-sentence answer using "rearrange to y = mx + c".