Chapter 3 · Functions
Model and compare two costs, locate every valid crossover, and make a recommendation consistent with quantities, units and budget.
Drag a handle or select an object, then tap its destination.
Give your reason before checking.
| n | P | Q | P − Q |
|---|---|---|---|
| 10 | 50 | 50 | 0 |
| 20 | 80 | 100 | -20 |
| 40 | 140 | 140 | 0 |
Tab to an object. Arrow keys move handles; Enter selects a card or places it at a destination.
Select an object
Select a destination
Works stay in this open page. Use the common notebook for consent-based saved notes.
Chapter 3 · Functions
Class ______ Name __________________ Date __________
Predict which rental plan is cheaper for short and long times, and whether the printing plans can exchange the cheaper position twice. Write a prediction before comparing the cost graphs.
For rental A(t) = 16t and B(t) = 120 + 10t with 0 < t ≤ 40 hours, record both costs and A − B at t = 10, 20, 30.
For whole-number printing quantities 1 ≤ n ≤ 80, compare P(n) = 20 + 3n with Q(n) = 5n for n ≤ 20 and Q(n) = 100 + 2(n − 20) for n ≥ 21. Record costs at n = 9, 10, 11, 20, 21, 39, 40, 41.
With a budget of 200, find the largest allowed integer n for each printing plan. Check the next integer to confirm that it exceeds the budget.
A class needs 35 books, then revises its request to 50 books. Recommend a printing plan for each request using exact costs. Explain why a recommendation without a quantity can be misleading.
For the printing plans, state both same-cost quantities and recommend the cheaper plan for 30 books. With a budget of 200, which plan allows more books, and how many?
Chapter 3 · Functions · Teacher reference
3.4.2(第14課時) · 14_用函數作出有根據的選擇.docx
5 min predict plan changes; 12 min inspect costs and both crossovers; 8 min justify difference signs; 12 min paper recommendations and budget; 3 min exit check.
Predict which rental plan is cheaper for short and long times, and whether the printing plans can exchange the cheaper position twice. Write a prediction before comparing the cost graphs.
Compare the difference on each valid branch. For rental on 0 < t ≤ 40, A − B = 6t − 120, so the plans tie at t = 20 hours and cost 320; A is cheaper for 0 < t < 20 and B for 20 < t ≤ 40. For printing, P − Q = 20 − 2n on 1–20 and n − 40 on 21–80. Both branch equations have valid roots, n = 10 and n = 40. Q is cheaper at 1–9 and 41–80; P at 11–39; they tie at 10 and 40. Under a budget of 200, P allows at most 60 books and Q at most 70. The recommendation depends on the intended quantity and objective; a plan that costs less at one quantity need not allow the most books under a fixed budget.
A class needs 35 books, then revises its request to 50 books. Recommend a printing plan for each request using exact costs. Explain why a recommendation without a quantity can be misleading.
For the printing plans, state both same-cost quantities and recommend the cheaper plan for 30 books. With a budget of 200, which plan allows more books, and how many?
They tie at 10 and 40 books. At 30 books P costs 110 and Q costs 120, so choose P. Under budget 200, P permits 60 books and Q permits 70; choose Q to maximise quantity.
