Zadanie 8d, Paper 1, listopad 2024, TZ2, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — listopad 2024, TZ2, Paper1 (zadania wzorowane), pytanie 8(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
Let f(x) = log5(25x) for x>0. The transformed graph in part (d) is restricted to x>0 as well.
Polecenie
For x>0, obtain the graph of y = f(125x3) from y = log5 x using one translation and one stretch. Describe both transformations and their order.
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Odpowiedź z klucza
Log rules give f(125x3) = log5(3125x3) = 5 + 3log5 x. Thus stretch vertically by factor 3, then translate upwards by 5 units. Equivalently translate upwards by 5/3 first and then stretch vertically by 3. A horizontal compression plus vertical stretch can produce the same function, but does not supply the requested translation.
Schemat punktowania
M1: apply the product/addition log rule. A1: correct split into a constant logarithm and a power term. M1: apply the power rule. A1: 5+3log_5x. A2: a correct ordered translation/stretch pair; award A1 if both operations are correct but order omitted. A valid description using only two stretches is capped at 5/6 because a translation is explicitly requested. For the two-stretch alternative, award M1 for attempting to express the argument as a power, A1 for log((k x)^3) with k=5^(5/3), M1 for the logarithmic power rule, A1 for 3log(k x), and A1 for vertical factor 3 with horizontal factor 1/k (either order), maximum 5/6.
Komentarz. Paper 1: no calculator. The AA SL formula booklet is allowed. Give exact answers and show the reasoning.