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Flip to reveal answersWrite the general sinusoidal model and name every parameter.
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All 16 Flashcards — Sinusoidal models
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Question
Write the general sinusoidal model and name every parameter.
Answer
f(t) = a sin(bt + c) + d. a = amplitude (half the range). Period = 2π/b. c = phase shift. d = midline (vertical shift).
Question
What is the amplitude of f(t) = 3 sin(2t) + 5?
Answer
Amplitude = 3. It is the coefficient of sin — the distance from the midline to the maximum or minimum.
Question
What is the period of f(t) = sin(πt/6)?
Answer
Period = 2π ÷ (π/6) = 2π × 6/π = 12.
Question
In f(t) = 4 cos(2πt/12) + 10, what is the midline and what values does f oscillate between?
Answer
Midline y = 10. Amplitude = 4, so f oscillates between 10 − 4 = 6 and 10 + 4 = 14.
Question
How do you find amplitude and midline from the max and min values?
Answer
Amplitude = (max − min) / 2. Midline = (max + min) / 2.
Question
A model has maximum 18 and minimum 4. Find the amplitude and midline.
Answer
Amplitude = (18 − 4)/2 = 7. Midline = (18 + 4)/2 = 11.
Question
Temperature oscillates between 8°C and 24°C daily. State the midline and amplitude.
Answer
Midline = (8 + 24)/2 = 16°C. Amplitude = (24 − 8)/2 = 8°C.
Question
The period of a sinusoidal model is 24 hours. Find b in f(t) = a sin(bt) + d.
Answer
2π/b = 24 → b = 2π/24 = π/12.
Question
IB asks for amplitude. Student writes "the maximum is 18." What is wrong?
Answer
Amplitude = (max − min)/2, not the maximum value alone. If min = 4, amplitude = (18 − 4)/2 = 7, not 18.
Question
What is the difference between period and frequency?
Answer
Period: how long one complete cycle takes (in time units, e.g. hours). Frequency: cycles per unit time = 1/period.
Question
A student says the period is b (the coefficient inside sin). What is wrong?
Answer
b is not the period — it is a parameter inside the argument. Period = 2π/b. For b = 2, period = π, not 2.
Question
f(t) = 5 sin(...) + 12. Student says maximum = 12 (reading the midline as max). What is the actual maximum?
Answer
Maximum = midline + amplitude = 12 + 5 = 17. The midline d is not the maximum.
Question
f(t) = 7 sin(πt/12) + 15. Find f(6).
Answer
f(6) = 7 sin(π · 6/12) + 15 = 7 sin(π/2) + 15 = 7(1) + 15 = 22.
Question
Tide height: h(t) = 3 sin(πt/6) + 5. Find h(3).
Answer
h(3) = 3 sin(π/2) + 5 = 3(1) + 5 = 8 m.
Question
A model predicts a value greater than the maximum. What does this indicate?
Answer
Either a calculation error, or the model is being used outside its valid range. A sinusoidal model never exceeds amplitude + midline.
Question
T(t) = 8 sin(πt/12) + 12. Find the first time after t = 0 when T = 20.
Answer
8 sin(πt/12) + 12 = 20 → sin(πt/12) = 1 → πt/12 = π/2 → t = 6 hours.
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