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3.7 · Big Picture
What are the four ways to build a sinusoidal model? What do you extract from each?
Graph · Verbal · Equation · Table
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3.7 · From Graph
A sinusoidal graph completes one full cycle from x=−2 to x=0. What is the period?
Period = horizontal width of one cycle
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3.7 · Finding c
g(x)=cos(2(x+c))−1 has its maximum at x=−π/2. How do you find c?
At max: argument of cos = 0
Set 2(x_max + c) = 0 and solve
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3.7 · Verbal → Period
A yo-yo completes 20 rotations in 5 seconds. What is the period? What is b?
Period = time ÷ rotations · b = 2π/P
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3.7 · Yo-yo Model
Yo-yo: 30-inch string, period=1/4 sec, starts at x=0 (t=0). Which form fits?
f(t) = 30sin(bt)
What is b? Does f(0)=0 hold?
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3.7 · Verbal → a and d
h(t)=a·sin(b(t+c))+d. h(0)=70 (max) and h(30)=52 (min). Find a and d.
Midline = avg of max and min
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3.7 · From Equation
T(m) = 25.7·sin(π/6·(m−4)) + 61.2. What are the max and min temperatures?
Max = d+a, Min = d−a
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3.7 · Max/Min Location
T(m) = 25.7·sin(π/6·(m−4)) + 61.2. At which month m does the minimum occur?
Set sin = −1: argument = −π/2
Solve π/6·(m−4) = −π/2
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✓ From a Graph
Period = width of one full cycle on the x-axis · a = (max−min)/2 · d = (max+min)/2 · c from key point location
4 ways: Graph → read period, a, d, c visually. Verbal → use rate and physical setup. Equation → read a,b,d directly; solve sin=±1 for max/min location. Table → estimate a,d from max/min data.
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✓ Big Picture — 4 Sources
Period = 0−(−2) = 2 units
Count from the start of one cycle to the start of the next. Here: cycle goes from x=−2 to x=0, so period=2.
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✓ Verbal → Period and b
Period = 5/20 = 1/4 sec · b = 2π ÷ (1/4) = 8π
20 rotations in 5 sec → 1 rotation per 1/4 sec. b = 2π/P = 2π/(1/4) = 8π.
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✓ Finding c from Max
c = π/2
At max of cosine: argument = 0. So 2(−π/2 + c) = 0 → −π/2 + c = 0 → c = π/2.
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✓ Verbal → a and d
a = 9 · d = 61
d = (70+52)/2 = 122/2 = 61. a = 70−61 = 9. The maximum is 70 (hand at top) and minimum is 52 (hand at bottom).
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✓ Yo-yo Model
f(t) = 30sin(8πt)
a=30 (string length). P=1/4 → b=8π. f(0)=30sin(0)=0 ✓ (starts at x=0). f(1/4)=30sin(2π)=0 ✓ (back after one rotation).
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✓ Max/Min Location
Min at m=1 · Max at m=7
π/6·(m−4)=−π/2 → m−4=−3 → m=1 (minimum). π/6·(m−4)=π/2 → m−4=3 → m=7 (maximum).
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✓ From Equation — Max/Min Values
Max = 86.9°F · Min = 35.5°F
a=25.7, d=61.2. Max=d+a=61.2+25.7=86.9. Min=d−a=61.2−25.7=35.5. Note: d=61.2 is the MIDLINE, not the max!
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✓ Regression Max
Model max ≈ 11 hours (a+d ≈ 3.2+8.2=11.4 → rounds to 11)
The regression gives a sinusoidal curve. Its maximum = a+d. Estimate from table: d=(11.4+5.0)/2=8.2, a=11.4−8.2=3.2. Max=11.4≈11.
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✓ From Table — Setup
d = (11.4+5.0)/2 = 8.2 · a = 11.4−8.2 = 3.2
Table max≈11.4 (t=1), min=5.0 (t=7). Midline d=(11.4+5.0)/2=8.2. Amplitude a=11.4−8.2=3.2. Model predicted max = a+d ≈ 11.