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3.7 Β· Big Picture
What are the four ways to build a sinusoidal model? What do you extract from each?
Graph Β· Verbal Β· Equation Β· Table
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3.7 Β· From Graph
A sinusoidal graph completes one full cycle from x=β2 to x=0. What is the period?
Period = horizontal width of one cycle
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3.7 Β· Finding c
g(x)=cos(2(x+c))β1 has its maximum at x=βΟ/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 Β· Verbal β Period
A yo-yo completes 20 rotations in 5 seconds. What is the period? What is b?
Period = time Γ· rotations Β· b = 2Ο/P
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3.7 Β· 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 Β· Verbal β a and d
h(t)=aΒ·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 Β· From Equation
T(m) = 25.7Β·sin(Ο/6Β·(mβ4)) + 61.2. What are the max and min temperatures?
Max = d+a, Min = dβa
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3.7 Β· Max/Min Location
T(m) = 25.7Β·sin(Ο/6Β·(mβ4)) + 61.2. At which month m does the minimum occur?
Set sin = β1: argument = βΟ/2
Solve Ο/6Β·(mβ4) = βΟ/2
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✓ From a Graph
Period = width of one full cycle on the x-axis Β· a = (maxβmin)/2 Β· d = (max+min)/2 Β· c from key point location
4 ways: Graph β read period, a, d, c visually. Verbal β use rate and physical setup. Equation β read a,b,d directly; solve sin=Β±1 for max/min location. Table β estimate a,d from max/min data.
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✓ Big Picture β 4 Sources
Period = 0β(β2) = 2 units
Count from the start of one cycle to the start of the next. Here: cycle goes from x=β2 to x=0, so period=2.
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✓ Verbal β Period and b
Period = 5/20 = 1/4 sec Β· b = 2Ο Γ· (1/4) = 8Ο
20 rotations in 5 sec β 1 rotation per 1/4 sec. b = 2Ο/P = 2Ο/(1/4) = 8Ο.
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✓ Finding c from Max
c = Ο/2
At max of cosine: argument = 0. So 2(βΟ/2 + c) = 0 β βΟ/2 + c = 0 β c = Ο/2.
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✓ Verbal β a and d
a = 9 Β· d = 61
d = (70+52)/2 = 122/2 = 61. a = 70β61 = 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(8Οt)
a=30 (string length). P=1/4 β b=8Ο. f(0)=30sin(0)=0 β (starts at x=0). f(1/4)=30sin(2Ο)=0 β (back after one rotation).
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✓ Max/Min Location
Min at m=1 Β· Max at m=7
Ο/6Β·(mβ4)=βΟ/2 β mβ4=β3 β m=1 (minimum). Ο/6Β·(mβ4)=Ο/2 β mβ4=3 β m=7 (maximum).
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✓ From Equation β Max/Min Values
Max = 86.9Β°F Β· Min = 35.5Β°F
a=25.7, d=61.2. Max=d+a=61.2+25.7=86.9. Min=dβa=61.2β25.7=35.5. Note: d=61.2 is the MIDLINE, not the max!
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3.7 Β· From Table
N(t) data: maxβ11.4, minβ5.0. What is the predicted max from the sinusoidal model?
Model max = a + d
Estimate a and d from table data
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3.7 Β· Regression Max
For N(t): estimated aβ3.2, dβ8.2. What is the model's predicted maximum (nearest hr)?
Max = a + d
Add amplitude to midline
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✓ Regression Max
Model max β 11 hours (a+d β 3.2+8.2=11.4 β 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=11.4β8.2=3.2. Max=11.4β11.
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✓ From Table β Setup
d = (11.4+5.0)/2 = 8.2 Β· a = 11.4β8.2 = 3.2
Table maxβ11.4 (t=1), min=5.0 (t=7). Midline d=(11.4+5.0)/2=8.2. Amplitude a=11.4β8.2=3.2. Model predicted max = a+d β 11.