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3.5 · Midline Formula
How do you find the midline of a sinusoidal function from its max and min?
Midline = ?
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3.5 · Amplitude Formula
How do you find the amplitude from the max and min values?
Amplitude = ?
Two equivalent methods
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3.5 · Period vs Frequency
How are period P and frequency related? What does each one measure?
They are reciprocals of each other
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3.5 · Sinusoidal Definition
What is a sinusoidal function? Is cos θ sinusoidal?
Think: transformations of sin θ
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3.5 · Example 1 — Read Graph
A graph has max=6, min=0, and completes one full cycle from 0 to π. Find all four properties.
max=6, min=0, period=π
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3.5 · Example 2 — Read Graph
A graph has max=3, min=−1, and completes one cycle from 0 to π/2. Find all four properties.
max=3, min=−1, period=π/2
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3.5 · Max to Min Rule
A sinusoidal function has max at θ=π and min at θ=3π. What is the period?
CRITICAL: max to min = HALF the period!
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3.5 · Example 3 — Full
h(θ): max at (π, 8), min at (3π, −2). Find period, frequency, midline, and amplitude.
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✓ Amplitude Formula
Amplitude = max − midline = (max − min) / 2
Both are equivalent. Distance from midline to the peak (or to the trough). Not the same as (max − min)!
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✓ Midline Formula
Midline = (max + min) / 2
The midline is the average of the maximum and minimum values. It is the horizontal axis of symmetry of the wave.
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✓ Sinusoidal Definition
Any additive/multiplicative transformation of sin θ. Yes, cos θ is sinusoidal: cos θ = sin(θ + π/2).
Sinusoidal functions have the same wave shape as sin θ but can have different midlines, amplitudes, periods, and phase shifts.
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✓ Period vs Frequency
Period = length of one full cycle · Frequency = 1/Period
Larger period = slower oscillation = smaller frequency. They are reciprocals: P × f = 1. E.g. period=π → frequency=1/π.
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✓ Example 2
Period=π/2 · Frequency=2/π · Midline y=1 · Amplitude=2
Midline=(3+(−1))/2=1. Amplitude=3−1=2. Period=π/2 (given). Frequency=1/(π/2)=2/π.
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✓ Example 1
Period=π · Frequency=1/π · Midline y=3 · Amplitude=3
Midline=(6+0)/2=3. Amplitude=6−3=3. Period=π (given). Frequency=1/π.
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✓ Example 3 — h(θ)
Period=4π · Frequency=1/(4π) · Midline y=3 · Amplitude=5
Half-period=3π−π=2π → period=4π. Midline=(8+(−2))/2=3. Amplitude=8−3=5.
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✓ Max to Min = HALF Period
Period = 2 × (3π − π) = 2 × 2π = 4π
The distance from max to the NEXT min is only half a period. Always multiply by 2! Distance = 3π − π = 2π → full period = 4π.
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3.5 · Example 4 — Setup
Clock: center=120 in above floor, hand=8 in, period=30 min. What are the amplitude and midline of h(t)?
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3.5 · Example 4 — 5 Points
For the clock problem (period=30 min, max=128, min=112), state the five key points F, G, J, K, P.
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3.5 · Example 4 — Frequency
In Example 4, the clock period is 30 minutes. What is the frequency of h(t)?
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3.5 · Example 4 — Decreasing & Concave Up
In Example 4, on which intervals is h(t) BOTH decreasing AND concave up?
Find where decreasing and concave-up overlap in the cycle
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✓ Five Key Points
F(0,128) G(7.5,120) J(15,112) K(22.5,120) P(30,128)
Quarter-period=7.5. At t=0: max (hand up). t=7.5: midline (hand right). t=15: min (hand down). t=22.5: midline (hand left). t=30: max again.
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✓ Amplitude & Midline
Amplitude = 8 in · Midline y = 120 in
Max=120+8=128 (hand points up). Min=120−8=112 (hand points down). Midline=(128+112)/2=120. Amplitude=128−120=8.
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✓ Decreasing & Concave Up
(7.5, 15) and (37.5, 45)
Decreasing: t=0 to 15. Concave up (lower half): t=7.5 to 22.5. Intersection = (7.5, 15). Repeats each period: (37.5, 45).
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✓ Frequency
Frequency = 1/30 (cycles per minute)
Frequency = 1/Period = 1/30. The clock completes 1/30 of a revolution per minute (because it makes one full revolution every 30 minutes).