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3.5 ยท Midline Formula
How do you find the midline of a sinusoidal function from its max and min?
Midline = ?
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3.5 ยท Amplitude Formula
How do you find the amplitude from the max and min values?
Amplitude = ?
Two equivalent methods
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3.5 ยท 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 ยท Sinusoidal Definition
What is a sinusoidal function? Is cos ฮธ sinusoidal?
Think: transformations of sin ฮธ
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3.5 ยท Example 1 โ Read Graph
A graph has max=6, min=0, and completes one full cycle from 0 to ฯ. Find all four properties.
max=6, min=0, period=ฯ
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3.5 ยท Example 2 โ Read Graph
A graph has max=3, min=โ1, and completes one cycle from 0 to ฯ/2. Find all four properties.
max=3, min=โ1, period=ฯ/2
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3.5 ยท Max to Min Rule
A sinusoidal function has max at ฮธ=ฯ and min at ฮธ=3ฯ. What is the period?
CRITICAL: max to min = HALF the period!
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3.5 ยท Example 3 โ Full
h(ฮธ): max at (ฯ, 8), min at (3ฯ, โ2). Find period, frequency, midline, and amplitude.
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โ Amplitude Formula
Amplitude = max โ midline = (max โ min) / 2
Both are equivalent. Distance from midline to the peak (or to the trough). Not the same as (max โ min)!
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โ 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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โ Sinusoidal Definition
Any additive/multiplicative transformation of sin ฮธ. Yes, cos ฮธ is sinusoidal: cos ฮธ = sin(ฮธ + ฯ/2).
Sinusoidal functions have the same wave shape as sin ฮธ but can have different midlines, amplitudes, periods, and phase shifts.
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โ Period vs Frequency
Period = length of one full cycle ยท Frequency = 1/Period
Larger period = slower oscillation = smaller frequency. They are reciprocals: P ร f = 1. E.g. period=ฯ โ frequency=1/ฯ.
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โ Example 2
Period=ฯ/2 ยท Frequency=2/ฯ ยท Midline y=1 ยท Amplitude=2
Midline=(3+(โ1))/2=1. Amplitude=3โ1=2. Period=ฯ/2 (given). Frequency=1/(ฯ/2)=2/ฯ.
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โ Example 1
Period=ฯ ยท Frequency=1/ฯ ยท Midline y=3 ยท Amplitude=3
Midline=(6+0)/2=3. Amplitude=6โ3=3. Period=ฯ (given). Frequency=1/ฯ.
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โ Example 3 โ h(ฮธ)
Period=4ฯ ยท Frequency=1/(4ฯ) ยท Midline y=3 ยท Amplitude=5
Half-period=3ฯโฯ=2ฯ โ period=4ฯ. Midline=(8+(โ2))/2=3. Amplitude=8โ3=5.
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โ Max to Min = HALF Period
Period = 2 ร (3ฯ โ ฯ) = 2 ร 2ฯ = 4ฯ
The distance from max to the NEXT min is only half a period. Always multiply by 2! Distance = 3ฯ โ ฯ = 2ฯ โ full period = 4ฯ.
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3.5 ยท Example 4 โ 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 ยท Example 4 โ 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 ยท Example 4 โ Frequency
In Example 4, the clock period is 30 minutes. What is the frequency of h(t)?
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3.5 ยท Example 4 โ 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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โ 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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โ Amplitude & Midline
Amplitude = 8 in ยท Midline y = 120 in
Max=120+8=128 (hand points up). Min=120โ8=112 (hand points down). Midline=(128+112)/2=120. Amplitude=128โ120=8.
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โ 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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โ 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).