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πŸ“š Topic 3.5 Β· Unit 3

Sinusoidal
Functions

Midline Β· amplitude Β· period Β· frequency Β· reading graphs Β· building sinusoidal models from properties.

πŸ“‹ Notes πŸƒ Flashcards 🧠 Practice Quiz
🌊 Topic 3.5 β€” Sinusoidal Functions
πŸ“–
Key Definitions
Midline Β· Amplitude Β· Period Β· Frequency Β· Sinusoidal Functions
TermDefinitionFormula
Midline Horizontal line halfway between the maximum and minimum values of the function y = (max + min) / 2
Amplitude Distance from the midline to the maximum (or minimum) value a = (max βˆ’ min) / 2
Period The length of one full cycle of the function P = (distance from max to next max)
Frequency The reciprocal of the period β€” how many cycles occur per unit f = 1P
🎯

Max to min = HALF a period

The distance from a maximum to the very next minimum is only half a period. So if max is at ΞΈ = Ο€ and min is at ΞΈ = 3Ο€, the half-period is 2Ο€ β†’ full period = 4Ο€. Always double the max-to-min distance to get the full period!

πŸ“Œ Sinusoidal Functions

A sinusoidal function is any function that involves additive and multiplicative transformations of f(ΞΈ) = sin ΞΈ.
Both sine and cosine are sinusoidal functions because: g(ΞΈ) = cos ΞΈ = sin(ΞΈ + Ο€2)
Cosine is just a sine function shifted left by Ο€2. This means any sinusoidal function can be written as either a sine or cosine transformation.
πŸ“Š
Reading Properties from Graphs β€” Examples 1 & 2
Find max and min β†’ compute midline, amplitude Β· find repeat distance β†’ period
y=3 Ο€/2 Ο€ 3Ο€/2 2Ο€ 6 3 0 ΞΈ f(ΞΈ)
Graph of f β€” max=6, min=0, midline y=3, amplitude=3, period=Ο€
πŸ“Œ Example 1 β€” Find period, frequency, amplitude, and midline for f(ΞΈ).
1
Period: One full cycle from 0 to Ο€ (the graph completes a full wave by ΞΈ=Ο€). P = Ο€
2
Frequency: 1P = 1Ο€
3
Max = 6, Min = βˆ’6 Β· Midline = (6 + (βˆ’6))/2 = y = 0
4
Amplitude: (max βˆ’ min)/2 = (6 βˆ’ (βˆ’6))/2 = 12/2 = 3... wait: max = 6 so amplitude = 6 βˆ’ 0 = 3? No β€” max is 6, midline is 0 (not 3). Amplitude = max βˆ’ midline = 6 βˆ’ 0 = 3. Hmm, but graph goes to 6...
Period
Ο€
Frequency
1Ο€
Midline
y = 0
Amplitude
3
From the graph: maximum = 6, minimum = 0. Midline = (6+0)/2 = 3. Amplitude = 6βˆ’3 = 3.
✏️

Corrected reading: max=6, min=0

The graph of f oscillates between max = 6 and min = 0 (it never goes below the x-axis). So: midline = (6+0)/2 = y = 3. Amplitude = 6 βˆ’ 3 = 3. Period = Ο€ (one complete cycle seen). Frequency = 1Ο€.

y=1 Ο€/2 Ο€ 3Ο€/2 2Ο€ 3 1 βˆ’1 ΞΈ g(ΞΈ)
Graph of g β€” max=3, min=βˆ’1, midline y=1, amplitude=2, period=Ο€/2
πŸ“Œ Example 2 β€” Find period, frequency, amplitude, and midline for g(ΞΈ). (max = 3, min = βˆ’1)
1
Period: One full cycle completes in Ο€2 (two cycles visible from 0 to Ο€). P = Ο€2
2
Frequency: 1P = 1/(Ο€2) = 2Ο€
3
Midline: (max + min)/2 = (3 + (βˆ’1))/2 = 2/2 = y = 1
4
Amplitude: max βˆ’ midline = 3 βˆ’ 1 = 2
Period
Ο€2
Frequency
2Ο€
Midline
y = 1
Amplitude
2
πŸ”’
Finding Properties from Max/Min Points β€” Example 3
Max to min = half period Β· midline = average of max and min values
πŸ“Œ Example 3 β€” h(ΞΈ) has a maximum at (Ο€, 8) and the first minimum after is at (3Ο€, βˆ’2). Find all properties.
1
Half-period: Distance from max to min = 3Ο€ βˆ’ Ο€ = 2Ο€ β†’ this is HALF the period.
2
Period: Full period = 2 Γ— (half period) = 2 Γ— 2Ο€ = 4Ο€
3
Frequency: 1P = 1/(4Ο€)
4
Midline: (max + min)/2 = (8 + (βˆ’2))/2 = 6/2 = y = 3
5
Amplitude: max βˆ’ midline = 8 βˆ’ 3 = 5
Period
4Ο€
Frequency
1/(4Ο€)
Midline
y = 3
Amplitude
5
βš™οΈ Step-by-step derivation
Max at ΞΈ=Ο€ (y=8) Β· Min at ΞΈ=3Ο€ (y=βˆ’2)
Half period = 3Ο€ βˆ’ Ο€ = 2Ο€ β†’ Full period P = 4Ο€
Midline = (8 + (βˆ’2))/2 = 6/2 = y = 3
Amplitude = 8 βˆ’ 3 = 5
πŸ•
Real-World Application β€” Example 4 (FRQ Model)
Clock problem Β· identify 5 key points Β· find all properties Β· intervals for decreasing and concave up
πŸ“‹

Setup

A clock is mounted on a wall. Center of clock = 120 inches above the floor. Minute hand = 8 inches long. Clock runs twice as fast as normal, so one revolution = 30 minutes. At t=0 the hand points straight up (12 o'clock). h(t) = distance from endpoint of hand to floor.

Part (A) β€” Five Key Points
t=0: Hand points up β†’ endpoint = 120 + 8 = 128 in
t=7.5: Hand points right (ΒΌ revolution) β†’ endpoint at midline height = 120 in
t=15: Hand points down (Β½ revolution) β†’ endpoint = 120 βˆ’ 8 = 112 in
t=22.5: Hand points left (ΒΎ revolution) β†’ endpoint at midline = 120 in
t=30: Back to top (full revolution) β†’ endpoint = 128 in
F: (0, 128)
G: (7.5, 120)
J: (15, 112)
K: (22.5, 120)
P: (30, 128)
Part (B) β€” Properties of h(t)
Max = 128 (hand points up)
Min = 112 (hand points down)
Midline = (128 + 112)/2 = y = 120
Amplitude = 128 βˆ’ 120 = 8
Period = 30 minutes (twice as fast)
Frequency = 130
Period
30
Frequency
130
Midline
y = 120
Amplitude
8
Part (C) β€” Find two intervals where h(t) is both decreasing AND concave up.
1
Decreasing: h is decreasing when the hand moves from the top (max) toward the bottom (min): from t=0 to t=15, and again from t=30 to t=45 in the second cycle.
2
Concave up: A sine curve is concave up in its lower half β€” from the midline crossing (going down) to the midline crossing (coming back up). That's from t=7.5 to t=22.5 for each cycle.
3
Intersection: Decreasing (0,15) ∩ Concave up (7.5,22.5) = (7.5, 15). In second cycle: (37.5, 45).
h is decreasing AND concave up on: (7.5, 15) and (37.5, 45)
πŸ’‘

How to find "decreasing and concave up" on a sinusoidal graph

Sketch one full cycle. The graph is decreasing from max to min. It is concave up in the bottom half (below the midline) β€” from the descending midline crossing to the ascending midline crossing. The intersection of these two behaviors is the second quarter of each cycle (from the descending midline crossing to the minimum).

⚑
Quick Reference
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πŸ”‘ Key Formulas
Midline = (max + min) / 2
Amplitude = max βˆ’ midline = (max βˆ’ min)/2
Period P = 2 Γ— (distance from max to min)
Frequency = 1P
cos ΞΈ = sin(ΞΈ + Ο€2) β€” sinusoidal!
⚠️ Common Mistakes
❌ max βˆ’ min = amplitude
(max βˆ’ min) = twice the amplitude. Divide by 2: amplitude = (max βˆ’ min)/2. Or: amplitude = max βˆ’ midline.
❌ Max-to-min distance = full period
Max to min is only HALF the period. Always multiply by 2 to get the full period.
❌ sin starts at max; cos starts at 0
It's the other way: sin(0)=0 (starts at zero), cos(0)=1 (starts at MAX). sin begins at the midline, cos begins at the peak.
❌ Frequency = period
Frequency = 1Period. They are reciprocals. Larger period β†’ smaller frequency (fewer cycles per unit).
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