• Start/End \( x \): 0 / _________

• Quarter points \( x \):

Q1: ______ | Q2: ______ | Q3: ______

y=3 y=2 y=1 0 y=-1 y=-2 y=-3 ____ ____ ____ ____

WAVE RIDERS: WORKSHEET PAGE 1 OF 3 LENNY LEARNING SYSTEMS © 2026

Part II: Graph-to-Equation Practice

Translating Wave Graphs to Equations

Deconstructing graphical waves into mathematical parameters

Topic 5.2 Practice Workspace

Directions: Analyze the graph of each transformed wave carefully. Determine its maximum/minimum heights, midline equation, amplitude, period, and frequency parameter \( b \). Then write the final equation.

Problem 5: Analyze and Model (Sine Wave, no Phase Shift)

5 y=2 0 -1 π 2π

Step 1: Max = _______ | Min = _______

Step 2: Midline \( k = \frac{\text{Max}+\text{Min}}{2} = \) _______

Step 3: Amplitude \( a = \frac{\text{Max}-\text{Min}}{2} = \) _______

Step 4: Period length = _______ | \( b = \frac{2\pi}{\text{Period}} = \) _______

Final Sine Model (h = 0):

\( y = \) _________________________________

Problem 6: Analyze and Model (Cosine Wave, starting at Peak)

3 y=1 0 -1 π/2 π

Step 1: Max = _______ | Min = _______

Step 2: Midline \( k = \frac{\text{Max}+\text{Min}}{2} = \) _______

Step 3: Amplitude \( a = \frac{\text{Max}-\text{Min}}{2} = \) _______

Step 4: Period length = _______ | \( b = \frac{2\pi}{\text{Period}} = \) _______

Final Cosine Model (h = 0):

\( y = \) _________________________________

WAVE RIDERS: WORKSHEET PAGE 2 OF 3 LENNY LEARNING SYSTEMS © 2026

Part III: Real-World Scenarios

Modeling Periodic Environments

Drafting dynamic trigonometric models for circular and seasonal structures

Topic 5.3 Practice Applied Math

Problem 7: The Giant Ocean Star Ferris Wheel

A tourist boards a massive Ferris wheel next to the ocean. The wheel has a diameter of 40 meters. The boarding platform is 2 meters above the ground. Once boarding is complete, the wheel spins and takes exactly 12 minutes to make one full rotation. Let \( t = 0 \) represent the boarding time at the lowest position.

A. Determine Heights:

Min rider height = ______ m

Max rider height = ______ m

Vertical Midline \( k \) = ______ m

B. Find Coefficients:

Amplitude \( a \) = ______ m

Rotation Period = ______ min

Frequency \( b = \frac{2\pi}{\text{Period}} = \) ______

C. Formulate the Trigonometric Equation Model

Hint: The rider starts at the lowest point at \( t=0 \). Therefore, a negative cosine model, \( h(t) = -a \cdot \cos(b \cdot t) + k \), perfectly represents the path without a horizontal phase shift!

\( h(t) = \) _________________________________________________

Problem 8: Climate Cycles in Surf City

The average monthly temperature of Surf City fluctuates periodically throughout the year. The average low temperature is 42°F in January (\( t = 1 \)), and the average high temperature is 86°F in July (\( t = 7 \)). A full year's cycle is exactly 12 months.

A. Temperature Midline & Amp:

Average Midline \( k \):
\( k = \frac{86 + 42}{2} = \) _______ °F

Amplitude \( a \):
\( a = \frac{86 - 42}{2} = \) _______ °F

B. Period & Frequency:

Temperature Cycle = _______ months

Calculate Frequency \( b \):
\( b = \frac{2\pi}{\text{Period}} = \) _______

C. Build a Transformed Model

Let's build a weather cycle starting at its lowest point in January (\( t = 1 \)). We can model this by shifting a negative cosine model right by 1 month (\( h = 1 \))!

Write your final Cosine wave model shifted right by 1 month (\( h=1 \)):
\( T(t) = -a \cdot \cos(b(t - 1)) + k \Rightarrow \) T(t) = _______________________________

WAVE RIDERS: WORKSHEET PAGE 3 OF 3 LENNY LEARNING SYSTEMS © 2026

WAVE RIDERS: WORKSHEET PAGE 1 OF 3 LENNY LEARNING SYSTEMS © 2026

Part II: Graph-to-Equation Practice

Translating Wave Graphs to Equations

Deconstructing graphical waves into mathematical parameters

Topic 5.2 Practice Workspace

Directions: Analyze the graph of each transformed wave carefully. Determine its maximum/minimum heights, midline equation, amplitude, period, and frequency parameter b. Then write the final equation.

Problem 5: Analyze and Model (Sine Wave, no Phase Shift)

5 y=2 0 -1 π 2π

Step 1: Max = _______ | Min = _______

Step 2: Midline k =

Max+Min 2

= _______

Step 3: Amplitude a =

Max−Min 2

= _______

Step 4: Period = _______ | b =

2π Period

= _______

Final Sine Model (h = 0):

y = _________________________________

Problem 6: Analyze and Model (Cosine Wave, starting at Peak)

3 y=1 0 -1 π/2 π

Step 1: Max = _______ | Min = _______

Step 2: Midline k =

Max+Min 2

= _______

Step 3: Amplitude a =

Max−Min 2

= _______

Step 4: Period = _______ | b =

2π Period

= _______

Final Cosine Model (h = 0):

y = _________________________________

WAVE RIDERS: WORKSHEET PAGE 2 OF 3 LENNY LEARNING SYSTEMS © 2026

Part III: Real-World Scenarios

Modeling Periodic Environments

Drafting dynamic trigonometric models for circular and seasonal structures

Topic 5.3 Practice Applied Math

Problem 7: The Giant Ocean Star Ferris Wheel

A tourist boards a massive Ferris wheel next to the ocean. The wheel has a diameter of 40 meters. The boarding platform is 2 meters above the ground. Once boarding is complete, the wheel spins and takes exactly 12 minutes to make one full rotation. Let t = 0 represent the boarding time at the lowest position.

A. Determine Heights:

Min rider height = ______ m

Max rider height = ______ m

Vertical Midline k = ______ m

B. Find Coefficients:

Amplitude a = ______ m

Rotation Period = ______ min

Frequency b =

2π Period

= ______

C. Formulate the Trigonometric Equation Model

Hint: The rider starts at the lowest point at t=0. Therefore, a negative cosine model, h(t) = −a · cos(b·t) + k, perfectly represents the path without a horizontal phase shift!

h(t) = _________________________________________________

Problem 8: Climate Cycles in Surf City

The average monthly temperature of Surf City fluctuates periodically throughout the year. The average low temperature is 42°F in January (t = 1), and the average high temperature is 86°F in July (t = 7). A full year's cycle is exactly 12 months.

A. Temperature Midline & Amp:

Average Midline k:
k =

86 + 42 2

= _______ °F

Amplitude a:
a =

86 − 42 2

= _______ °F

B. Period & Frequency:

Temperature Cycle = _______ months

Calculate Frequency b:
b =

2π Period

= _______

C. Build a Transformed Model

Let's build a weather cycle starting at its lowest point in January (t = 1). We can model this by shifting a negative cosine model right by 1 month (h = 1)!

Write your final Cosine wave model shifted right by 1 month (h=1):
T(t) = −a · cos(b(t − 1)) + k ⇒ T(t) = _______________________________

WAVE RIDERS: WORKSHEET PAGE 3 OF 3 LENNY LEARNING SYSTEMS © 2026

Wave Riders Master Key (Page 2)

Graphing Calculations and Graphic Walkthrough Solutions

SOLUTIONS Classroom Guide

1. Guided Walkthrough: Graphing \( y = 2\sin(2(x - \frac{\pi}{4})) + 1 \)

Calculated Parameters:

Amplitude \( a \) = 2

Frequency \( b \) = 2

Period = \(\pi\)

Phase Shift \( h \) = \(\pi/4\) Right

Vertical Shift \( k \) = 1

Critical Bounds:

Midline: y = 1

Max Height: 1 + 2 = 3

Min Height: 1 - 2 = -1

Horizontal Bounds: \(\pi/4\) to \(5\pi/4\)

Quarter Width: \(\pi/4\)

Critical 5-Points Coordinate Table:

Start

(\(\pi/4\), 1)

Quarter

(\(\pi/2\), 3)

Mid

(\(3\pi/4\), 1)

3/4

(\(\pi\), -1)

End

(\(5\pi/4\), 1)

2. Worksheet (Page 2) Problem 5 & 6 Solutions

Problem 5 Sine Graph Answers: y = 3sin(x) + 2

• Step 1 Peak/Trough: Max = 5 | Min = -1

• Step 2 Midline (k): (5 + -1)/2 = 2 (y = 2)

• Step 3 Amplitude (a): (5 - -1)/2 = 3

• Step 4 Period & b: Pd = 2π | b = 2π/2π = 1

Problem 6 Cosine Graph Answers: y = 2cos(2x) + 1

• Step 1 Peak/Trough: Max = 3 | Min = -1

• Step 2 Midline (k): (3 + -1)/2 = 1 (y = 1)

• Step 3 Amplitude (a): (3 - -1)/2 = 2

• Step 4 Period & b: Pd = π | b = 2π/π = 2

UNIT 5: ANSWER KEYS PAGE 2 OF 3 LENNY LEARNING SYSTEMS © 2026

Teacher Resource • Answer Key

Wave Riders Master Key (Page 3)

Applied Real-World trigonometric Modeling Solutions

SOLUTIONS Classroom Guide

1. Guided Notes Page 3: Harbor Tides Model Solution

• Max Depth = 16 meters | Min Depth = 4 meters

• Midline Average Depth \( k = \frac{16 + 4}{2} = \) 10 meters

• Amplitude Fluctuation \( a = \frac{16 - 4}{2} = \) 6 meters

• Time High-to-Low is 6 hrs \(\to\) Full Cycle Period = 12 hours

• Frequency Coefficient \( b = \frac{2\pi}{12} = \) \(\pi/6\)

Final Tides Equation: \( d(t) = 6\cos(\frac{\pi}{6}t) + 10 \)

2. Practice Worksheet Problem 7: Ocean Star Ferris Wheel Solution

• Height Bounds: Min height = 2 m | Max height = 2 + 40 = 42 m

• Vertical Center Midline \( k = \frac{42 + 2}{2} = \) 22 m

• Radius Amplitude \( a = \frac{42 - 2}{2} = \) 20 m

• Rotation Period = 12 minutes

• Frequency \( b = \frac{2\pi}{12} = \) \(\pi/6\)

Final Ferris Wheel Equation: \( h(t) = -20\cos(\frac{\pi}{6}t) + 22 \)

3. Practice Worksheet Problem 8: Climate Cycles in Surf City Solution

• Average Temperatures: Min = 42°F (January) | Max = 86°F (July)

• Yearly Temperature Midline \( k = \frac{86 + 42}{2} = \) 64°F

• Temperature Amplitude \( a = \frac{86 - 42}{2} = \) 22°F

• Cycle Period = 12 months | Frequency \( b = \frac{2\pi}{12} = \) \(\pi/6\)

• Phase Shift: Minimum occurs at January (\( t = 1 \)), shifting negative cosine right by 1.

Final Temperature Equation: \( T(t) = -22\cos(\frac{\pi}{6}(t - 1)) + 64 \)

UNIT 5: ANSWER KEYS PAGE 3 OF 3 LENNY LEARNING SYSTEMS © 2026

Frequency b = 2

Period = π

Phase Shift h = π/4 (Right)

Vertical Shift k = 1

Critical Bounds:

Midline: y = 1

Max Height: 3

Min Height: -1

Horizontal Bounds: π/4 to 5π/4

Quarter Width: π/4

Critical 5-Points Coordinate Table:

Start

(π/4, 1)

Quarter

(π/2, 3)

Mid

(3π/4, 1)

3/4

(π, -1)

End

(5π/4, 1)

2. Worksheet (Page 2) Problem 5 & 6 Solutions

Problem 5 Sine Graph Answers: y = 3·sin(x) + 2

• Step 1 Peak/Trough: Max = 5 | Min = -1

• Step 2 Midline (k): (5 + -1)/2 = 2 (y = 2)

• Step 3 Amplitude (a): (5 − -1)/2 = 3

• Step 4 Period & b: Pd = 2π | b = 1

Problem 6 Cosine Graph Answers: y = 2·cos(2x) + 1

• Step 1 Peak/Trough: Max = 3 | Min = -1

• Step 2 Midline (k): (3 + -1)/2 = 1 (y = 1)

• Step 3 Amplitude (a): (3 − -1)/2 = 2

• Step 4 Period & b: Pd = π | b = 2

UNIT 5: ANSWER KEYS PAGE 2 OF 3 LENNY LEARNING SYSTEMS © 2026

Teacher Resource • Answer Key

Wave Riders Master Key (Page 3)

Applied Real-World trigonometric Modeling Solutions

SOLUTIONS Classroom Guide

1. Guided Notes Page 3: Harbor Tides Model Solution

• Max Depth = 16 meters | Min Depth = 4 meters

• Midline Average Depth k = (16+4)/2 = 10 meters

• Amplitude Fluctuation a = (16-4)/2 = 6 meters

• Time High-to-Low is 6 hrs → Full Cycle Period = 12 hours

• Frequency Coefficient b = 2π/12 = π/6

Final Tides Equation: d(t) = 6·cos((π/6)t) + 10

2. Practice Worksheet Problem 7: Ocean Star Ferris Wheel Solution

• Height Bounds: Min height = 2 m | Max height = 2 + 40 = 42 m

• Vertical Center Midline k = (42+2)/2 = 22 m

• Radius Amplitude a = (42-2)/2 = 20 m

• Rotation Period = 12 minutes

• Frequency b = 2π/12 = π/6

Final Ferris Wheel Equation: h(t) = −20·cos((π/6)t) + 22

3. Practice Worksheet Problem 8: Climate Cycles Solution

• Average Temperatures: Min = 42°F (Jan) | Max = 86°F (July)

• Yearly Temperature Midline k = (86+42)/2 = 64°F

• Temperature Amplitude a = (86-42)/2 = 22°F

• Cycle Period = 12 months | Frequency b = π/6

• Phase Shift: Minimum occurs in Jan (t=1), shifting negative cosine right by 1.

Final Temperature Equation: T(t) = −22·cos((π/6)(t − 1)) + 64

UNIT 5: ANSWER KEYS PAGE 3 OF 3 LENNY LEARNING SYSTEMS © 2026