MathIsimple
Lesson 4-3

Trigonometric Function Transformations & Modeling

Understand amplitude, period, phase, and vertical shifts. Compose transformations and fit sinusoidal models to real periodic phenomena.

Learning Objectives Back to Unit 4

Learning Objectives

  • Identify amplitude, period, phase shift, and vertical shift in y = A sin(ωx + φ) + k.
  • Compose transformations and graph key points efficiently.
  • Fit sinusoidal models to real data using peak/trough/zero-crossing features.
  • Interpret model parameters in context and validate unit consistency.
  • Compare sin and cos forms and convert using phase relations.

Core Transformations

Amplitude & Vertical Shift

y=Asin⁡(ωx+ϕ)+ky=A\sin(\omega x+\phi)+k

Amplitude = |A|, vertical shift = k.

Period & Phase

T=2πω,  phase shift=−ϕωT=\tfrac{2\pi}{\omega},\; \text{phase shift}= -\tfrac{\phi}{\omega}

Period compresses for ω > 1, stretches for 0 < ω < 1.

Worked Examples

Example 1: Describe Transformation

For y=2sin⁡(3x−π2)+1y=2\sin(3x-\tfrac{\pi}{2})+1, list amplitude, period, phase shift, vertical shift.

Amplitude 2; period 2π3\tfrac{2\pi}{3}; phase shift 13π/3=π6\tfrac{1}{3}\pi/3=\tfrac{\pi}{6} right; vertical shift +1.

Example 2: Fit a Sinusoid

Monthly temperature model with max 40°C at June (m=6), min 10°C at December (m=12). Find T(m)=Asin⁡(π6(m−δ))+kT(m)=A\sin(\tfrac{\pi}{6}(m-\delta))+k.

Midline k = (40+10)/2 = 25; amplitude A = (40-10)/2 = 15; period 12 ⇒ ω = π/6; choose phase so peak at m=6.

Example 3: Phase Conversion

Write Asin⁡(ωx+ϕ)A\sin(\omega x+\phi) as Bcos⁡(ωx+ψ)B\cos(\omega x+\psi).

Use sin⁡(θ)=cos⁡(π2−θ)\sin(\theta)=\cos(\tfrac{\pi}{2}-\theta), then absorb constants into phase.

Extended Practice Sets

Set 1: Basic Transformations

  1. Sketch y=3sin⁡(2x−π4)+1y=3\sin(2x-\tfrac{\pi}{4})+1 showing amplitude, period, phase shift.
  2. Identify key points: max, min, zeros, and midline.
  3. Compare with parent function y = sin x.
  4. State domain and range of transformed function.

Set 2: Temperature Modeling

  1. City temperature: max 35°C (July), min 5°C (January). Model T(t).
  2. Determine amplitude A = (35-5)/2 = 15°C.
  3. Find vertical shift k = (35+5)/2 = 20°C.
  4. Calculate period and phase for July peak.

Set 3: Daylight Hours

  1. Model daylight hours: 14h (summer), 10h (winter).
  2. Use cosine function with 12-month period.
  3. Account for location's latitude effect.
  4. Verify model against real astronomical data.

Set 4: Phase Conversions

  1. Convert y=4sin⁡(3x+π6)y=4\sin(3x+\tfrac{\pi}{6}) to cosine form.
  2. Use identity sin⁡θ=cos⁡(π2−θ)\sin\theta = \cos(\tfrac{\pi}{2}-\theta).
  3. Verify equality by checking key points.
  4. Graph both forms to confirm equivalence.

Set 5: Tidal Modeling

  1. High tide: 8.2m at 3:00 PM, low tide: 2.4m at 9:00 PM.
  2. Model water depth h(t) over 12-hour cycle.
  3. Find amplitude, period, and phase shift.
  4. Predict depth at any given time.

Set 6: Composite Functions

  1. Analyze y=2sin⁡(x)+cos⁡(2x)y=2\sin(x)+\cos(2x) behavior.
  2. Find period of composite function.
  3. Identify maximum and minimum values.
  4. Sketch approximate graph using addition of ordinates.

Set 7: Amplitude Modulation

  1. Model AM radio signal: y=(1+0.5cos⁡(ωmt))cos⁡(ωct)y=(1+0.5\cos(\omega_m t))\cos(\omega_c t).
  2. Identify carrier frequency ωc\omega_c and modulation ωm\omega_m.
  3. Analyze envelope function behavior.
  4. Calculate modulation index and efficiency.

Set 8: Biorhythm Modeling

  1. Physical cycle: 23-day period, emotional: 28-day, intellectual: 33-day.
  2. Model each as sinusoidal starting from birth.
  3. Find when all three cycles align positively.
  4. Analyze combined biorhythm score function.

Set 9: Oscillation Analysis

  1. Spring-mass system: x(t)=Acos⁡(ωt+ϕ)x(t)=A\cos(\omega t + \phi).
  2. Given initial position and velocity, find A and φ.
  3. Calculate period and frequency of oscillation.
  4. Determine velocity and acceleration functions.

Set 10: Economic Cycles

  1. Model seasonal sales with 12-month period.
  2. Peak sales in December, lowest in July.
  3. Include long-term growth trend component.
  4. Predict future sales using combined model.

Set 11: Data Fitting

  1. Given periodic data points, estimate sinusoidal parameters.
  2. Use least squares or visual fitting methods.
  3. Validate model by calculating residuals.
  4. Refine parameters for better fit quality.

Set 12: Advanced Applications

  1. Model musical pitch as sinusoidal frequency.
  2. Analyze beat frequencies from two close pitches.
  3. Calculate harmonic content in complex tones.
  4. Apply Fourier analysis concepts to decomposition.

Data Fitting Mini-Projects

Project 1: Sunspot Activity

Analyze 11-year solar cycle data to model sunspot numbers over time.

  • Extract amplitude from historical maximum and minimum values.
  • Determine 11-year period and recent cycle timing.
  • Account for irregular cycle variations in model.
  • Predict next solar maximum using fitted parameters.

Project 2: Stock Market Seasonality

Model seasonal patterns in stock returns using historical monthly data.

  • Identify amplitude from seasonal variation range.
  • Determine 12-month period with appropriate phase.
  • Separate seasonal component from long-term trend.
  • Validate model against out-of-sample data.

Project 3: Human Circadian Rhythms

Model body temperature variation over 24-hour cycles.

  • Determine amplitude from normal daily temperature range.
  • Establish 24-hour period with phase for peak times.
  • Account for individual variations and age effects.
  • Compare model predictions with actual temperature data.

Project 4: Predator-Prey Oscillations

Model population cycles using Lotka-Volterra-inspired sinusoidal approximations.

  • Extract amplitude from population boom and bust cycles.
  • Estimate period from historical cycle length data.
  • Model phase relationship between predator and prey.
  • Test model validity against ecological observations.

Project 5: Electrical Power Demand

Model daily and seasonal electricity consumption patterns.

  • Identify dual cycles: 24-hour daily and 365-day annual.
  • Determine amplitude for peak vs. off-peak demand.
  • Phase alignment for summer cooling and winter heating.
  • Combine cycles for comprehensive demand forecasting.

Project 6: Ocean Wave Modeling

Create sinusoidal models for wave height based on wind and tidal data.

  • Measure amplitude from significant wave height statistics.
  • Determine period from dominant wave frequency analysis.
  • Account for tidal and storm surge phase effects.
  • Validate against buoy measurement data.

Project 7: Biomedical Signal Analysis

Model heart rate variability and respiratory patterns as periodic functions.

  • Extract amplitude from heart rate variation range.
  • Identify breathing period and cardiac cycle interactions.
  • Determine phase relationships between signals.
  • Use model for health monitoring applications.

Project 8: Climate Oscillations

Model El Niño/La Niña cycles and their climate impact patterns.

  • Determine amplitude from sea surface temperature anomalies.
  • Estimate 2-7 year ENSO cycle period variability.
  • Phase alignment with regional precipitation patterns.
  • Correlate model with climate index observations.

Identities & Phase Toolbox

Sum-to-Product

sin⁡a+sin⁡b=2sin⁡a+b2cos⁡a−b2\sin a + \sin b = 2 \sin\tfrac{a+b}{2} \cos\tfrac{a-b}{2}

Phase Shift

Use identity sin⁡(x+ϕ)=sin⁡xcos⁡ϕ+cos⁡xsin⁡ϕ\sin(x+\phi)=\sin x\cos\phi+\cos x\sin\phi to rewrite in cos-form and read phase.

Practice Bank

Bank 1: Identify Parameters

  1. Given y = -3 sin(2x + π/3) - 2, find amplitude/period/phase/shift
  2. Sketch one period indicating key points
  3. Give domain/range

Bank 2: Convert Forms

  1. Rewrite 4 sin(3x - π/6) as cosine form
  2. Verify by comparing zeros/max/min
  3. Explain phase interpretation

Bank 3: Fit Through Points

  1. Through (0,0), (π/4, 2), (π/2, 0) fit y = A sin(ωx + φ)
  2. Solve for A, ω, φ
  3. Validate with residual check

Bank 4: Mixed Composition

  1. Analyze y = 2 sin x + cos 2x
  2. Find overall period
  3. Locate approximate maxima/minima

Bank 5: Temperature Model

  1. Max 38°C in July, min 6°C in Jan
  2. Build T(m) with 12-month period
  3. Predict April temperature

Bank 6: Tidal Cycle

  1. High tide 7.8m at 4 PM; low 2.1m at 10 PM
  2. Create depth model over 12h
  3. Time of next high tide

Bank 7: Phase From Data

  1. Given two consecutive peaks and a midline crossing
  2. Estimate ω and φ
  3. Discuss measurement errors

Bank 8: Auxiliary Angle

  1. Rewrite 3 sin x + 4 cos x = R sin(x + φ)
  2. Find R and φ
  3. Explain geometric meaning

Bank 9: Damped Sinusoid (concept)

  1. Qualitatively sketch y = e^{-0.1x} sin x
  2. Explain amplitude envelope
  3. When is amplitude halved?

Bank 10: Beat Frequency

  1. y = sin(5x) + sin(6x): write as product
  2. Identify carrier and beat frequencies
  3. Sketch one beat window

Bank 11: Phase Comparison

  1. Compare y1 = sin(x) and y2 = sin(x - π/3)
  2. Find relative shift and lag/lead
  3. Plot key points alignment

Bank 12: Inverse Modeling

  1. Given amplitude 2, period π, midline -1, peak at x = π/6
  2. Construct y = A sin(ωx + φ) + k
  3. Check against conditions

Bank 13: Phase-lock Task

  1. Find φ so that sin(2x + φ) zeroes at x = 0 and π
  2. List all φ in [0, 2π)
  3. Explain periodicity of solutions

Bank 14: Parameter Sensitivity

  1. How does small δω change period?
  2. Estimate ΔT when ω → ω + δω
  3. Linear approximation discussion

Bank 15: Piecewise Fit

  1. Given noisy periodic data over one cycle
  2. Pick 3-4 key points to fit quickly
  3. Report uncertainty of parameters

Bank 16: Phase Wrap

  1. Normalize φ to (−π, π]
  2. Given φ = 5π/3, compute wrapped value
  3. Explain why wrapping helps comparison
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