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Precalculus practice

Sinusoidal Modeling

Build exact sine or cosine models from noiseless extrema, tables, or graphs in circular, tide, seasonal, and harmonic contexts.

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State the modeling assumptions, identify amplitude, midline, period and phase, then answer the contextual query with units.

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These worksheets pair exact algebra with labeled graph scales, explicit domains, bounded intervals, and model assumptions. The generated teacher key uses the same resolved task document as the student preview.

Learning objective

What this activity builds

Derive amplitude, midline, period, frequency, and phase, then interpret a bounded contextual query.

Worked example

A tide ranges from 2 m to 8 m every 12 hours.

Amplitude 3 m, midline 5 m, period 12 h.

Amplitude is half the range and midline is the average of the extrema.

Teaching notes

Make the practice count

  • Mark a reference maximum or midline crossing before choosing sine or cosine.
  • Keep angular frequency separate from period.
  • State that the supplied data are exact and periodic.

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Math SheetNameScoreTeacherDateSinusoidal ModelingState the modeling assumptions, identify amplitude, midline, period and phase, then answer the contextual query with units.1. Use the displayed graph of water depth against time. Build a sine model on 2 <= t <= 9 hours and interpret its amplitude, midline, period, and angularfrequency.Assumptions: The tide data are noiseless and exactly periodic on the bounded interval 2 <= t <= 9 hours; no regression or extrapolation is intended.263.87.55.597.310.59122. Use the displayed graph of rider height against time. Build a cosine model on 0 <= t <= 6 seconds and interpret its amplitude, midline, period, andangular frequency.Assumptions: The circular motion data are noiseless and exactly periodic on the bounded interval 0 <= t <= 6 seconds; no regression or extrapolation is intended.021.563104.514618Show exact work and preserve every stated domain restriction.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateSinusoidal ModelingState the modeling assumptions, identify amplitude, midline, period and phase, then answer the contextual query with units.3. Use the displayed graph of rider height against time. Build a sine model on 2 <= t <= 12 seconds and find the rider height at t=4.5 seconds.Assumptions: The circular motion data are noiseless and exactly periodic on the bounded interval 2 <= t <= 12 seconds; no regression or extrapolation is intended.274.5117159.51912234. The rider height ranges from 5 to 17 meters, repeats every 6 seconds, and at t=1 crosses the midline while increasing. Build a sine model on 1 <= t <= 7seconds and find the first maximum with t>1.Assumptions: The circular motion data are noiseless and exactly periodic on the bounded interval 1 <= t <= 7 seconds; no regression or extrapolation is intended.Show exact work and preserve every stated domain restriction.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateSinusoidal ModelingState the modeling assumptions, identify amplitude, midline, period and phase, then answer the contextual query with units.5. The average temperature ranges from 3 to 11 degrees Celsius, repeats every 5 months, and at t=0 is at a maximum. Build a cosine model on 0 <= t <= 5months and find the average temperature at t=1.25 months.Assumptions: The seasonal data are noiseless and exactly periodic on the bounded interval 0 <= t <= 5 months; no regression or extrapolation is intended.6. Use the displayed graph of average temperature against time. Build a cosine model on 2 <= t <= 7 months and interpret its amplitude, midline, period,and angular frequency.Assumptions: The seasonal data are noiseless and exactly periodic on the bounded interval 2 <= t <= 7 months; no regression or extrapolation is intended.273.39.54.5125.814.5717Show exact work and preserve every stated domain restriction.math-sheet.comPage 3 of 3

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