Modeling with Sinusoidal Functions
Introduced| English | Meaning |
|---|---|
| amplitude/ˈæmplɪtjuːd/ | The maximum displacement from an equilibrium position. |
| period/ˈpɪərɪəd/ | The time taken for one complete oscillation or cycle. |
| midline/ˈmɪdlaɪn/ | A reference line halfway between the maximum and minimum values of an oscillating graph. |
| sinusoidal model/ˌsaɪnəˈsɔɪdl ˈmɒdl/ | sinusoidal model |
| sinusoidal regression/ˌsaɪnəˈsɔɪdl rɪˈɡreʃn/ | sinusoidal regression |
Predicting the tides
- Tide charts, daily temperatures, and hours of daylight all rise and fall on a cycle.
- To forecast the next high tide, we fit a sine wave to past measurements.
- The four sinusoid dials — amplitude, period, midline, shift — get read straight from the data.
- Once fitted, the model predicts every future cycle.
Reading the model from data
- A sinusoidal model 正弦型模型 is $y = a\sin(b(x - c)) + d$ fitted to periodic data.
- The midline 中线 $d$ is the average of the highest and lowest values.
- The amplitude 振幅 $|a|$ is half the distance between the high and the low.
- These two come straight from the peaks and troughs of the data.
A sinusoidal model is the right choice for data that…
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Repeating highs and lows — tides, temperatures, daylight hours — call for a sinusoidal model.
Period and shift from data
- The period 周期 is how long one full cycle takes; then $b = \tfrac{2\pi}{\text{period}}$.
- The phase shift $c$ lines the wave up with when the data first peaks.
- Tides repeat about every $12.4$ hours; a year of temperatures repeats every $12$ months.
- Match the cycle length and the model locks onto the rhythm.
Tune a sinusoid to match periodic data
y = a·sin(b(x − c)) + d
Set the midline d and amplitude a from the data's high and low, then set b from the period until the wave fits.
The midline of a fitted sinusoid is the ____ of the highest and lowest data values.
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Midline $= \dfrac{\text{max} + \text{min}}{2}$; the amplitude is half the distance between them.
Tides run from a low of $1$ m to a high of $5$ m. What is the amplitude?
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Amplitude $= \dfrac{5 - 1}{2} = 2$ m — half the total swing.
Select all true statements about sinusoidal models.
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Sinusoids repeat; they do not fit unbounded growth. The other three are correct.
Sinusoidal regression
- With many data points, use technology to run a sinusoidal regression 正弦型回归.
- It returns the best-fit $a, b, c, d$ all at once.
- The curve threads the data's cycle without hitting every point exactly.
- The result is a ready-to-use periodic formula.
Technology can compute a sinusoidal regression to fit periodic data.
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A sinusoidal regression returns the best-fit $a, b, c, d$ for a data set, just like other regressions.
Using the model
- Plug in a future input to predict, say, the tide height at 3 p.m. tomorrow.
- Because it repeats, one well-fitted cycle forecasts them all.
- Keep the domain sensible — the pattern holds only while conditions stay steady.
- Interpret each parameter: amplitude is the tidal range, midline is the mean level.
A sinusoidal model only fits data that genuinely repeats. Do not force one onto data that trends upward overall — a rising, wobbling series needs a trend plus a wave, not a pure sinusoid.
A harbour's tide is low ($1$ m) at midnight and high ($5$ m) six hours later.
- Midline $d = \tfrac{1 + 5}{2} = 3$ m; amplitude $|a| = \tfrac{5 - 1}{2} = 2$ m.
- Low-to-high is half a cycle in $6$ h, so the period is $12$ h and $b = \tfrac{2\pi}{12}$.
- Model: $h = 3 - 2\cos\!\left(\tfrac{2\pi}{12}t\right)$, starting low at $t = 0$.
A sinusoidal model $y = a\sin(b(x-c)) + d$ fits repeating data: the midline is the average of high and low, the amplitude is half their gap, and the period is one cycle's length. A sinusoidal regression finds the best fit for you.
Write from an amplitude of 3
- Tides run from a low of 2 m to a high of 8 m.
- Find the amplitude.
Tides run from a low of 2 m to a high of 8 m. What is the amplitude, in metres?
Half of 6 is 3.
What is the amplitude?
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Half the range is the amplitude. Other options — 2 m (the lesson amplitude); 8 m (the high tide).