Algebra 2
Trigonometric functions (introduction)
Periodic phenomena and modelling
20 practice questions
0 video lessons
Theory + worked examples
Periodic Phenomena and Modelling
California Algebra 2 • Standard F-TF.5 • Trigonometric Functions
Periodic Phenomena and Modelling is a topic in Trigonometric Functions in the California Common Core State Standards. It is aligned to Standard F-TF.5, which requires students to choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Periodic data such as tides and temperature is modeled by \(y=a\sin(b(t-c))+d\), with amplitude \(a\), midline \(d\), and period \(\dfrac{2\pi}{b}\).
Theory
Periodic phenomena repeat, so they are modeled by a sinusoid:
\[y=a\sin\big(b(t-c)\big)+d,\]
- Amplitude \(a=\dfrac{\text{max}-\text{min}}{2}\).
- Midline \(d=\dfrac{\text{max}+\text{min}}{2}\).
- Period \(=\dfrac{2\pi}{b}\); \(c\) is the phase shift.
Read amplitude and midline from the maximum and minimum values.
A model with midline \(4\) and amplitude \(3\).
Building a sinusoidal model.
Amplitude and midline:
\[a=\dfrac{\text{max}-\text{min}}{2},\qquad d=\dfrac{\text{max}+\text{min}}{2}\]
Find \(b\) from the period: \(b=\dfrac{2\pi}{\text{period}}\).
How to build a model
- Find the max and min from the data.
- Amplitude \(=\dfrac{\text{max}-\text{min}}{2}\).
- Midline \(=\dfrac{\text{max}+\text{min}}{2}\).
- Get \(b\) from the period, then write the sinusoid.
Example 1 — Amplitude from data
A tide rises to \(7\) ft and falls to \(1\) ft. Find the amplitude.
Solution
Amplitude is half the range.
| \(a\) | \(=\) | \(\dfrac{7-1}{2}=3\) |
Example 2 — Midline
For the same tide, find the midline.
Solution
Midline is the average of the extremes.
| \(d\) | \(=\) | \(\dfrac{7+1}{2}=4\) |
Example 3 — Period to b
A cycle repeats every \(12\) hours. Find \(b\).
Solution
Use period \(=\dfrac{2\pi}{b}\).
| \(12\) | \(=\) | \(\dfrac{2\pi}{b}\) |
| \(b\) | \(=\) | \(\dfrac{\pi}{6}\) |
Example 4 — Write the model
Combine those values into a model (ignore the shift).
Solution
Use \(y=a\sin(bt)+d\).
| \(y\) | \(=\) | \(3\sin\!\left(\dfrac{\pi}{6}t\right)+4\) |
Common pitfalls
Amplitude is half the range, not the maximum.
The midline is the average, not zero.
\(b=\dfrac{2\pi}{\text{period}}\) — invert carefully.
Frequently asked questions
How do you find the amplitude from data?
Half the difference between the max and min.
How do you find the midline?
The average of the max and min.
How do you get b from the period?
\(b=\dfrac{2\pi}{\text{period}}\).
What kind of data suits a sinusoidal model?
Anything periodic — tides, daylight, temperature.
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Graphs of sine, cosine, and tangent
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Pythagorean identity (sin² + cos² = 1)
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