Modelling Periodic Phenomena (without Calculus)
Modelling periodic phenomena fits a sine or cosine model \(y=a\sin\!\big(b(t+c)\big)+d\) to repeating real-world data such as tides, temperature and sound, then uses it to predict values and times.
Part of the NSW Year 12 Mathematics Advanced course, in the Functions area of study (Further graph transformations and modelling focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
Periodic phenomena that repeat in a regular cycle — tides, temperature, a Ferris wheel — are modelled by transformed sine or cosine functions \(y=a\sin\!\big(b(t-c)\big)+d\). This Year 12 Mathematics Advanced topic (MAV-12-02) reads amplitude, period, midline and maximum/minimum from a model, without calculus.
A repeating quantity is modelled by \(y=a\sin\!\big(b(t-c)\big)+d\) (or \(\cos\)). The amplitude \(|a|\) is half the gap between highest and lowest; the period \(\dfrac{2\pi}{b}\) is the time for one cycle.
The midline \(y=d\) is the average value, so the maximum is \(d+|a|\) and the minimum is \(d-|a|\). To find when a value occurs, substitute and solve within the given time interval.
Method
- Identify \(a,b,c,d\) from the model (or from the described features).
- Read amplitude \(|a|\), period \(\dfrac{2\pi}{b}\) and midline \(d\).
- Answer in context: max/min \(=d\pm|a|\); substitute a time to evaluate, or solve to find a time.
| \(\text{max}\) | \(=\) | \(20+5=25\) |
| \(\text{min}\) | \(=\) | \(20-5=15\) |
Max \(25^{\circ}\)C, min \(15^{\circ}\)C.
| \(\text{period}\) | \(=\) | \(\dfrac{2\pi}{\pi/20}=40\text{ s}\) |
| \(\text{max}\) | \(=\) | \(12+10=22\text{ m}\) |
| \(d\) | \(=\) | \(3+2\cos\tfrac{\pi}{2}\) |
| \(=\) | \(3+0=3\text{ m}\) |
| \(a\) | \(=\) | \(\dfrac{18-2}{2}=8\text{ m}\) |
| \(d\) | \(=\) | \(\dfrac{18+2}{2}=10\text{ m}\) |
| \(\text{period}\) | \(=\) | \(60\text{ s}\) |
Common pitfalls
Frequently asked questions
How do you find the amplitude of a periodic model?
The amplitude is half the distance between the maximum and minimum values. If the maximum is 18 and the minimum is 2, the amplitude is (18 minus 2) divided by 2, which is 8.
How do you find the period of a sine model?
The period is 2 pi divided by b, where b is the number multiplying t inside the function. For example if b is pi over 20 the period is 40.
What is the midline of a trig model?
The midline is the horizontal line y = d about which the graph oscillates. It is the average of the maximum and minimum values, and it is the vertical shift d in the model.
How do you find when a periodic model reaches a certain value?
Substitute the value for y and solve the resulting trig equation for t, keeping only the times that lie within the given interval.