Predict future behaviour from periodic data

Ontario Grade 11 Mathematics

Trigonometric Functions

Recognize repeating patterns and use a simple sine model to make estimates

Tides rise and fall, and daylight changes through the year. These situations can show patterns that repeat. Periodic data are data that follow a pattern that repeats after a regular interval. A model uses the pattern to estimate what may happen next. First check that the data really do repeat at a reasonably steady rate. A prediction is an estimate, not a promise. It may be less reliable if conditions change or if it reaches far beyond the recorded data.

What you will learn

Recognize when data show a reasonably regular repeating pattern.
Estimate the period, midline, and amplitude from periodic data.
Use a simple sine model to predict a value when its starting point fits the data.
Check a prediction against the data and explain its limits.

1. Look for a repeating pattern

A graph shows how one quantity changes as another quantity changes. Often, time is on the horizontal axis and the measured quantity is on the vertical axis. A maximum is the greatest value in a cycle. A minimum is the least value.

A cycle is one complete repeat of a pattern. The period is the time or horizontal distance needed for one cycle. For example, if high water occurs about every 12 hours, the estimated period is 12 hours. To estimate a period, compare matching points, such as one high point with the next high point.

Measurements in real situations may not repeat exactly. Small differences can come from measurement or changing conditions. Look across several cycles if possible. Repeated high points, low points, or crossings in the same direction can help you decide whether a regular pattern is present.

Compare matching points to estimate the period.
Use several cycles when possible; a single gap may be unusual.
A repeating pattern supports a model, but does not guarantee that it will continue forever.

2. Describe the cycle

The midline is the horizontal level halfway between the maximum and minimum. It is the centre of the pattern's upward and downward movement. The amplitude is the vertical distance from the midline to either extreme.

Suppose the minimum is 1 and the maximum is 5. The midline is 3, halfway between them. The amplitude is 2, the distance from 3 to either extreme. The total distance from minimum to maximum is twice the amplitude.

A table can show where values occur during a cycle. A graph makes the repeating shape easier to see. A sine graph is a smooth wave that repeats. A sine model can represent periodic data when its shape and timing fit reasonably well.

Technology can help display data and compare a curve with the points. You still need to judge whether the model makes sense. Check its period and its highest and lowest values against the data.

M=\frac{\text{maximum}+\text{minimum}}{2},\quad A=\frac{\text{maximum}-\text{minimum}}{2}

The midline is halfway between the maximum and minimum.
The amplitude is half the distance from minimum to maximum.
A graph or table can help reveal the cycle before you write an equation.

3. Use a simple sine model

A basic sine curve begins at its midline and rises. If the data begin there, this makes a useful starting point for a model. Let

t

represent time measured from that starting point, and let

P

be the period. The input angle completes one full turn, or 360 degrees, during one period.

For a cycle with midline

M

, amplitude

A

, and period

P

, a suitable model is

y=M+A\sin\left(\frac{360^\circ}{P}t\right)

. Here,

y

is the predicted measurement. The factor

\frac{360^\circ}{P}

sets how quickly the angle changes as time passes.

This simple model applies when the pattern starts at the midline and rises. If the data start at a different point in the cycle, do not use this starting point without checking the fit. In this model, predicted values stay between the minimum and maximum set by the midline and amplitude.

Before using the model, check the units for time and measurement. Also check that the starting point and direction match the situation. A periodic model is not a good choice if the data lack a reasonably regular repeat.

y=M+A\sin\left(\frac{360^\circ}{P}t\right)

Use the period to set how quickly the model completes a cycle.
The midline and amplitude set the centre and vertical size of the pattern.
A prediction depends on the pattern continuing in a similar way.

4. Make and judge a prediction

A useful prediction states the estimated value, its units, and when it applies. Keep the time variable and measured quantity clear. A predicted water level is a level at a particular time, not a prediction of the time of high water.

Check whether the predicted value falls within the observed range. Also check its place in the cycle. For example, after the midline crossing while rising, the model should move toward its maximum before turning downward.

A real pattern may be affected by conditions that a simple model does not include. The model captures the repeating feature; it does not explain every cause. If conditions change, or the prediction is far beyond the data, the estimate may be less reliable. State that limit rather than treating the value as certain.

Give the prediction with context and units.
Compare the result with the data range and the pattern's direction.
Treat future values as estimates, especially when conditions may change.

One cycle of the water-level model

Time (hours)	Modelled level (metres)	Position in cycle
0	3	Midline, rising
3	5	Maximum
6	3	Midline, falling
9	1	Minimum
12	3	Cycle repeats

Worked example

Predicting a repeating water level

A simplified water-level record has a minimum of 1 metre and a maximum of 5 metres. It crosses 3 metres while rising at time 0, and it repeats about every 12 hours. Use a sine model to predict the level at 5 hours.

Find the midline and amplitude
The midline is halfway between the low and high levels. The amplitude is half their difference. These values set the centre and height of the model.
$M=3,\quad A=2$
Set the rate of the cycle
One cycle takes 12 hours, so the model's angle must increase by 360 degrees during those 12 hours. This is 30 degrees per hour. The data cross the midline while rising at time 0, so measure time from that crossing.
$\frac{360^\circ}{12\text{ h}}=30^\circ/\text{h}$
Write the model
The sine curve begins at the midline and rises. Add its scaled wave to the midline, using time in hours.
$y=3+2\sin(30^\circ t)$
Evaluate at five hours
At 5 hours, the angle is 150 degrees. The sine of 150 degrees is 0.5, so the predicted level is 4 metres.
$y=3+2\sin(150^\circ)=3+2(0.5)=4$

Answer: The model predicts a water level of about 4 metres at 5 hours.

Check: The predicted value is between the minimum of 1 metre and maximum of 5 metres. At 5 hours, the pattern is past the halfway point of the cycle and is approaching its high level, so 4 metres is reasonable.

Common mistakes and how to avoid them

Calling the full distance from maximum to minimum the amplitude.

Correction: The amplitude is half that distance. It measures from the midline to an extreme.

Using the time from a maximum to the next minimum as the full period.

Correction: For a regular cycle, that is half a period. Compare matching points, such as consecutive maxima.

Using a model far beyond the data without considering changes.

Correction: Treat the result as an estimate. Explain that it may be less reliable if conditions change.

Using a sine model when the data do not repeat regularly.

Correction: Look for several reasonably consistent cycles first. If they are not present, the data do not support this periodic model.

Lesson summary

Periodic data show a pattern that repeats after a regular interval.
Estimate the period by comparing matching points in different cycles.
Use the maximum and minimum to find the midline and amplitude.
A simple sine model can predict values when its starting point, period, and range fit the data.
Check predictions against the pattern and describe them as estimates.

Check your understanding

Question 1

A repeating measurement ranges from 2 to 10 units. What are its midline and amplitude?

Midline 6; amplitude 4
Midline 4; amplitude 6
Midline 8; amplitude 4
Midline 6; amplitude 8

Show answer and explanation

Midline 6; amplitude 4

The halfway value between 2 and 10 is 6. The distance from 6 to either extreme is 4.

Question 2

A cycle reaches a maximum at hour 2 and the next maximum at hour 14. What is the estimated period?

6 hours
12 hours
14 hours
16 hours

Show answer and explanation

12 hours

The period is the time between consecutive matching points. The difference between 14 and 2 is 12 hours.

Question 3

A data set rises and falls, but the gaps between repeated high points change greatly. What is the best conclusion?

A simple periodic model may not fit the data well.
The period must be the largest gap.
The amplitude is the same as the average value.
The pattern is guaranteed to repeat.

Show answer and explanation

A simple periodic model may not fit the data well.

A periodic model needs a reasonably regular repeat. Strongly changing gaps make its predictions less reliable.

Key terms

Periodic: Repeating in a regular pattern after a fixed interval.
Cycle: One complete repeat of a pattern.
Period: The time or horizontal distance for one complete cycle.
Midline: The horizontal centre halfway between the maximum and minimum.
Amplitude: The distance from the midline to a maximum or minimum.
Sine model: An equation using the sine function to represent a smooth repeating pattern.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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