D2.2 Predict future behaviour from periodic data

Ontario Grade 11 Mathematics

Trigonometric Functions

MCR3U study topic D2.2

Some measurements rise and fall in patterns that repeat. Water depth can change with a regular tide, and average temperatures can follow a yearly pattern. Data with a repeating pattern is called periodic data. If the pattern continues, a value from one cycle can help predict a value in a later cycle. These predictions are estimates: the data supports a pattern, but it cannot guarantee that the real situation will repeat exactly.

What you will learn

Identify a cycle and period in repeating data.
Use a period to predict a future value.
Check a prediction against the pattern, its range, and the situation.

1. Recognize a cycle and its period

A data table often pairs an input, such as time, with an output, such as depth or temperature. A pattern is periodic when it repeats after equal intervals of the input. One complete repeat is called a cycle. The period is the input interval for one complete cycle.

For example, suppose a measurement follows the same pattern every 12 hours. Its period is 12 hours. A time 12 hours later is at the same point in the repeating pattern. It is not necessarily the same number of hours or units in the output; the period describes the input interval.

To look for a period, compare the shape of the data over time. Watch for a rise, a high point, a fall, and a low point returning in the same order. Repeated high or low values can help, but one matching pair alone may not be enough to establish a reliable pattern. Use the period given or supported by the data, and think about whether the situation makes sense.

A cycle is one complete repeat.
The period measures the input interval for one cycle.
A few repeated measurements suggest a pattern; they do not prove it will continue forever.

2. Carry a known value into a later cycle

If two input times are separated by one full period, the pattern predicts the same output at both times. This also works when the times are separated by several whole periods. To predict, compare the future time with a time whose value is known. Add or subtract whole periods from the input time until you reach a matching point in a known cycle.

For instance, if the period is 12 hours, times 5 hours and 17 hours are one period apart. The predicted outputs match because the pattern has completed one full repeat between those times. A simple table can make this relationship easier to see.

A repeating rise-and-fall pattern may also be described by a model. When a model is provided, substitute the time you want to predict and evaluate it. Some models use sine or cosine. In this course, angles in such models are measured in degrees. A full turn is 360 degrees, so adding or subtracting full turns gives an equivalent angle and the same trigonometric value.

The maximum is the highest value in a cycle, and the minimum is the lowest. The range of the repeating model runs from that minimum to that maximum. These values help you decide whether a prediction is reasonable. The period tells when the pattern repeats; it does not tell how large the output is.

f(t+T)=f(t)

Input times separated by whole periods have matching predicted outputs.
Use a known point in the pattern or evaluate a supplied model.
A model's predicted values should stay between its minimum and maximum.

3. Check what the prediction means

Check the prediction in three ways. First, confirm that the future time and period were used correctly. Second, check the arithmetic and compare the answer with the model's range. Third, ask whether the repeating pattern is plausible in the real situation.

A prediction is an estimate based on the assumption that the pattern continues. A yearly temperature pattern may be useful for estimating a later month's average, but unusual weather can make an actual temperature different. A tide pattern can also be affected by conditions that are not represented in a simple model.

The farther a prediction is beyond the data that was observed, the more it relies on the pattern continuing unchanged. State the prediction as an estimate and note the assumption when it matters. Do not treat a mathematical repeat as a guarantee about the real world.

Check the time, calculation, and output range.
A prediction assumes the pattern continues in a similar way.
Real conditions can make an observed value differ from a model's estimate.

Matching times in a 12-hour cycle

Known time	Period added	Matching future time	Predicted outputs
5 hours	12 hours	17 hours	Same
Month 8	12 months	Month 20	Same

Worked example

Example 1: Predicting a water depth

A harbour's water depth follows a pattern with a period of 12 hours. A model is

D(t)=2.25\cos(30^\circ(t-4))+3.75

, where

t

is time in hours and

D(t)

is depth in metres. Predict the depth at 15 hours.

Locate the future time in the cycle
The model reaches a maximum at $t=4$ . The next maximum is 12 hours later, at $t=16$ . So $t=15$ is one hour before that maximum. This suggests that the depth should be near, but below, the model's highest value.
$15-4=11$
Evaluate the model
Substitute $t=15$ into the model. The angle is $330^\circ$ , and its cosine is about $0.8660$ . Multiply by the amplitude and add the midline to find the predicted depth.
$D(15)=2.25\cos(330^\circ)+3.75\approx5.70\text{ m}$

Answer: The predicted depth at 15 hours is about 5.70 m.

Check: The model's minimum is 1.50 m and its maximum is 6.00 m, so the prediction is in range. It is close to the maximum, which fits the fact that the time is one hour before the next peak. The prediction assumes that the model continues to describe the harbour.

Worked example

Example 2: Predicting a monthly temperature

A city's average temperature follows a yearly pattern with a period of 12 months. January is month 1, and a model is

T(t)=-17\cos(30^\circ(t-1))+9

, where

t

is the month number and

T(t)

is temperature in degrees Celsius. Predict the temperature in month 20.

Find a matching month
Month 20 is 12 months after month 8. Since the period is 12 months, those two months occupy the same point in the model's repeating pattern. We can predict the month 20 value by evaluating the model at month 8.
$20-12=8$
Evaluate the model at the matching month
At month 8, the angle is $210^\circ$ , whose cosine is about $-0.8660$ . Multiplying by the negative coefficient gives a positive contribution. Adding the midline gives the estimated temperature.
$T(8)=-17\cos(210^\circ)+9\approx23.7^\circ\text{C}$

Answer: The predicted average temperature in month 20 is about 23.7°C.

Check: Month 20 and month 8 are one period apart, so the model assigns them the same value. The model's minimum is

-8^\circ\text{C}

and its maximum is

26^\circ\text{C}

. The prediction lies between them. It is an estimate based on the yearly pattern continuing.

Common mistakes and how to avoid them

Treating the period as the measured output.

Correction: The period is an interval in the input, often time. It tells how long one complete repeat takes.

Adding the period to the output value.

Correction: Add or subtract periods from the input time. The output is predicted to repeat; it does not increase by the period.

Assuming a pattern will continue forever because it has repeated so far.

Correction: A prediction depends on the assumption that the pattern continues. Consider whether that assumption fits the situation.

Lesson summary

Periodic data repeats its pattern at regular input intervals.
The period is the input interval for one complete cycle.
Use matching points in different cycles or a supplied model to predict a future value.
Check the calculation, the model's range, and the assumption that the pattern continues.

Check your understanding

Question 1

A pattern repeats every 8 days. If its value on day 3 is 14, what does the periodic pattern predict for day 11?

6
14
22
The period alone does not allow a prediction

Show answer and explanation

14

Day 11 is 8 days after day 3, exactly one period later. The pattern predicts the same value, 14.

Question 2

A repeating model has a maximum of 20 and a minimum of 8. Which value is outside its range?

8
14
20
22

Show answer and explanation

22

The model's values range from 8 to 20, including both endpoints. The value 22 is above the maximum.

Question 3

A pattern has a period of 6 hours. A known output occurs at hour 4. Which later time is one full period after hour 4?

Hour 6
Hour 8
Hour 10
Hour 24

Show answer and explanation

Hour 10

One full period after hour 4 is hour 10, because the input time increases by 6 hours.

Key terms

Periodic data: Data that follows the same pattern again at regular input intervals.
Cycle: One complete repeat of a pattern.
Period: The input interval for one complete cycle.
Prediction: An estimate of a future value based on a pattern or model.
Maximum and minimum: The highest and lowest values in a cycle.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D2.2. It is a study resource, not an official curriculum publication.

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D2.2 · Predict future behaviour from periodic data

Ontario Grade 11 Mathematics

MCR3U study topic D2.2

What you will learn

1. Recognize a cycle and its period

2. Carry a known value into a later cycle

3. Check what the prediction means

Matching times in a 12-hour cycle

Example 1: Predicting a water depth

Example 2: Predicting a monthly temperature

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCR3U

About this lesson