D. Trigonometric Functions · Sinusoidal Functions

Predicting the Future: Extrapolating Periodic Behaviour

D2.2 D2.1About 65 minutes

Learning goals

I can use the period to find values of a periodic function far beyond the data I have.
I can extend a periodic graph or table to predict future behaviour.
I can explain why predictions from real periodic data are only approximate.

Words to know

extrapolate
To predict values outside the range of the data you have, by extending the pattern.

interpolate
To estimate values between data points you already have.

f(x + np) = f(x)
For a periodic function with period p, moving any whole number n of periods left or right gives the same y-value.

remainder
What is left over after dividing by the period; it tells you where you are inside a cycle.

Will it still be light at 8 p.m. next June?

You cannot look up next year's sunset yet, because next year has not happened. But you already know the answer: yes. Daylight follows the same pattern every year, so next June will look a lot like this June.

That is the power of a periodic function: once you know one cycle, you know them all. Predicting beyond your data by extending a pattern is called extrapolating.

The key fact

If a function has period p, then for any whole number n (positive or negative)

f(x + np) = f(x)

So to find a value far away, subtract whole periods until you land inside a cycle you know.

A quick way: divide x by the period. The remainder tells you where you are in the cycle. For example, with period 60, 1000 ÷ 60 = 16 remainder 40, so f(1000) = f(40).

For the Ferris wheel from the last lesson (period 60 s): three jumps of one period take t = 220 back to t = 40, so h(220) = h(40).

Worked example

Example 1: A long Ferris wheel ride

The Ferris wheel from the last lesson has period 60 s. Its rider's height is 2 m at t = 0 and the table shows one cycle:

t (s): 0, 10, 20, 30, 40, 50, 60
h (m): 2, 12, 32, 42, 32, 12, 2

Find the rider's height at t = 1000 s and at t = 250 s.

Divide by the period: 1000 ÷ 60 = 16 remainder 40 (16 × 60 = 960, and 1000 − 960 = 40).
So h(1000) = h(40) = 32 m.
250 ÷ 60 = 4 remainder 10 (4 × 60 = 240), so h(250) = h(10) = 12 m.
Check that the answers make sense: 32 m and 12 m are between the minimum (2 m) and the maximum (42 m).

h(1000) = 32 m and h(250) = 12 m.

Extending a graph from real data

Real data are rarely perfectly periodic, but many are close. To extrapolate from real data:

Graph the data and look for a repeating cycle.
Estimate the period (often a day, a year or a tide cycle).
Extend the graph by copying the cycle, or by averaging several cycles if they are slightly different.
Read off the prediction, and say that it is approximate.

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Year 1 (m³)	280	250	200	130	70	40	30	30	50	110	180	250
Year 2 (m³)	290	255	190	125	75	35	30	35	55	105	185	260

Sample data: natural gas used each month by one Ontario home that heats with gas.

Two years of sample data repeat a cycle with period 12 months: high use in the cold winter, low use in summer. The dashed line predicts year 3 by averaging the two years for each month.

Worked example

Example 2: Predicting gas use

Use the sample gas data. (a) What is the period? (b) Predict the gas used in January and in June of year 3. (c) Why is the prediction only approximate?

(a) Both years have their peak in January and their lowest values in July and August. The cycle repeats every 12 months.
(b) January of year 3 is month 25 = 1 + 2(12). Average the two Januaries: (280 + 290) ÷ 2 = 285 m³.
June of year 3 is month 30 = 6 + 2(12). Average: (40 + 35) ÷ 2 = 37.5 m³, about 38 m³.
(c) Gas use depends on the weather. A colder-than-usual January would use more gas, so the data repeat only approximately. The family could also change the thermostat setting or get a more efficient furnace.

Period 12 months; about 285 m³ in January and about 38 m³ in June of year 3.

Date (21st of)	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Day number	21	52	80	111	141	172	202	233	264	294	325	355
Daylight (h)	9.5	10.7	12.2	13.8	15.0	15.4	15.0	13.8	12.1	10.6	9.4	9.0

Approximate hours of daylight in southern Ontario (rounded to 0.1 h).

Daylight data for one year (dots) with a smooth curve through them. Copying the cycle predicts next year (dashed). Day 172 + 365 = 537 is June 21 next year, so its prediction is about 15.4 h.

Worked example

Example 3: Daylight next year

Use the daylight table. (a) Predict the hours of daylight on June 21 next year. (b) Day 400 is in February next year. Which day this year has the same expected daylight? (c) Predict the daylight on December 21 two years from now.

The period is about 365 days (ignoring leap years).
(a) June 21 next year is day 172 + 365 = 537. Same point in the cycle, so about 15.4 h.
(b) 400 − 365 = 35, so day 400 should match day 35 (February 4). Day 35 is between Jan 21 (9.5 h) and Feb 21 (10.7 h), so expect roughly 10 h.
(c) Two periods later: 355 + 2(365) = 1085. Same point as day 355, so about 9.0 h.

About 15.4 h; day 35; about 9.0 h.

How far can you trust it?

Extrapolation assumes the pattern continues. That is very safe for things driven by the Earth's motion (daylight, tides) and for machines (a Ferris wheel). It is less safe for things that depend on weather or on people's choices (gas use, temperature on one particular day). Always state your prediction as approximate and give a reason.

Just for fun

Ontario's famous groundhog, Wiarton Willie, makes a weather prediction every February 2. Your periodic model cannot predict an early spring either, but it can promise that the days will keep getting longer until June.

Hands-on activity

Predict and test with a pendulum

You need: a piece of string about 50 cm long, a key or a metal washer, tape, and a phone timer.

Tie the key to the string and tape the other end to the edge of a table so the key hangs freely.
Pull the key a little to one side (about 10 cm) and let go. Time 10 complete swings (out and back is one swing). Divide by 10 to get the period.
Predict: at what times will the key be back at its starting side for the 20th, 30th and 40th time?
Release it again and test your prediction for the 20th swing.
Watch the swings carefully. Do they get smaller? Does the period change?

What to look for: a 50 cm pendulum has a period of about 1.4 s, so the 20th return should come at about 28 s. The swings slowly get smaller (friction and air resistance), but the period stays almost the same. That is why your time prediction works even though the size of the swing changes.

Your turn

A Ferris wheel has period 60 s, and the rider is at a height of 22 m at t = 15 s. What is the height at t = 255 s?

A function has period 12 and f(3) = 8. Find f(51) and f(−9).

Use the sample gas data in this lesson to predict the gas used in July of year 3.

A lighthouse beam turns once every 10 s. It shines on your boat at t = 4 s. When are the next three times it shines on your boat?

A full Moon happens about every 29.5 days. If there is a full Moon on day 10 of the year, predict the day numbers of the next two full Moons.

Day 420 is in February next year. Which day of this year should have about the same amount of daylight?

Give two reasons why a prediction of next January's gas use could be wrong.

More practice: warm-up

A function has period 4 and f(1) = 5. Find f(9).

The temperature in a greenhouse follows the same daily pattern (period 24 h). It is 18 °C at 3 p.m. Monday. Predict the temperature at 3 p.m. Tuesday.

Find the quotient and remainder of 500 ÷ 60. What does this tell you about h(500) for a wheel with period 60 s?

A store's daily customer count repeats each week (period 7 days). Day 3 had 120 customers. Predict day 31.

What does extrapolate mean?

More practice: on your own

One cycle of a Ferris wheel ride (period 40 s) is given.

t (s): 0, 5, 10, 15, 20, 25, 30, 35, 40
h (m): 3, 6.5, 15, 23.5, 27, 23.5, 15, 6.5, 3

Find h(95) and h(130).

Two high tides at a harbour are at 3:10 a.m. and 3:35 p.m. Predict the next high tide.

Use the daylight table in this lesson to predict the daylight on day 537.

The pattern 2, 5, 9, 5, 2, 5, 9, 5, … gives f(0), f(1), f(2), … Find f(102).

Mira says, "It was 30 °C on July 15 last year, so it will be 30 °C on July 15 this year." Is this a good use of a periodic model? Explain.

Challenge

A periodic function has period 15, with f(1) = 3 and f(4) = −2. Find f(−26) and f(2026).

A Ferris wheel turns once every 60 s and a carousel turns once every 45 s. Two friends start at the loading points of each ride at t = 0. When are both back at their loading points at the same moment for the first time?

There is a full Moon on day 3 of a 365-day year, and full Moons come every 29.5 days. How many full Moons are in that year?

Check your understanding

Quiz

A function has period 8 and f(2) = −3. What is f(26)?

A Ferris wheel has period 50 s. Which time has the same height as t = 20 s?

What is the remainder when 730 is divided by a period of 60?

Which prediction is the most reliable?

Extrapolating means…

Daylight has a period of about 365 days. Day 100 has about 13 h of daylight. About how much daylight is expected on day 830?

For parents and teachers

Big idea: D2.2 asks students to predict future behaviour by extrapolating a periodic pattern, and D2.1 to read properties from real data. The tool is simple: f(x + np) = f(x). Subtract whole periods to land in a known cycle.

How to help: Use a weather app's sunrise and sunset times, or your gas or hydro bills (many show 12 months of use), to look for yearly cycles together. Ask, "What would you predict for this month next year, and why might it be different?"

Common difficulties:

Subtracting the wrong amount, for example subtracting the time of one maximum instead of a whole period.
Dividing correctly but using the quotient instead of the remainder to locate the point in the cycle.
Treating predictions from weather-dependent data as exact. Encourage your teen to say "about" and give a reason.
Forgetting leap years when counting days far into the future; for this course a 365-day period is fine, but it is good to notice.