Grade by Grade
C. Trigonometric Functions · Sine Functions

Reading Periodic Data and Making Predictions

C2.1 C2.2About 70 minutes
Learning goals
  • I can describe the period, maximum, minimum, axis and amplitude of real-world periodic data from a table or graph.
  • I can explain what each key property means in the context.
  • I can use the period to predict (extrapolate) future values.
  • I can explain why a prediction from real data is an estimate.

Words to know

extrapolate
To predict a value beyond the range of the known data.
interpolate
To estimate a value between known data values.
tide
The regular rise and fall of the sea level, caused mainly by the moon's gravity. Many coasts have two high tides about every 24 h 50 min.
solstice
The day with the most (summer, about June 21) or least (winter, about December 21) daylight in the year.
equinox
A day near March 20 or September 22 when day and night are close to 12 h each.

Using the past to predict the future

If a pattern repeats, then knowing one cycle lets you predict every other cycle. Sailors in the Bay of Fundy plan around tide tables. Gas utilities plan for the cold months because furnaces in Ontario homes use much more natural gas in winter than in summer. Farmers and outdoor workers plan around the hours of daylight.

Today you will read real-world periodic data from tables and graphs, describe its key properties, and use the period to extrapolate: predict values outside the data you have.

Remember from Lesson 1

Period = horizontal length of one cycle. Axis: y = (max + min) ÷ 2. Amplitude = (max − min) ÷ 2. If f has period p, then f(x + p) = f(x) for every x.

Hours of daylight in Ottawa

The table shows the approximate number of hours from sunrise to sunset in Ottawa on the 21st of each month. (Values were calculated from the sun's position and rounded to 0.1 h; a weather service's table may differ slightly.)

MonthJanFebMarAprMayJunJulAugSepOctNovDec
Daylight (h)9.310.712.113.815.115.715.213.912.310.79.38.7
Approximate hours of daylight in Ottawa on the 21st of each month.
monthhours of daylight3691215182124246810121416180≈ 12.2period = 12 monthsmonth 1 = Jan year 1; month 13 = Jan year 2
Two years of the Ottawa daylight data. The pattern repeats every 12 months. The highest value (June) and the lowest (December) are half a year apart.
Worked example

Example 1: Describing real periodic data

Use the Ottawa daylight data. Find the period, maximum, minimum, equation of the axis and amplitude, and explain what each means.

  1. Period: the pattern repeats every year, so the period is 12 months. Meaning: the daylight pattern is the same each year.
  2. Maximum: 15.7 h (June 21, near the summer solstice). Minimum: 8.7 h (December 21, near the winter solstice).
  3. Axis: d = (15.7 + 8.7) ÷ 2 = 24.4 ÷ 2 = 12.2. Meaning: the 'middle' amount of daylight, close to 12 h, which happens near the March and September equinoxes.
  4. Amplitude: (15.7 − 8.7) ÷ 2 = 7.0 ÷ 2 = 3.5 h. Meaning: the daylight swings 3.5 h above and below the middle value.

Period 12 months, max 15.7 h, min 8.7 h, axis d = 12.2, amplitude 3.5 h.

Extrapolating with the period

Interpolating means estimating between known data points. Extrapolating means predicting beyond them. For a periodic function, extrapolation is easier than for most relationships, because you can subtract (or add) whole periods until you land inside a cycle you already know:

  1. Find the period, p.
  2. Write the new input as (known input) + (whole number) × p.
  3. The output is the same as at the known input.

This works only if the pattern really keeps repeating in the same way. Real data are rarely perfect, so a prediction is an estimate.

0510152025303540+8+8+8+8
If the period is 8 and f(3) is known, then f(11), f(19), f(27) and f(35) all have the same value as f(3). Each jump adds one period.
Worked example

Example 2: Predicting tides

At a wharf on the Bay of Fundy, a model of the water depth has a high tide of 13 m at 2:00 a.m. The period of the tide is about 12.4 h, and the low-tide depth is 1 m. Predict (a) the times of the next three high tides, (b) the time of the first low tide, (c) the amplitude and axis.

  1. (a) Add one period at a time: 12.4 h = 12 h 24 min. 2:00 a.m. + 12 h 24 min = 2:24 p.m. Add again: 2:24 p.m. + 12 h 24 min = 2:48 a.m. the next day. Again: 3:12 p.m. the next day.
  2. (b) Low tide comes half a period after high tide: 12.4 ÷ 2 = 6.2 h = 6 h 12 min. 2:00 a.m. + 6 h 12 min = 8:12 a.m.
  3. (c) Amplitude = (13 − 1) ÷ 2 = 6 m. Axis: d = (13 + 1) ÷ 2 = 7, so d = 7.

High tides near 2:24 p.m., 2:48 a.m. and 3:12 p.m.; first low tide near 8:12 a.m.; amplitude 6 m; axis d = 7.

t (h)depth (m)4812162024283236402468101214037.2 h24.8 h12.4 hmeasuredpredicted
A tide model. The solid curve covers 24.8 h (two cycles) of data. The dashed part is extrapolated: the pattern is assumed to keep repeating every 12.4 h. The dots mark high tides. Two cycles take 24.8 h, so high tide comes about 48 min later each day.
Worked example

Example 3: Predicting natural gas use

A sample Ontario household (made-up data for practice) used this much natural gas each month, in cubic metres:

Jan 300; Feb 270; Mar 220; Apr 150; May 90; Jun 50; Jul 40; Aug 45; Sep 70; Oct 140; Nov 210; Dec 280

The pattern was similar the year before. (a) Is the data periodic? (b) Predict the use in February two years from now. (c) Give one reason the prediction might be wrong.

  1. (a) Use is high in winter (heating) and low in summer, and the pattern repeats each year. It is periodic with a period of about 12 months. It is not a perfect wave (real use depends on each month's weather), so it is only roughly sinusoidal.
  2. (b) February two years from now = February now + 2 periods. Predict about 270 m³.
  3. (c) A much colder or warmer winter, a new furnace, better insulation, or a change in the household would change the use. Extrapolation assumes the pattern stays the same.

Periodic (period 12 months); predicted February use about 270 m³, with the warning that weather and changes in the home can change it.

How good is an extrapolation?

  • The closer to the data, the more trustworthy the prediction.
  • Real periodic data drift: tides follow the moon, so their timing shifts a little each day, and weather changes energy use.
  • Always say what you assumed ("the pattern repeats every 12.4 h").
Just for fun

Why are tides so good at keeping promises?

Because they always come back, roughly twice a day.

Hands-on activity

Predict the full moon

You need: a printed or online calendar that shows full moon dates for the past several months, paper and a pencil.

  1. Write down the dates of at least four full moons in a row.
  2. Count the days between each pair. Find the average gap. This is the period of the moon's phases, in days.
  3. Use the period to predict the next three full moons. Write each date.
  4. Check your predictions on the calendar (or wait and watch the sky on a clear night).

What to look for: gaps of 29 or 30 days, an average close to 29.5 days, and predictions within about a day of the real dates. Adding 29.5 days each time keeps the predictions from drifting.

Your turn

1

Daylight in a town has max 16.0 h and min 8.4 h. Find the amplitude and the equation of the axis.

2

A periodic function has period 12 and f(5) = 20. Find f(29) and f(−7).

3

High tide at a harbour is at 6:00 a.m. The period is 12.4 h. When is the next high tide?

4

Using the same harbour, when is the first low tide after 6:00 a.m.?

5

Using the Ottawa table, predict the hours of daylight on April 21 three years from now.

6

What is the difference between interpolating and extrapolating?

More practice: warm-up

1

Period 4, f(1) = 3. Find f(5).

2

Period 4, f(1) = 3. Find f(13).

3

12.4 h is how many hours and minutes?

4

The Ottawa data peak in which month?

5

Name one quantity that is periodic with a period of about one year.

More practice: on your own

1

A Ferris wheel seat is at the top at t = 10 s and next at the top at t = 46 s. Predict three more times it is at the top, and the first time after t = 10 s that it is at the bottom.

2

A wave makes a moored boat bob with a period of 5 s. The boat is at its highest at t = 2 s. Will it be at its highest at t = 47 s? Explain.

3

Using the Ottawa table, between which two consecutive months does the amount of daylight change the most? Why might this be?

4

Low tide at a wharf is at 9:30 a.m. with period 12.4 h. Find the first high tide after that and the next low tide.

5

Using the gas table in Example 3, find the amplitude and axis, and explain what the axis tells you.

Challenge

1

A periodic function has period 15. You know f(0) = 2, f(5) = 9 and f(10) = 4. Find f(100), f(110) and f(−20).

2

High tide is at 1:00 a.m. on a Monday. With a period of 12.4 h, how many high tides happen from Monday 1:00 a.m. until Wednesday 1:00 a.m. (48 h later), counting the first one?

3

A student predicts the Ottawa daylight on December 21, 2046 from this year's table. Explain why this is reasonable, and why predicting natural gas use for December 2046 the same way is less reliable.

Check your understanding

Quiz

1.

A periodic function has period 6 and f(2) = −1. Which is also equal to −1?

2.

High tide is at 4:00 p.m. and the period is 12.4 h. When is the next high tide?

3.

In the same harbour, how long after a high tide is the next low tide?

4.

Hours of daylight have max 15.7 h and min 8.7 h. What is the amplitude?

5.

What does extrapolate mean?

6.

Why is a prediction of next winter's natural gas use only an estimate?

For parents and teachers

Big idea: Describing real periodic data (C2.1) and predicting by extrapolating with the period (C2.2).

How to help: Look up a tide table, sunrise and sunset times, or the full-moon calendar together and make predictions. Ask your teen to explain each prediction by naming the period they used.

Common difficulties:

  • Converting decimal hours to minutes: 0.4 h = 0.4 × 60 = 24 min, not 40 min.
  • Adding a whole period when only half a period is needed (high tide to low tide is half a period).
  • Treating predictions as certain. Encourage phrases like "about" and "assuming the pattern continues".