Use sinusoidal functions to represent data and solve problems

Alberta Grade 12 Mathematics

Relations and Functions

Represent a repeating pattern, make predictions, and check what the model can tell you

Many measurements rise and fall in a repeating pattern. Examples include water levels, daylight hours, and the height of a rider on a Ferris wheel. A sine function can model this kind of data. In this lesson, you will connect the pattern to a graph, build a model, and use it to answer a question. A model is a mathematical representation of a real situation. It is useful only when its assumptions and limits fit that situation.

What you will learn

Identify the midline, amplitude, and period of a repeating pattern.
Build a sine model from contextual information and state what each parameter means.
Use a sine model to calculate or estimate values in context.
Check units, the model’s domain, and whether a prediction is reasonable.

1. Review the pattern before using symbols

A graph has a horizontal axis and a vertical axis. The horizontal axis often represents time, and the vertical axis represents a measured quantity. Check the units on both axes before calculating.

A repeating pattern returns to the same part of its cycle. One complete cycle is the interval from one point in the pattern to the next matching point, such as one maximum to the next maximum. The period is the length of that interval.

The maximum is the greatest value in a cycle. The minimum is the least. The midline is halfway between them, and the amplitude is the distance from the midline to either extreme. Amplitude is a distance, so it is non-negative.

D=\frac{y_{\max}+y_{\min}}{2},\quad A=\frac{y_{\max}-y_{\min}}{2}

The midline is the average of the maximum and minimum.
The amplitude is half the difference between the maximum and minimum.
The period has the same units as the horizontal-axis quantity.

2. Build a sine model

A sine model can be written as

y=a\sin(b(x-c))+d

. In this lesson, use the angle in radians as part of the model. You do not need to convert between degrees and radians.

The value

a

controls the amplitude, so the amplitude is |a|. The value

d

gives the midline. The value

b

controls how quickly the cycle repeats. If the period is

P

, then

b=\frac{2\pi}{P}

. The value

c

locates a point where the sine pattern crosses its midline while rising. Use the given pattern and graph to identify that point; do not rely on a phase-shift direction rule.

For a sine pattern starting at a rising midline, the key points occur at quarter-period intervals: rising midline, maximum, falling midline, minimum, and rising midline again. This gives a practical way to sketch a model before substituting values.

y=a\sin(b(x-c))+d,\quad b=\frac{2\pi}{P}

The vertical range predicted by a model is from $d-|a|$ to $d+|a|$ .
If $x$ is measured in hours, the period is in hours and $b$ is in radians per hour.
The model describes a repeating pattern; real measurements may not repeat perfectly.

3. Use and check the model

To use a model, first identify the requested quantity and its units. Substitute the given input, evaluate the sine value, and interpret the result in context. Keep enough digits during the calculation, then round the final answer to a sensible precision.

A prediction should stay within the model’s range. It should also use an input from the interval where the model is meant to apply. This interval is called the domain. For example, a model based on one day of observations may not be reliable for a different season.

Graphing technology can help fit a sine curve to a set of data or estimate where the curve reaches a target value. A close fit does not prove that the real-world pattern will continue forever. Consider the data source, the measurement interval, and whether the situation is genuinely repeating.

State units with contextual answers.
Check that an answer lies within the model’s range and relevant domain.
Treat a model as an estimate, not as a guarantee about future measurements.

4. Practice the model-building routine

A useful routine is to identify the maximum and minimum, calculate the midline and amplitude, find the period, and then determine a suitable rising midline point for the sine pattern. Write the model with units, substitute only values in the stated domain, and check the result against the context.

When data are given as a table, look for repeated highs or lows to estimate a period. If the measurements are uneven or noisy, use graphing technology to fit a sine model. Report that the model is an estimate and use the graph window to judge whether the curve follows the supplied data.

Do not confuse amplitude with the full maximum-to-minimum difference.
Use the time units stated in the question when finding the period.
A model’s output should make sense for the measured quantity.

Worked example

Daylight hours through a year

At a location, daylight is about 15.5 hours at month 6 and 8.5 hours at month 12. Model the repeating yearly pattern with a sine function. Estimate the daylight at month 9.

Daylight model

A window of the model for one year; the curve is a model, not additional measured data.

Find the vertical pattern
The maximum is 15.5 hours and the minimum is 8.5 hours. Their average is the midline; half their difference is the amplitude.
$d=12,\quad a=3.5$
Use the yearly period
The pattern repeats every 12 months, so the period is 12. The maximum occurs at month 6. A sine pattern reaches its maximum one quarter-period after a rising midline point, so that point is month 3.
$b=\frac{2\pi}{12}=\frac{\pi}{6}$
Write the model and evaluate
With month $x$ as the input, the model predicts daylight hours. At month 9, the sine value is zero, so the prediction is the midline.
$H(x)=3.5\sin\left(\frac{\pi}{6}(x-3)\right)+12,\quad H(9)=12$

Answer: The model is

H(x)=3.5\sin\left(\frac{\pi}{6}(x-3)\right)+12

. It estimates 12 hours of daylight at month 9.

Check: The model ranges from 8.5 to 15.5 hours, matching the stated minimum and maximum. The predicted 12 hours lies in that range.

Worked example

Estimating a water level

A sensor records a water level that reaches 4.8 m at 2:00 a.m. and 0.8 m at 8:00 a.m. The cycle repeats every 12 hours. Let

x

be hours after midnight. Estimate the level at 5:00 a.m.

Water level model

A window of the model over one 12-hour cycle; no additional measurements are plotted.

Calculate amplitude and midline
The maximum and minimum are 4.8 m and 0.8 m. The midline is 2.8 m, and the amplitude is 2.0 m.
$d=2.8,\quad a=2$
Determine the repeating rate
The period is 12 hours, so the model uses $b=\frac{2\pi}{12}$ . The maximum at hour 2 is one quarter-period after a rising midline point, which is hour -1.
$b=\frac{\pi}{6}$
Evaluate at hour 5
Substitute $x=5$ . The angle is $\pi$ , where the sine value is zero. The level is therefore at the model’s midline.
$L(x)=2\sin\left(\frac{\pi}{6}(x+1)\right)+2.8,\quad L(5)=2.8$

Answer: The model estimates a water level of 2.8 m at 5:00 a.m.

Check: The result is between the model’s minimum of 0.8 m and maximum of 4.8 m. The input is within the stated daily time interval.

Worked example

Checking a model against observed data

A greenhouse sensor records temperature in a repeating daily pattern. A sine model is

T(x)=4\sin\left(\frac{\pi}{12}(x-6)\right)+20

, where

x

is hours after midnight. Estimate the temperature at 12:00 p.m., and state the model’s range.

Greenhouse temperature model

A window of the model for one day; the curve shows model values.

Substitute the time
Noon is 12 hours after midnight. Substituting this input gives an angle of $\frac{\pi}{2}$ , whose sine value is 1.
$T(12)=4\sin\left(\frac{\pi}{2}\right)+20=24$
State the model range
The midline is 20°C and the amplitude is 4°C. The model varies 4°C below and above the midline.
$16\leq T(x)\leq 24$

Answer: The estimated temperature at noon is 24°C. The model’s range is 16°C to 24°C.

Check: The result is at the upper end of the model’s range, as expected when the sine value is 1. The model assumes the daily pattern continues over the stated time interval.

Common mistakes and how to avoid them

Using the maximum-to-minimum difference as the amplitude.

Correction: Divide that difference by 2. The amplitude is the distance from the midline to one extreme.

Using the wrong time units when finding the period.

Correction: Use the same units as the horizontal axis. A 12-hour period requires a rate measured in radians per hour.

Reporting a prediction without checking its range or context.

Correction: Compare the result with the model’s minimum and maximum, and confirm that the input is in the relevant domain.

Assuming real measurements will follow the model exactly forever.

Correction: Treat the sine curve as an estimate based on available data and the assumption of repetition.

Lesson summary

A sine model represents a repeating pattern.
The midline is the average of the maximum and minimum; the amplitude is half their difference.
The period sets the repeating interval, with $b=\frac{2\pi}{P}$ .
Substitute inputs with correct units, then check the prediction against the model’s range and domain.

Check your understanding

Question 1

A repeating measurement has a maximum of 18 units and a minimum of 10 units. What are its midline and amplitude?

Midline 14; amplitude 4
Midline 14; amplitude 8
Midline 4; amplitude 14
Midline 28; amplitude 8

Show answer and explanation

Midline 14; amplitude 4

The average of 18 and 10 is 14. Half their difference is 4.

Question 2

A sine model repeats every 8 hours. What value of

b

matches that period?

$\frac{\pi}{4}$
$\frac{\pi}{8}$
$8\pi$
$4\pi$

Show answer and explanation

\frac{\pi}{4}

Use

b=\frac{2\pi}{P}

. With

P=8

,

b=\frac{2\pi}{8}=\frac{\pi}{4}

radians per hour.

Question 3

A model has midline 7 and amplitude 2. Which statement gives its range?

From 5 to 9
From 2 to 7
From 7 to 14
From -2 to 2

Show answer and explanation

From 5 to 9

The minimum is

7-2=5

and the maximum is

7+2=9

.

Key terms

Amplitude: The distance from a repeating pattern’s midline to its maximum or minimum.
Midline: The horizontal value halfway between the maximum and minimum.
Period: The horizontal interval for one complete repetition.
Domain: The set or interval of input values for which a model is being used.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation RF8. 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.

Official curriculum reference

Report a correction or ask a question

RF8 · Use sinusoidal functions to represent data and solve problems

Alberta Grade 12 Mathematics

Represent a repeating pattern, make predictions, and check what the model can tell you

What you will learn

1. Review the pattern before using symbols

2. Build a sine model

3. Use and check the model

4. Practice the model-building routine

Daylight hours through a year

Estimating a water level

Checking a model against observed data

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-2

About this lesson and its review