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D3.1 · Collect and graph sinusoidal data

Learn to collect and graph sinusoidal data through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Trigonometric Functions

Recognizing and Plotting Real-World Periodic Patterns

Have you ever watched the height of a tide rise and fall, or noticed that the number of daylight hours changes and then comes back every year? Both of these are examples of patterns that repeat smoothly and regularly — and both can be graphed as a wave. In this lesson you will learn how to collect data that behaves this way, organize it in a table, and draw it as a graph. Along the way you will name the key features that describe every wave-shaped graph. This kind of graph is called sinusoidal, a word that simply means "shaped like a sine wave." No prior knowledge of sine or cosine functions is assumed here; you only need to be comfortable plotting points on a coordinate grid and reading a table of values — skills from Grade 9 and 10 mathematics.

What you will learn

Identify whether a set of real-world data shows a sinusoidal (wave-like) pattern.
Collect or read data that repeats in a regular cycle and organize it in a table.
Plot sinusoidal data accurately on a labelled graph using degree measure on the horizontal axis.
Identify the key features of a sinusoidal graph: maximum, minimum, midline, amplitude, and period.

Prerequisite Bridge: Plotting Points and Reading Tables

Before graphing sinusoidal data, recall two skills from earlier courses. First, a coordinate grid has a horizontal axis (the input axis) and a vertical axis (the output axis). Each data point is written as an ordered pair (input, output) and placed where the two values meet on the grid.

Second, when you read a table of values, the left column usually holds the input — such as time or angle — and the right column holds the measured output, such as height or temperature. You will use both of these skills constantly in this lesson.

One important convention for this course: angles and positions in a cycle are always measured in degrees. Whenever you see a number on the horizontal axis representing position in a cycle, treat it as a degree measure. For example, one full rotation of a wheel is 360°, and a quarter-turn is 90°.

Input goes on the horizontal axis; output goes on the vertical axis.
Each row of a table becomes one plotted point.
All angle or cycle measurements in this course use degrees.

What Makes Data Sinusoidal?

Data is called sinusoidal when it rises smoothly to a highest value, falls smoothly to a lowest value, and then repeats that pattern over and over in equal intervals. The shape of the graph resembles a rolling wave — never jagged, never flat for long, always curving.

Three real-world situations commonly produce sinusoidal data: the height of a point on a rotating wheel, the depth of water at a harbour as the tide comes in and out, and the hours of daylight at a fixed location as the seasons change. In each case, the measured quantity goes up, comes back down, and repeats.

Not every repeating pattern is sinusoidal. A bouncing ball slows at the top and speeds up at the bottom, but it stops abruptly at the floor — that creates a sharp corner in the graph, not a smooth curve. Sinusoidal data always produces a smooth, continuous wave with no corners or sudden jumps.

When you are given a table of data and asked whether it is sinusoidal, look for two things: (1) the output values rise to a clear maximum and fall to a clear minimum, and (2) the pattern repeats with a consistent spacing between peaks.

Sinusoidal data rises and falls smoothly and repeats at equal intervals.
The graph is a smooth wave — no sharp corners or sudden jumps.
Common sources: rotating objects, tides, seasonal daylight changes.
Check for a consistent maximum, minimum, and repeat interval before labelling data sinusoidal.

Key Features of a Sinusoidal Graph

Every sinusoidal graph shares four measurable features. Learning their names and how to find them from a table or graph is essential for everything that follows in this unit.

The maximum value is the highest output the data reaches, and the minimum value is the lowest. The midline is the horizontal line exactly halfway between the maximum and minimum. You can calculate the midline value by averaging the maximum and minimum.

The amplitude is the distance from the midline up to the maximum (or equivalently, from the midline down to the minimum). It is always a positive number. You calculate it as half the difference between the maximum and minimum values.

The period is the horizontal distance required to complete exactly one full cycle — that is, to travel from one peak back to the next peak (or from any point back to the identical point one cycle later). On a time axis the period might be measured in seconds, hours, or days. On a degree axis it is measured in degrees.

\text{amplitude} = \frac{\text{max} - \text{min}}{2}

Maximum: highest output value; Minimum: lowest output value.
Midline value = (maximum + minimum) ÷ 2.
Amplitude = (maximum − minimum) ÷ 2.
Period = horizontal distance from one peak to the next peak.
These four features fully describe the scale and shape of any sinusoidal graph.

Collecting and Organizing Sinusoidal Data

Collecting sinusoidal data means recording a quantity at regular intervals and writing the results in a table. The key word is regular: you measure at equally spaced inputs so the wave pattern is not distorted by uneven gaps.

For example, you might record the height of a nail on a rotating bicycle wheel every 45° of rotation, or record the average daily temperature of a city every month for a full year. In both cases the input increases by a fixed amount each time.

Once your data is in a table, check whether it looks sinusoidal before graphing. Scan the output column: does it increase, reach a peak, decrease, reach a valley, and then start increasing again? If yes, and if the distance between peaks looks consistent, the data is likely sinusoidal.

When the data passes this check, set up your coordinate grid. Label the horizontal axis with the input variable and its units (for example, 'Angle (degrees)' or 'Month'). Label the vertical axis with the output variable and its units (for example, 'Height (cm)'). Choose a scale that fits all your data points comfortably, with equal spacing between gridlines.

Collect data at equally spaced input values to preserve the wave shape.
Scan the output column for a rise-peak-fall-valley-rise pattern before graphing.
Label both axes with variable names and units.
Choose a consistent scale so the wave shape is clearly visible.

Drawing and Interpreting the Graph

Once you have plotted all the data points, connect them with a smooth, continuous curve — not straight line segments. The curve should flow without corners. If your data is truly sinusoidal, the result will look like a rolling wave.

After drawing the curve, add a dashed horizontal line at the midline. This helps you see the amplitude visually and confirms that the wave rises and falls by equal amounts above and below the centre.

To read the period from your finished graph, find two consecutive peaks (the two highest points of neighbouring cycles) and subtract their horizontal positions. Alternatively, find two consecutive valleys (lowest points) and subtract. Both methods give the same period because the wave is symmetric.

Interpreting the graph means using it to answer questions. For example: 'What is the height after a rotation of 180°?' Simply locate 180° on the horizontal axis, read up to the curve, and read across to the vertical axis. This is the same skill as reading any function graph, applied to a wave-shaped curve.

T = x_{\text{peak 2}} - x_{\text{peak 1}}

Connect plotted points with a smooth curve, never with straight line segments.
Draw the midline as a dashed horizontal line through the centre of the wave.
Period = (horizontal position of second peak) − (horizontal position of first peak).
Read values from the graph by moving vertically to the curve, then horizontally to the output axis.

Key Features of a Sinusoidal Graph at a Glance

Feature	What it measures	How to calculate it
Maximum	Highest output value	Read directly from graph or table
Minimum	Lowest output value	Read directly from graph or table
Midline	Centre height of the wave	(max + min) ÷ 2
Amplitude	Distance from midline to max (or min)	(max − min) ÷ 2
Period	Horizontal length of one complete cycle	Position of peak 2 − position of peak 1

Worked example

Example 1 — Graphing the Height of a Rotating Point

A point on the rim of a wheel is tracked as the wheel rotates. The height of the point above the centre of the wheel is recorded every 45° of rotation. The data are: (0°, 0 cm), (45°, 71 cm), (90°, 100 cm), (135°, 71 cm), (180°, 0 cm), (225°, −71 cm), (270°, −100 cm), (315°, −71 cm), (360°, 0 cm). (a) Plot these points and draw a smooth sinusoidal curve. (b) Identify the maximum, minimum, midline, amplitude, and period from the graph.

Set up the coordinate grid
Draw a horizontal axis labelled 'Angle (degrees)' running from 0° to 360°, with gridlines every 45°. Draw a vertical axis labelled 'Height (cm)' running from $-100$ to $100$ , with gridlines every 20 cm. Equal spacing between gridlines on each axis is essential so the wave is not stretched or squashed.
Plot each ordered pair
Place a dot at each ordered pair from the table. For example, the point $(90°, 100)$ sits directly above the 90° mark at the 100 cm level, and $(270°, -100)$ sits below the horizontal axis at the −100 cm level. Plot all nine points carefully.
Connect with a smooth curve
Draw one continuous, smooth curve through all nine points without lifting your pencil. The curve rises from $(0°, 0)$ , peaks at $(90°, 100)$ , falls back through $(180°, 0)$ , dips to $(270°, -100)$ , and returns to $(360°, 0)$ . Do not use straight line segments between points — the curve must bend smoothly.
Identify maximum and minimum
Read the highest point on the curve: the maximum value is 100 cm, occurring at 90°. The lowest point gives the minimum value of −100 cm, occurring at 270°.
$\text{max} = 100 \text{ cm}, \quad \text{min} = -100 \text{ cm}$
Calculate the midline
The midline value is the average of the maximum and minimum. Add 100 and −100, then divide by 2. The midline sits at 0 cm, meaning the horizontal axis itself is the midline for this graph. Draw a dashed line along the horizontal axis to mark it.
$\text{midline} = \frac{100 + (-100)}{2} = 0 \text{ cm}$
Calculate the amplitude
The amplitude is half the difference between the maximum and minimum. Subtract the minimum from the maximum to get the total range of the wave, then divide by 2. This tells you how far the wave reaches above (or below) its centre line.
$\text{amplitude} = \frac{100 - (-100)}{2} = \frac{200}{2} = 100 \text{ cm}$
Identify the period
The peak of the wave occurs at 90°. The data runs from 0° to 360° and completes exactly one full rise-and-fall cycle in that span. Because the curve starts and ends at the same height (0 cm) after completing one complete wave, the period is 360°.
$T = 360° - 0° = 360°$

Answer: Maximum:

100

cm; Minimum:

-100

cm; Midline:

0

cm; Amplitude:

100

cm; Period: CAD 360°.

Check: Midline check:

(100 + (-100)) \div 2 = 0

✓. Amplitude check:

(100 - (-100)) \div 2 = 100

✓. The data point at 180° has a height of 0 cm, which equals the midline — a sinusoidal wave always crosses the midline halfway between a peak and a valley ✓.

Worked example

Example 2 — Reading Features from Harbour Tide Data

A harbour records the water depth, in metres, at the end of every 3-hour block over one 24-hour day. The readings are: 0 h → 4.0 m, 3 h → 7.5 m, 6 h → 9.0 m, 9 h → 7.5 m, 12 h → 4.0 m, 15 h → 0.5 m, 18 h → −1.0 m, 21 h → 0.5 m, 24 h → 4.0 m. (a) Decide whether the data is sinusoidal. (b) Plot the data and draw a smooth curve. (c) Find the midline, amplitude, and period.

Check for a sinusoidal pattern
Scan the depth column: the depth rises from 4.0 m to a peak of 9.0 m at hour 6, then falls to a low of −1.0 m at hour 18, then rises again to 4.0 m at hour 24. The pattern shows one smooth rise-and-fall cycle over 24 hours with no sudden jumps or flat sections. This is consistent with sinusoidal behaviour.
Set up the coordinate grid
Draw a horizontal axis labelled 'Time (h)' from 0 to 24, with gridlines every 3 h. Draw a vertical axis labelled 'Water depth (m)' from −2 to 10, with gridlines every 1 m. Starting the vertical axis slightly below −1 m ensures the minimum point is clearly visible on the graph.
Plot and connect
Plot each of the nine ordered pairs as dots on the grid. Then connect them with a single smooth, continuous curve. The curve peaks near hour 6 and dips near hour 18, with gradual, rounded transitions — no straight segments and no sharp corners.
Find maximum and minimum
Read the highest and lowest points from the graph or directly from the table. The maximum depth is 9.0 m, occurring at $t = 6$ h. The minimum depth is −1.0 m, occurring at $t = 18$ h.
$\text{max} = 9.0 \text{ m}, \quad \text{min} = -1.0 \text{ m}$
Calculate the midline
Add the maximum and minimum values, then divide by 2. The midline is the average of the two extremes. Draw a dashed horizontal line across the graph at this depth.
$\text{midline} = \frac{9.0 + (-1.0)}{2} = \frac{8.0}{2} = 4.0 \text{ m}$
Calculate the amplitude
Subtract the minimum from the maximum to get the full range of the wave, then divide by 2. The amplitude tells you how far the water level rises or falls from the midline depth.
$\text{amplitude} = \frac{9.0 - (-1.0)}{2} = \frac{10.0}{2} = 5.0 \text{ m}$
Identify the period
The depth at $t = 0$ h is 4.0 m and rising; the depth at $t = 24$ h is again 4.0 m and rising, having completed exactly one full cycle. The period is therefore 24 h. As a check: the peak is at $t = 6$ h, so the next peak would be at $6 + 24 = 30$ h, which is one full period later.
$T = 24 \text{ h}$

Answer: The data is sinusoidal. Midline:

4.0

m; Amplitude:

5.0

m; Period:

24

Check: Midline + amplitude

= 4.0 + 5.0 = 9.0

m (matches the recorded maximum) ✓. Midline − amplitude

= 4.0 - 5.0 = -1.0

m (matches the recorded minimum) ✓. The data point at

t = 0

h has depth 4.0 m, equal to the midline, confirming the wave crosses its centre line at the start of the cycle ✓.

Common mistakes and how to avoid them

Connecting plotted data points with straight line segments instead of a smooth curve.

Correction: A sinusoidal graph is always a smooth, continuous wave. Use a freehand curve that bends gently through each point — never ruler-straight segments between them.

Confusing amplitude with the maximum value. For example, calling the amplitude 100 cm when the wave ranges from −100 cm to 100 cm.

Correction: The amplitude is the distance from the midline to the maximum (or minimum), not the maximum value itself. Calculate it as (max − min) ÷ 2.

Measuring the period as the distance from the start of the data to the first peak, rather than from one peak to the next.

Correction: The period is the distance for one full cycle: from peak to peak, valley to valley, or from any point back to the exact same stage of the next cycle.

Forgetting to label the axes with both the variable name and its units.

Correction: Always write the variable name and units on each axis — for example, 'Time (h)' or 'Angle (degrees)'. Missing units make a graph impossible to interpret.

Deciding data is sinusoidal after seeing only one rise or one fall, without confirming a complete cycle.

Correction: A sinusoidal pattern requires at least one full cycle — a rise to a peak, a fall to a valley, and a return to the starting level — before you can confirm the data is sinusoidal.

Lesson summary

Sinusoidal data rises and falls smoothly in a regular, repeating cycle. Real-world examples include tidal depths, heights of rotating points, and seasonal daylight lengths.
Collect sinusoidal data by recording measurements at equally spaced input values (time, angle in degrees, etc.) and organizing them in a clearly labelled table.
To graph sinusoidal data: set up a labelled grid with appropriate scales, plot each ordered pair as a dot, and connect the points with a smooth continuous curve — never with straight segments.
The four key features of every sinusoidal graph are the maximum, minimum, midline (= (max + min) ÷ 2), amplitude (= (max − min) ÷ 2), and period (= horizontal distance from one peak to the next).
To verify your graph, check that midline + amplitude equals the maximum, and that midline − amplitude equals the minimum.
Angle and position values in this course are always expressed in degrees. One full rotation equals 360°.

Check your understanding

Question 1

A sinusoidal wave has a maximum value of 14 and a minimum value of 2. What is the amplitude?

Show answer and explanation

Amplitude = (max − min) ÷ 2 = (14 − 2) ÷ 2 = 12 ÷ 2 = 6. The amplitude is 6, not 14 (that is the maximum) and not 8 (which would come from incorrectly dividing the maximum by 2 instead of the full range).

Question 2

A student plots tide data and connects the points with straight line segments. What is wrong with this graph?

The axes are not labelled correctly.
The period cannot be found from a tide graph.
Sinusoidal data must be connected with a smooth curve, not straight segments.
Tide data is never sinusoidal.

Show answer and explanation

Sinusoidal data must be connected with a smooth curve, not straight segments.

A sinusoidal graph is a smooth, continuously curving wave. Connecting data points with straight segments creates a jagged, angular shape that does not represent the gradual rise and fall of sinusoidal data.

Question 3

A sinusoidal graph has its first peak at 90° and its second peak at 450°. What is the period?

90°
360°
450°
540°

Show answer and explanation

360°

Period = position of second peak − position of first peak = 450° − 90° = 360°. The period is 360°, which represents one complete rotation of the wheel.

Question 4

Which of the following best describes sinusoidal data?

Data that increases steadily without repeating.
Data that jumps sharply up and down at irregular intervals.
Data that rises smoothly to a maximum, falls smoothly to a minimum, and repeats this cycle at equal intervals.
Data that stays constant and then suddenly rises once.

Show answer and explanation

Data that rises smoothly to a maximum, falls smoothly to a minimum, and repeats this cycle at equal intervals.

Sinusoidal data must rise and fall smoothly (no sharp jumps), reach consistent maximum and minimum values, and repeat the pattern at equal intervals. The other options describe linear growth, random variation, and a step function — none of which are sinusoidal.

Key terms

Sinusoidal: Shaped like a smooth, repeating wave that rises to a maximum, falls to a minimum, and repeats this pattern at equal intervals.
Maximum: The highest output value on a sinusoidal graph.
Minimum: The lowest output value on a sinusoidal graph.
Midline: The horizontal line exactly halfway between the maximum and minimum values of a sinusoidal graph; calculated as (max + min) ÷ 2.
Amplitude: The distance from the midline to the maximum (or equivalently, from the midline to the minimum); calculated as (max − min) ÷ 2.
Period: The horizontal distance needed to complete one full cycle of a sinusoidal wave; found by subtracting the position of one peak from the position of the next peak.
Cycle: One complete repetition of the wave pattern, from any starting point back to the identical point at the same stage of the next repetition.
Degree: The unit of angle measurement used in this course. One full rotation equals 360 degrees, written 360°.

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