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C3.1 · Collect and graph data modelled by a sine function

Learn to collect and graph data modelled by a sine function through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Trigonometric Functions

Use repeated measurements to recognize and display a smooth cycle

Many quantities rise and fall in a repeating pattern. A point on a rotating wheel moves up and down. The height rises to a maximum, falls to a minimum, and then repeats. A sine function can model data with a smooth, regular cycle. In this lesson, you will practise collecting or organizing measurements and graphing them so you can decide whether a sine model is suitable. The focus is the data and its graph, not a complicated equation.

What you will learn

  • Collect or organize measurements from a repeating situation.
  • Choose suitable graph axes and plot the measured data accurately.
  • Recognize features of a sine-shaped cycle, including its high point, low point, and period.
  • Use a graphing tool to display and check data modelled by a sine function.

1. Start with data and graphs you already know

A data point records a pair of values. For example, a measurement might be recorded at a particular time. The time is the input, and the measurement is the output. A graph shows these pairs as points.
Before graphing, review the axes. The horizontal axis usually shows the input, such as time. The vertical axis shows the measured quantity, such as height. Each axis needs a label and a scale. A scale tells you what each grid interval represents.
A pattern is periodic if it repeats after a fixed amount of time or distance. The time or distance for one complete repeat is called the period. A sine-shaped pattern is a smooth periodic rise and fall. Its graph has rounded high and low points rather than sharp corners.
  • Record each input together with its measured output.
  • Label both axes and use a scale that fits the data.
  • A repeating pattern has a period: the length of one complete cycle.

2. Collect measurements that can show a cycle

Choose something that changes repeatedly and can be measured. Decide what you will measure and what each input value means. For example, you could record the height of a point on a turning wheel at regular time intervals.
Use a consistent interval between measurements. If measurements are too far apart, you may miss important parts of the cycle, such as a high point. If they are unevenly spaced, the graph can still be made, but it may be harder to compare one cycle with the next.
Record the measurements in a table before graphing. Include units, such as seconds and centimetres. Repeat measurements across enough time to see the pattern recur. Real measurements may not make a perfectly smooth curve because of timing, measuring, or equipment errors.
A sine model is reasonable when the data rise and fall smoothly and repeat in a similar way. A single rise and fall is not enough to confirm a repeating pattern. Look for another cycle or for repeated high and low points.
  • Measure at regular intervals when possible.
  • Record units and collect enough data to see repetition.
  • Expect small differences in real measurements.

3. Graph the data and read its features

Plot each table entry as a point. Put the input on the horizontal axis and the measured output on the vertical axis. Choose scales that show all the points clearly without wasting most of the graph.
For data that represent a continuous change, draw a smooth curve through or near the points. Do not join the points with sharp straight segments if the quantity changes smoothly. A graphing tool or spreadsheet can plot the points and help show a smooth trend. Keep the measured points visible so the curve does not hide the evidence.
The highest output is the maximum, and the lowest output is the minimum. The midpoint between these two output values is often called the midline. For example, if a measurement ranges from about 2 to 10, the midpoint is about 6. A sine-shaped cycle moves above and below this level.
Estimate the period by finding the horizontal distance between matching points in consecutive cycles. For example, measure from one high point to the next high point. Check that the distance between low points gives a similar estimate. Use units on your estimate.
A sine curve can begin at different places in its cycle. Therefore, its first measured point does not have to be a high point or a midline crossing. Judge the shape across the full graph rather than expecting every data set to start at the same place.
midline level=maximum+minimum2\text{midline level} = \frac{\text{maximum} + \text{minimum}}{2}
  • Plot ordered pairs and keep the scale consistent.
  • A smooth sine-shaped graph repeatedly rises and falls.
  • Estimate the period between matching points in consecutive cycles.
  • Use the maximum and minimum to describe the range of the data.

4. Use a graphing tool to check the model

A spreadsheet or graphing tool can make plotting a large data set easier. Enter the input values in one column and the measurements in another. Select a scatter plot so each measured pair appears as a point. Add a suitable smooth curve only if the tool supports it and the data pattern justifies it.
Use the graph to ask practical questions. Does the pattern repeat? Are the high and low points spaced regularly? Is the change smooth? Do later cycles look similar to earlier cycles? If the points do not follow a regular, smooth cycle, a sine model may not describe them well.
A graph is a model of the situation, not the situation itself. Measurements can vary, and a smooth curve may only approximate the points. In a report, identify which values were measured and make clear when you are describing an estimated curve or period.
  • A scatter plot keeps the measured points visible.
  • A suitable model should follow the overall pattern without hiding mismatches.
  • Describe estimates as estimates, especially with real measurements.

Sample height measurements

Time (s)Height (cm)
06
110
38
52
74
910
118

Worked example

Graphing a rotating point's height

A student records the height of a point on a rotating display, in centimetres, at equal time intervals. The table shows one cycle and part of the next. Graph the data and decide whether a sine model is reasonable. Estimate the period and the midline.
  1. Set up the axes
    Time is the input, so place it on the horizontal axis. Height is the measured output, so place it on the vertical axis. Label the axes in seconds and centimetres, then choose scales that include all the values.
  2. Plot the measured pairs
    Plot each time with its matching height. The points rise, reach a high value, fall to a low value, and then begin rising again. Connect the pattern with a smooth curve because the point's height changes continuously.
  3. Estimate the period
    The graph has high points near 1 second and 9 seconds. The horizontal distance between these matching points estimates one period. The low points are also about 8 seconds apart, which supports the estimate.
    9−1=8 s9-1=8\text{ s}
  4. Estimate the midline
    The highest recorded height is 10 centimetres, and the lowest is 2 centimetres. Their midpoint is the level halfway between the top and bottom of the cycle.
    10+22=6 cm\frac{10+2}{2}=6\text{ cm}
  5. Decide whether a sine model fits
    The measured points show a smooth rise and fall, and a high point recurs about 8 seconds later. A sine model is reasonable for describing the overall pattern. Small differences from a perfectly smooth curve are expected in measurements.
Answer: The graph is a repeating, smooth rise-and-fall pattern. A sine model is reasonable. The estimated period is 8 seconds, and the midline is about 6 centimetres.
Check: The estimated distance from one low point to the next should also be about 8 seconds. If it is very different, recheck the plotted points or reconsider whether the data repeat regularly.

Common mistakes and how to avoid them

Putting the measured height on the horizontal axis.
Correction: Put the input, such as time, on the horizontal axis and the measured output on the vertical axis.
Calling the time between a high point and the next low point one full period.
Correction: That distance is about half a cycle. Estimate one period from a high point to the next high point, or from a low point to the next low point.
Drawing a sharp zigzag through measurements of a smoothly changing quantity.
Correction: Keep the plotted points, then use a smooth curve to show the overall pattern when the situation changes continuously.
Assuming any set of rising and falling data must be sine-modelled.
Correction: Check for a smooth, regular repetition across more than one cycle. If the pattern does not repeat, a sine model may not fit.
Forcing the curve to pass exactly through every measured point.
Correction: Real measurements can vary. Show the points honestly and judge a smooth curve by how well it represents the overall pattern.

Lesson summary

  • Collect paired input and output measurements, with units and consistent intervals when possible.
  • Plot inputs horizontally and outputs vertically, using clear labels and scales.
  • Look for a smooth, repeating rise and fall before choosing a sine model.
  • Estimate one period between matching points in consecutive cycles.
  • Use the data points as evidence and treat the smooth curve as a model.

Check your understanding

Question 1

A graph has a high point at 2 seconds and the next high point at 14 seconds. What is the estimated period?
  1. 6 seconds
  2. 12 seconds
  3. 14 seconds
  4. 16 seconds
Show answer and explanation
12 seconds
The period is the horizontal distance between consecutive matching points. The difference is 12 seconds.

Question 2

Which graph feature gives a useful estimate of the midline level?
  1. The difference between the input values
  2. The midpoint of the maximum and minimum output values
  3. The first point in the table
  4. The number of measurements
Show answer and explanation
The midpoint of the maximum and minimum output values
The midline is halfway between the maximum and minimum output levels.

Question 3

A set of measurements rises once but has no repeated pattern. What is the best conclusion?
  1. It must be a sine model because it changes smoothly.
  2. It proves the period is one measurement interval.
  3. There is not enough evidence of repetition to support a sine model.
  4. The vertical axis should be changed to time.
Show answer and explanation
There is not enough evidence of repetition to support a sine model.
A sine model describes a repeating pattern. One rise alone does not show that the data repeat.

Key terms

Input
The value chosen or recorded first, such as time.
Output
The measured value that goes with an input, such as height.
Periodic
Repeating after a fixed amount of time, distance, or another input.
Period
The horizontal distance or time for one complete repeat.
Maximum and minimum
The greatest and least output values in the data or cycle.
Midline
The level halfway between a cycle's maximum and minimum outputs.
Sine model
A sine function used to represent data with a smooth, repeating rise-and-fall pattern.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C3.1. 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.

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