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C2.5 · Connect changes in periodic situations with graph transformations

Learn to connect changes in periodic situations with graph transformations through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Trigonometric Functions

Connect changes in a repeating situation to changes in its graph

A periodic situation repeats in a regular pattern. Examples include the height of a moving seat on a ride and the level of water that rises and falls. A graph can show both the repeating pattern and how the situation changes. In this lesson, you will connect three features of a periodic situation to graph transformations: the size of its variation, the length of its cycle, and its average level.

What you will learn

1. Prerequisite bridge: reading a repeating graph

A graph shows how one quantity changes as another quantity changes. The horizontal axis often shows time. The vertical axis shows a measurement, such as height or water level.
A periodic situation repeats after the same amount of time or input distance. That amount is called the period. For example, if a complete pattern repeats every 12 hours, its period is 12 hours.
A sine graph is a smooth, repeating wave. In degree measure, the basic graph of y=sin⁡xy=\sin x completes one cycle as xx goes from 0∘0^\circ to 360∘360^\circ. It rises from zero to a high point, returns to zero, falls to a low point, and returns to zero.
The highest and lowest values describe the graph’s range. The amplitude is half the distance between those values. The midline is halfway between them. These features help describe a graph before writing an equation.
amplitude=maximum−minimum2\text{amplitude}=\frac{\text{maximum}-\text{minimum}}{2}

2. Connect situation changes to graph changes

Imagine a situation that rises and falls around a steady average level. If the rises and falls become larger while the average level stays the same, the graph becomes taller. Its amplitude increases. If the rises and falls become smaller, its amplitude decreases.
If a situation completes each cycle in less time, its graph repeats more quickly. If each cycle takes more time, the graph repeats more slowly. The period describes this cycle length.
If every measurement moves up or down by the same amount, the graph shifts vertically. Its midline changes, but its amplitude and period can stay the same. For example, changing the reference level for a water measurement could move the whole graph up or down.
For a sine graph with no horizontal shift, these features can be shown by y=asin⁡(bx)+dy=a\sin(bx)+d. In this form, the amplitude is |a|, the period in degrees is 360∘∣b∣\frac{360^\circ}{|b|}, and the midline is y=dy=d. The equation is one way to represent the three graph features; you can also compare them directly on graphs.
y=asin⁡(bx)+dy=a\sin(bx)+d

3. Use tables, graphs, equations, and technology

Each representation can help in a different way. A table lists selected input and output values. A graph shows the repeating shape between those values. An equation gives a compact rule for the values.
To read a graph, first find its highest and lowest points. Use them to identify the amplitude and midline. Next, measure the input distance between matching points in consecutive cycles, such as two high points. That distance is the period.
Technology, such as graphing software, can display an equation’s graph and help compare it with measured data. Check that the input uses degrees if you use the degree-based period rule. Also check the units. If time is measured in hours, label the horizontal axis in hours.
When two periodic graphs are compared, ask what stayed the same. If their midlines and periods match but one wave is taller, their amplitudes differ. If their amplitudes and midlines match but one repeats sooner, their periods differ. If the waves have the same size and cycle length but one is higher, the difference is a vertical shift.
period=360∘∣b∣\text{period}=\frac{360^\circ}{|b|}

4. Apply the connections

Suppose a machine makes a repeated motion. If the motion’s range gets larger but its average position stays the same, its graph has greater amplitude. If the machine completes its motion more quickly, its graph has a shorter period. If the whole measured motion is recorded from a higher reference level, the graph shifts up.
These connections help explain a graph in terms of the real situation. They also help you choose a representation. A table can show key values in a cycle, a graph can show the repeating pattern, and an equation can summarize the pattern and its features.

Worked example

Model a repeating water level

A water level varies smoothly through a repeating cycle. Its average level is 8 metres. It reaches a maximum of 11 metres and a minimum of 5 metres. One complete cycle takes 12 hours. At time 0, the level is at its average and rising. Write a sine model in degrees, make a table for one cycle, and explain what changing the cycle length would do.
  1. Find the amplitude and midline
    The maximum is 11 metres and the minimum is 5 metres. Half their difference gives the amplitude. Their average gives the midline, which matches the stated average level.
    11−52=3,11+52=8\frac{11-5}{2}=3,\qquad \frac{11+5}{2}=8
  2. Find the period factor
    One cycle takes 12 hours. In a sine equation with input measured in degrees, the period is 360∘b\frac{360^\circ}{b}. Set this equal to 12 hours and solve for the positive value of bb.
    b=360∘12 h=30∘/hb=\frac{360^\circ}{12\text{ h}}=30^\circ/\text{h}
  3. Write the model
    At time 0, the level is on its midline and rising. A basic sine graph starts at zero and rises, so it matches that starting behaviour. The amplitude sets the variation, the period factor sets a 12-hour cycle, and the midline sets the average level.
    y=3sin⁡(30t)+8y=3\sin(30t)+8
  4. Make a table and interpret
    Use times spaced 3 hours apart. Each step is one quarter of the 12-hour cycle, so the sine values move through 0, 1, 0, negative 1, and 0. The water level moves from its average to its maximum, back to average, to minimum, and back to average.
    t (h)036912y (m)811858\begin{array}{c|ccccc}t\text{ (h)}&0&3&6&9&12\\\hline y\text{ (m)}&8&11&8&5&8\end{array}
Answer: A suitable model is y=3sin⁡(30t)+8y=3\sin(30t)+8, where tt is time in hours and yy is the water level in metres. The table shows one complete cycle. If the cycle took less than 12 hours, the graph would repeat more quickly. If it took more than 12 hours, it would repeat more slowly.
Check: The model’s maximum is 8+3=118+3=11 metres and its minimum is 8−3=58-3=5 metres. Its period is 360∘/(30∘/h)=12360^\circ/(30^\circ/\text{h})=12 hours. These values match the situation.

Common mistakes and how to avoid them

Calling the maximum value the amplitude.
Correction: Amplitude is half the difference between the maximum and minimum. The maximum also depends on the midline.
Assuming a larger period means a faster repeat.
Correction: A larger period means each cycle takes more time or input distance, so the graph repeats more slowly.
Changing the amplitude when the whole graph moves up.
Correction: A vertical shift changes the midline. It does not change the distance from the midline to a high or low point.

Lesson summary

Check your understanding

Question 1

A periodic graph has amplitude 4 and midline y=10y=10. What are its maximum and minimum values?
  1. Maximum 14 and minimum 6
  2. Maximum 10 and minimum 4
  3. Maximum 8 and minimum 2
  4. Maximum 14 and minimum 10
Show answer and explanation
Maximum 14 and minimum 6
The graph rises 4 above the midline and falls 4 below it. The values are 10+4=1410+4=14 and 10−4=610-4=6.

Question 2

A graph is represented by y=2sin⁡(45t)+3y=2\sin(45t)+3, with angles in degrees. What is its period in units of tt?
  1. 4
  2. 8
  3. 45
  4. 360
Show answer and explanation
8
The period is 360∘/45=8360^\circ/45=8. The added 3 shifts the graph but does not change its period.

Question 3

A repeating situation keeps the same amplitude and period, but its average level rises by 2 units. Which change best represents this?
  1. Increase the amplitude by 2.
  2. Shorten the period.
  3. Shift the graph up by 2.
  4. Make the period twice as long.
Show answer and explanation
Shift the graph up by 2.
Raising every value by 2 moves the graph and its midline up by 2 without changing the wave’s size or cycle length.

Key terms

Periodic situation
A situation whose pattern repeats regularly.
Period
The time or input distance needed for one complete repeat.
Amplitude
Half the difference between the maximum and minimum values of a repeating graph.
Midline
The horizontal level halfway between the maximum and minimum; it represents the graph’s average level.
Vertical shift
Moving every point on a graph up or down by the same amount.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.5. It is a study resource, not an official curriculum publication.

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