Difficulty: Intermediate | Prerequisites: Sine and cosine transformations (amplitude, period, frequency, phase shift, midline)
This topic takes the transformation rules from the previous lesson and puts them to work. The question shifts from "what do these parameters do to a graph?" to "given a real-world scenario that repeats, how do I choose the right parameters?" You will model vibrating tuning forks, oscillating springs, and Ferris wheel rides by extracting amplitude, period, and shifts from a word problem and assembling them into a sine or cosine equation. If you can identify amplitude, period, and midline from a general equation, you are ready.
To model a periodic scenario, decide whether to use sine or cosine based on the initial value, then read amplitude, period, and any shifts directly from the problem. Plug those into y = a sin[b(x − h)] + k or y = a cos[b(x − h)] + k. Most exam problems follow a predictable six-step process.
Periodic function
A function that repeats its values at regular intervals. The smallest such interval is the period.
In simple terms, any pattern that cycles (a heartbeat, a pendulum, a season) can be described by a periodic function.
Displacement
In oscillation problems, the signed distance from the equilibrium (rest) position. Positive means one direction, negative means the other.
Think of it as how far and in which direction the object has moved from where it would naturally sit still.
Equilibrium
The rest position or midline of an oscillating system. For a spring, it is the point where the spring is neither stretched nor compressed.
In simple terms, equilibrium is "home base" for the oscillation.
Hertz (Hz)
A unit of frequency equal to one cycle per second.
Think of it as: 440 Hz means the thing vibrates 440 times every second.
The decision rests on what happens at time t = 0:
Use sine when the function value is 0 (at the midline) at t = 0, because sin(0) = 0.
Use cosine when the function value is at a maximum or minimum at t = 0, because cos(0) = 1.
If the scenario starts at a minimum, use −a with cosine (reflecting across the x-axis) so the curve begins at its lowest point.
Scenario: A tuning fork playing A above middle C vibrates at 440 Hz with a maximum displacement of 0.5 mm. Find d as a function of t.
At t = 0 the fork is at rest (d = 0), so sine is the right choice.
General equation: d = a sin(bt)
Amplitude: a = 0.5 (maximum displacement is 0.5 mm)
Finding b from frequency:
Frequency = b / (2π) = 440
b = 880π
Final model: d = 0.5 sin(880πt)
Graphing note: because the frequency is so high, set the viewing window to roughly −0.01 ≤ t ≤ 0.01. The first maximum occurs at approximately t = 0.000568 seconds.
Scenario: A weight on a spring starts 6 inches below equilibrium at t = 0, oscillates, and returns to its starting position at t = 5 seconds.
At t = 0 the distance is −6 (a minimum, not zero), so cosine is the right choice.
General equation: d = a cos(bt)
Amplitude: |−6| = 6. Because the motion starts at a minimum, use −a, giving a = −6 in the equation.
Finding b from the period:
The spring returns to its original position after 5 seconds, so the period is 5.
5 = 2π / b → b = 2π / 5
Final model: d = −6 cos[(2π/5)t]
Graphing notes:
The first equilibrium crossing (d = 0) occurs at t = 1.25 seconds.
At t = 2.5 seconds the spring is 6 inches above equilibrium (the opposite extreme).
Scenario: A Ferris wheel ride is described by the table: (0, 2), (2, 52), (4, 102), (8, 2), with h(t) in metres.
y-intercept: h(0) = 2 m, so the rider boards 2 metres above ground.
Period: h(t) returns to 2 at t = 8, so one full revolution takes 8 minutes.
Maximum height: 102 m. The total vertical travel from minimum to maximum is 102 − 2 = 100 m, giving a diameter of 100 m for the wheel.
Amplitude: (102 − 2) / 2 = 50 m.
Midline: (102 + 2) / 2 = 52 m above ground. This is the vertical shift, k = 52.
The rider starts at the bottom, so the model would use −cos (inverted cosine starting at a minimum): h(t) = −50 cos[(2π/8)t] + 52, which simplifies to h(t) = −50 cos[(π/4)t] + 52.
General sine model: y = a sin[b(x − h)] + k
General cosine model: y = a cos[b(x − h)] + k
Period from b: Period = 2π / |b|
Frequency from b: Frequency = |b| / (2π)
Finding b from period T: b = 2π / T
Finding b from frequency f (in Hz): b = 2πf
Check the initial value to decide between sine and cosine.
Find a using the amplitude or total displacement; decide whether a needs to be negative (graph inverted / starts at a minimum).
Find b by identifying the period and using b = 2π / T (or from frequency: b = 2πf).
Find k by locating the midline: k = (max + min) / 2.
Find h by comparing the scenario's starting behaviour to the standard curve's starting behaviour.
Assemble into y = a sin[b(x − h)] + k or y = a cos[b(x − h)] + k.
Springs, pendulums, and shock absorbers all follow sinusoidal models, which is why engineers use these equations when designing suspension systems. In music and acoustics, every pitch corresponds to a sine wave at a specific frequency; the model d = 0.5 sin(880πt) is not a classroom exercise, it is the literal physics of a tuning fork vibrating at concert A.
Students often confuse "returns to its original position" with "completes half a cycle." If the problem says the object returns to where it started, that is one full period, not half.
Students forget to negate a when the scenario starts at a minimum. Writing a = 6 instead of a = −6 for the spring example flips the entire motion upside down.
Students sometimes use frequency in place of b directly. If a problem states 440 Hz, b is not 440. You must multiply by 2π first: b = 2π × 440 = 880π.
Students mix up what goes inside the function argument and what goes outside. Horizontal changes (b and h) live inside; vertical changes (a and k) live outside.
⚠️ Word problems that give frequency in Hz and ask for a sine model are extremely common. Practise converting Hz to a b value quickly.
⚠️ Expect a Ferris wheel or spring problem where you must extract amplitude, period, and midline from a table or graph, then write the equation.
⚠️ Be ready to justify your choice of sine vs. cosine. Markers often award a point for the reasoning ("at t = 0 the value is at the midline, so sine is appropriate").
⚠️ If the problem gives a graph, read the midline and the extremes carefully. The amplitude is half the distance from peak to trough, and the midline is the average of the two.
True or False: If a weight starts at its lowest point at t = 0, you should use a sine model.
A spring completes one full oscillation in 3 seconds. The value of b is __________.
True or False: For a tuning fork at 440 Hz, b = 440.
The midline of a function with a maximum of 80 and a minimum of 20 is y = __________.
True or False: In the six-step process, you find k before b.
Answers: 1. False (use cosine, since the value starts at an extreme). 2. b = 2π/3. 3. False (b = 880π). 4. y = 50. 5. False (find b at step 3, k at step 4).
Q: A pendulum swings with a period of 2 seconds and a maximum angular displacement of 15°. At t = 0 the pendulum is at its maximum displacement. Write a cosine model for the angular displacement θ as a function of time.
A: Amplitude = 15, period = 2, so b = 2π/2 = π. Starts at maximum, so positive cosine. Model: θ = 15 cos(πt).
Q: A buoy bobs up and down in the ocean. It reaches a high point of 8 feet above sea level and a low point of 2 feet above sea level. One complete bob takes 6 seconds, and at t = 0 the buoy is at sea level (5 feet) and rising. Write a sine model for the height h(t).
A: Amplitude = (8 − 2)/2 = 3. Midline = (8 + 2)/2 = 5. Period = 6, so b = 2π/6 = π/3. Starts at midline and rises, so sine with positive a. Model: h(t) = 3 sin[(π/3)t] + 5.
Q: From the Ferris wheel data (0, 2), (2, 52), (4, 102), (8, 2), what is the rider's height at t = 6 minutes?
A: Using h(t) = −50 cos[(π/4)t] + 52: h(6) = −50 cos(6π/4) + 52 = −50 cos(3π/2) + 52 = −50(0) + 52 = 52 metres. The rider is at the midline (centre of the wheel).
Q: Why is −a used in the spring example instead of simply writing a = −6?
A: Both approaches give the same equation. The notes use −a to emphasise that the amplitude (a positive quantity, |a| = 6) is being negated to invert the cosine curve. Writing d = −6 cos[(2π/5)t] means the same as saying a = 6 and then applying a reflection.
This connects back to Sine and Cosine Transformations, where you learnt what each parameter does to the graph. It also links forward to solving trigonometric equations: once you have a model, exam questions often ask "at what time does the height first reach 80 m?" which requires setting the model equal to a value and solving for t. In physics, simple harmonic motion uses exactly these models, so the spring and pendulum examples here will reappear in mechanics courses.
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