Mathematics › 🌌 Grade 11 › Phase Shift Studio: Amplitude, Period, Phase, and Midline

Phase Shift Studio: Amplitude, Period, Phase, and Midline

Four numbers place a sine wave anywhere you like — taller or shorter, faster or slower, slid sideways, and raised or lowered.

Grade 11Algebra 2 / Pre-Calculus

Phase Shift Studio: Amplitude, Period, Phase, and Midline — illustration

💡

The big idea: The general sinusoid y = a·sin(b(x − c)) + d has four independent controls. Amplitude |a| sets its height, the period 2π/|b| sets its length, the horizontal shift c slides it left or right (the phase shift), and d raises or lowers the whole midline. Because so many real signals — tides, daylight hours, sound, alternating current — are sinusoids, being able to read and write these four parameters is the key to modeling cyclic phenomena.

🎯 By the end, you'll be able to

Identify amplitude, period, phase shift, and midline in y = a sin(b(x − c)) + d
Compute the phase shift as the horizontal translation c
Find the midline y = d and the maximum and minimum values
Build a sinusoidal model from a described real-world cycle

📎 You should already know

Amplitude and period of sine and cosine
Transformations of functions

From one wave to any wave

The plain sin x wave is fixed: it swings between −1 and 1, repeats every 2π, and sits centered on the x-axis. Real signals are not so tidy. To describe a tide, a temperature cycle, or a musical tone, we need to move and reshape that basic wave freely.

Four parameters do exactly that, and each acts independently.

\[ y = a\,\sin\!\big(b(x - c)\big) + d \]

The general sinusoid: a sets amplitude, b sets period, c is the phase (horizontal) shift, d is the vertical shift (midline).

🔑 What each parameter does

a → amplitude |a| (height of the swing). b → period 2π/|b| (length of one cycle). c → phase shift: the wave slides right by c (left if c is negative). d → the midline y = d that the wave oscillates around.

The midline and the max/min

The midline is the horizontal center of the wave, y = d. The wave rises a full amplitude above it and falls a full amplitude below it, so the maximum is d + |a| and the minimum is d − |a|. This is often the fastest way to recover the parameters from a graph: the midline is the average of the peak and trough, and the amplitude is half their difference.

🎮 Phase-Shift Studio LIVE

Shift a sine wave left/right and up/down: amplitude, period, phase and midline together.

The phase shift needs the factored form

To read the phase shift, the argument must be written as b(x − c) — with b factored out. Only then is c the true horizontal shift. If you see something like sin(2x − π), factor first: 2x − π = 2(x − π/2), so the phase shift is π/2 to the right, not π.

\[ \text{phase shift} = c = \dfrac{\text{(constant inside)}}{b} \]

Divide the inside constant by b to get the true horizontal shift; a right shift is positive c.

📝 Worked example: For y = 3 sin(2(x − π/4)) + 5, give the amplitude, period, phase shift, midline, and maximum.

Amplitude: \( |a| = 3 \).
Period: \( \dfrac{2\pi}{|b|} = \dfrac{2\pi}{2} = \pi \).
Phase shift: the form is already \( b(x - c) \) with \( c = \pi/4 \), so it shifts \( \pi/4 \) to the right.
Midline is \( y = 5 \); the maximum is \( d + |a| = 5 + 3 = 8 \).

✓ Amplitude 3, period π, phase shift π/4 right, midline y = 5, maximum 8.

📝 Worked example: A Ferris wheel platform is 2 m off the ground, has radius 10 m, and completes a turn every 40 s. Model a rider's height starting at the bottom.

Amplitude is the radius, \( a = 10 \); the midline is the hub height, \( d = 2 + 10 = 12 \).
Period is 40 s, so \( b = \dfrac{2\pi}{40} = \dfrac{\pi}{20} \).
Starting at the bottom (a minimum) suggests using a downward-shifted cosine: \( y = -10\cos\!\big(\tfrac{\pi}{20}t\big) + 12 \), which equals \( 2 \) m at \( t = 0 \).

✓ h(t) = −10 cos((π/20)t) + 12, giving 2 m at the bottom and 22 m at the top.

⚠️ Factor out b before reading the shift

The phase shift is not the raw constant inside the sine. In sin(bx − k) you must divide: the shift is k / b. Forgetting to factor out b is the single most common phase-shift error.

Check your understanding

1. In y = a sin(b(x − c)) + d, which parameter sets the midline?

d shifts the whole wave vertically, so the midline is y = d.

2. For y = 2 sin(x − π/3), what is the phase shift?

Here b = 1 and c = π/3, and a positive c shifts the graph π/3 to the right.

3. A sinusoid has midline y = 4 and amplitude 6. What is its maximum value?

Maximum = d + |a| = 4 + 6 = 10.

4. What is the phase shift of y = sin(3x − π)?

Factor: 3x − π = 3(x − π/3), so the shift is π/3 to the right, not π.

5. A wave oscillates between a maximum of 9 and a minimum of 1. What is its amplitude?

Amplitude is half the peak-to-trough distance: (9 − 1)/2 = 4 (and the midline is 5).

✅ Key takeaways

The general sinusoid y = a sin(b(x − c)) + d has four independent controls.
Amplitude is |a|, period is 2π/|b|, the phase shift is c (right if positive), and the midline is y = d.
Maximum = d + |a| and minimum = d − |a|; the midline is the average of the two.
To read the phase shift, first factor b out of the argument, then the shift is c.
Sinusoids model real cycles — tides, daylight, Ferris wheels — by fitting these four parameters.