AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.6B — Sinusoidal Function Transformations
Notes — Horizontal Scaling and Phase Shifts
💡 Learning Objectives (3.6.A Part 2)
By the end of this lesson you will be able to:
- Apply horizontal stretches and compressions to sin(x) and cos(x)
- Apply phase shifts (horizontal translations) correctly
- Combine horizontal and vertical transformations to write a complete sinusoidal model
- Read a model from a graph involving all four parameters
1. Horizontal Scaling by b
In y = sin(b · x), the input is scaled. Because the input cycle 0 → 2π now happens when x goes from 0 to 2π/b, the PERIOD changes:
Period of y = sin(bx) is 2π / |b|
- b = 2: period π — waves twice as fast
- b = 1/2: period 4π — waves half as fast
- b = 3π: period 2π/(3π) = 2/3 — a very rapid oscillation
📝 Why bx and not x/b?
It feels backwards: putting a BIGGER number inside SPEEDS UP the wave (smaller period). That's because b is the angular frequency — how many radians of ‘true angle’ the function sweeps through per unit of x.
2. Phase Shift c
Replace x with (x − c) to translate the graph horizontally to the right by c units (or left if c < 0). In y = sin(b(x − c)):
- The entire wave slides right by c units
- The ‘start’ of a sine wave (midline going up) now happens at x = c
- The ‘start’ of a cosine wave (max) now happens at x = c
⚠️ Common mistake — the b must factor
y = sin(2x − π/3) is NOT a phase shift of π/3. Factor first: y = sin(2(x − π/6)). The phase shift is π/6. When in doubt, always put the model into the form y = a · sin(b(x − c)) + d.
3. Combining All Four Parameters
The general sinusoidal model
y = a · sin(b(x − c)) + d
has the following graph:
- Amplitude |a|, midline y = d, period 2π/|b|, phase shift c
- Max = d + |a|, Min = d − |a|
- For a > 0: starts at the midline going UP at x = c
- For a < 0: starts at the midline going DOWN at x = c
4. Worked Example — Sketching y = 3 sin(π(x − 2)/2) + 1
📘 Example — Full transformation sketch
Rewrite to make b explicit: y = 3 sin((π/2)(x − 2)) + 1
- Amplitude |a| = 3; midline d = 1
- Max: 1 + 3 = 4; Min: 1 − 3 = −2
- Period: 2π ÷ (π/2) = 4
- Phase shift: c = 2 (right by 2)
- Five key points: x = 2 (mid, up), x = 3 (max), x = 4 (mid, down), x = 5 (min), x = 6 (mid, up)
5. Reading a Model from a Graph
Given a sinusoidal graph, extract parameters in this order:
- Find the max and min. Compute d = (max + min)/2 and |a| = (max − min)/2.
- Measure one period from matching features. Compute b = 2π / period.
- Decide sine or cosine. Use whichever has an easy-to-spot reference point.
- Read c: for sine, it is the x-value of a midline-going-up crossing; for cosine, it is the x-value of a max.
📘 Example — Cosine model from a graph
Graph has max y = 7 at x = π/4, min y = −1 at x = π/4 + π (so period = 2π).
- d = (7 + (−1))/2 = 3, |a| = 4
- Period 2π, so b = 2π/2π = 1
- Cosine model with max at x = π/4: c = π/4
- y = 4 cos(x − π/4) + 3
6. Transformations Can Obscure Each Other
A horizontal shift of π and a sign flip on amplitude BOTH flip sine vertically. So y = sin(x + π) = −sin(x). Infinitely many (a, c) pairs can describe the same graph — but only ONE form will match a required representation (for instance, ‘write in the form y = a cos(b(x − c)) + d with a > 0 and 0 ≤ c < period’).
7. Period Change with Phase Shift
Phase shift does NOT affect period. A wave of period 4 stays period 4 no matter how far you shift it. Period is set purely by b.
8. Summary
- In y = sin(bx), the period becomes 2π/|b|
- Replacing x with x − c shifts the graph right by c
- Always factor b before identifying the phase shift
- The five key points split one period into four equal segments of length (period)/4
- Phase shift does not change period; the two parameters are independent