AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.5 — Sinusoidal Functions
Notes — The General Sine and Cosine Model
💡 Learning Objectives (3.5.A)
By the end of this lesson you will be able to:
- Identify the amplitude, period, midline, and phase shift of a sinusoidal function
- Interpret each parameter in the general form y = a · sin(b(x − c)) + d
- Read these parameters from a graph of a sinusoidal function
- Distinguish cosine-based and sine-based sinusoidal models of the same curve
1. The General Sinusoidal Form
A SINUSOIDAL FUNCTION is any function that can be written in either of these forms:
y = a · sin(b(x − c)) + d or y = a · cos(b(x − c)) + d
Every sinusoidal function is a stretched, shifted copy of sine or cosine. The four parameters a, b, c, and d each control one aspect of the graph.
2. What Each Parameter Does
💡 Meaning of the Four Parameters
- a — Amplitude factor. Amplitude = |a|. If a < 0 the graph is reflected across the midline.
- b — Frequency factor. Period = 2π/|b|. Larger |b| means faster oscillation, shorter period.
- c — Phase shift. The graph shifts right by c (or left if c is negative).
- d — Vertical shift. Midline is y = d. Max is d + |a|; min is d − |a|.
3. Reading Parameters from a Graph
Given a graph, you can extract every parameter by inspection:
- Midline d = (max + min) / 2
- Amplitude |a| = (max − min) / 2
- Period = distance between two consecutive matching features (peak-to-peak or midline-up-to-midline-up); then b = 2π / period
- Phase shift c: for a sine model, locate the nearest MIDLINE CROSSING going UP — that x-value is c (plus any multiple of the period)
📘 Example — Extracting parameters
A graph has max 9, min 1, one period of length π, and crosses the midline going upward at x = π/3.
- d = (9 + 1)/2 = 5
- |a| = (9 − 1)/2 = 4, and the graph starts going up, so a = 4
- Period = π, so b = 2π/π = 2
- Phase shift: c = π/3
- Model: y = 4 · sin(2(x − π/3)) + 5
4. Sine vs. Cosine Form
Any sinusoidal graph can be written as either a sine model OR a cosine model — you just need to pick a different reference point. Cosine hits its MAX at x = c; sine hits its MIDLINE (going up) at x = c.
- If it is easier to locate a max, use cosine: y = a · cos(b(x − c)) + d with c at the max
- If it is easier to locate a midline-crossing-going-up, use sine: y = a · sin(b(x − c)) + d with c at the crossing
The two forms are related by c_sine = c_cosine − (period/4).
5. Negative Amplitude
If a is negative, the wave is flipped vertically about the midline. A sine graph with a < 0 starts by going DOWN from the midline instead of up. Equivalently, y = −sin(x) = sin(−x) (by odd symmetry) or = sin(x + π) (by phase shift of π).
⚠️ Common mistake
Amplitude is always NON-NEGATIVE. ‘Amplitude = −3’ is meaningless; write ‘a = −3, so amplitude = 3.’
6. Interpreting b
The period of y = sin(x) is 2π. Putting b inside (multiplying x) compresses horizontally by factor |b|. So:
- b = 2 doubles the speed: period = π
- b = 1/3 slows by a factor of 3: period = 6π
- b < 0 reflects horizontally, but since sin(−u) = −sin(u), it's equivalent to making a negative
7. Numerical Snapshot
For y = 3 · sin(π(x − 1)) + 2:
Parameter | Value |
|---|---|
Amplitude |a| | 3 |
Midline d | 2 |
Max / Min | 5 / −1 |
Period 2π/|b| | 2 |
Phase shift c | 1 (to the right) |
8. Summary
- y = a · sin(b(x − c)) + d (or cosine equivalent) is the general sinusoidal model
- |a| sets amplitude; 2π/|b| sets period; c shifts horizontally; d sets the midline
- Any sinusoidal graph can be written in either sine or cosine form
- Negative a flips the curve vertically; negative b can be absorbed into a
- Read parameters by extracting max, min, period, and a reference x-value from the graph