Mathfolis

Sinusoidal Functions

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

The general sinusoidal function is f(x)=Asin⁡(B(x−h))+kf(x) = A \sin(B(x - h)) + k (or with cosine in place of sine). Four parameters carry geometric meaning: ∣A∣|A| is the amplitude, 2π/∣B∣2\pi/|B| is the period, hh is the phase shift, and kk is the midline. Together they specify every key feature of the graph.

Parameters of f(x)=Asin⁡(B(x−h))+kf(x) = A \sin(B(x - h)) + k

Amplitude
∣A∣|A|
Period
2π∣B∣\frac{2\pi}{|B|}
Phase shift
hh
Midline
y=ky = k
Caution: Period is 2π/∣B∣2\pi / |B|, not 2πB2\pi B. If B=2B = 2, the period is π\pi — the function repeats twice as often.
Type 1

Identify parameters from the equation

Read AA, BB, hh, kk directly. Compute period as 2π/∣B∣2\pi / |B|.

Example 1
For f(x)=3sin⁡(2(x−1))+5f(x) = 3 \sin(2(x - 1)) + 5, state the amplitude, period, phase shift, and midline. (A) Amp 3, period π\pi, phase 1, midline y=5y = 5 (B) Amp 3, period 2π2\pi, phase 1, midline y=5y = 5 (C) Amp 6, period π\pi, phase 1, midline y=5y = 5 (D) Amp 3, period 4π4\pi, phase 2, midline y=5y = 5

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Type 2

Predict graph features from parameters

Use ∣A∣|A| and kk to find max and min: max = k+∣A∣k + |A|, min = k−∣A∣k - |A|.

Example 2
What are the maximum and minimum values of f(x)=4sin⁡x+1f(x) = 4 \sin x + 1? (A) Max 4, min −4-4 (B) Max 5, min −3-3 (C) Max 5, min 11 (D) Max 4, min 11

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Type 3

Build f from amplitude, period, midline

Set AA, kk directly. Solve 2π/∣B∣=period2\pi/|B| = \text{period} for BB.

Example 3
Build a sinusoid with amplitude 2, period 6, and midline y=3y = 3 (no phase shift, using sine). (A) f(x)=2sin⁡(π3x)+3f(x) = 2 \sin(\tfrac{\pi}{3} x) + 3 (B) f(x)=2sin⁡(6x)+3f(x) = 2 \sin(6 x) + 3 (C) f(x)=3sin⁡(2x)+6f(x) = 3 \sin(2 x) + 6 (D) f(x)=2sin⁡(π6x)+3f(x) = 2 \sin(\tfrac{\pi}{6} x) + 3

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