3.4 Sinusoidal Functions

Anatomy of a sinusoidal function

A sinusoidal function has the form f(x) = A·sin(B(x−C)) + D (or cosine). |A| is the amplitude (half the distance between max and min), D is the midline value, and the period is 2π/|B|. C is the phase shift, sliding the graph left or right.

Reading key features from data

Given a max value and min value, midline D = (max+min)/2 and amplitude A = (max−min)/2. The period is the distance between two consecutive maxima (or minima). Use these values plus a known point to solve for B and C when building an equation from a table or context.

Modeling periodic behavior

Sinusoids model real-world periodic phenomena: tides, temperature cycles, Ferris wheels, sound waves. Identify whether the context starts at a max, min, or midline to choose sine vs. cosine and the sign of A, then use period and amplitude data to finish the model.

Key terms

Amplitude

|A|, half the distance between the maximum and minimum values of a sinusoid.

Midline

D, the horizontal line halfway between the max and min: D = (max+min)/2.

Period formula

Period = 2π/|B|, the length of one complete cycle.

Phase shift

C in A·sin(B(x−C))+D; shifts the graph horizontally left or right.

Choosing sine vs cosine for a model

Use cosine if the cycle starts at a max/min with no shift; use sine if it starts at the midline.

Try an AP-style question

For y = 3 sin(2x) + 1, what are the amplitude, period, and midline?

1. 3, π, y = 1
2. 3, 2π, y = 1
3. 2, π, y = 3
4. 3, π/2, y = 1