Identify the amplitude, period, midline, and phase shift of a sinusoidal function from its equation; graph sine and cosine functions; and build a sinusoidal model of a periodic real-world situation.

What this topic is asking

The Regents Algebra II exam (the Trigonometric Functions, F-TF, and Interpreting Functions, F-IF, clusters) wants you to read the amplitude, period, midline, and phase shift of a sinusoidal function from its equation, graph sine and cosine functions, and build a sinusoidal model of a periodic real-world situation such as tides, temperature, or a Ferris wheel. Sinusoidal modeling is a frequent Part III task.

The four parameters

A sinusoidal function is fully described by four numbers in its equation.

The amplitude is the height from the midline to a peak; the period is the horizontal length of one full cycle; the midline is the horizontal line the curve oscillates around; the phase shift moves the whole curve left or right. Note that the period depends on $B$ through $\frac{2\pi}{B}$, so a larger $B$ compresses the cycle.

Graphing a sinusoid

To graph, start from the midline $y = D$, mark the maximum $D + |A|$ and minimum $D - |A|$, and lay out one period of length $\frac{2\pi}{B}$ divided into quarters (the sine pattern is midline, max, midline, min, midline across one period). Apply any phase shift by sliding the starting point.

Modeling a periodic situation

To build a model from data such as a tide table, extract the four parameters from the situation:

• Midline $D$ = the average of the maximum and minimum values.
• Amplitude $A$ = half the difference between the maximum and minimum.
• $B$ from the period: $B = \frac{2\pi}{\text{period}}$.

A clarifying point that protects credit is the distinction between the amplitude and the maximum value: the amplitude is the distance from the midline to a peak, while the maximum is the midline plus the amplitude. A tide with amplitude 4 and midline 7 has a maximum of 11, not 4. The Regents frequently tests this by asking for the maximum height, and answering with the amplitude alone is a common, costly slip. Likewise, the period and the value of $B$ are reciprocally related through the factor $2\pi$, so keep them straight.

A second useful idea is the difference between sine and cosine starting points, which matters when you choose a model. A cosine curve $y = A\cos(Bx) + D$ starts at its maximum when $x = 0$, while a sine curve $y = A\sin(Bx) + D$ starts at the midline and rises. So when a real situation begins at its highest value (a Ferris wheel rider boarding at the top, a temperature peaking at noon), a cosine model is the natural fit; when it begins at the average value and increases, sine fits directly. You can always convert between them with a phase shift, but choosing the function that matches the starting behavior keeps the model simpler and the parameters easier to read.

Try this

Q1. State the amplitude and period of $y = 5\sin(3x)$. [2 credits]

• Cue. Amplitude $5$; period $\frac{2\pi}{3}$.

Q2. A sinusoid has maximum 10 and minimum 2. Find its midline and amplitude. [2 credits]

• Cue. Midline $\frac{10 + 2}{2} = 6$; amplitude $\frac{10 - 2}{2} = 4$.