Topic 3.5 Sinusoidal Functions: write a sinusoidal function in the form a*sin(b(x - c)) + d (or with cosine) and relate amplitude, period, phase shift and vertical shift to the parameters.

What this topic is asking

The College Board (Topic 3.5) wants you to work with the general sinusoidal function and connect its four parameters to the graph's features. The standard form is $y = a\sin(b(x - c)) + d$ (or with cosine), where $a$ controls amplitude, $b$ controls period, $c$ is the phase (horizontal) shift, and $d$ is the vertical shift (the midline).

The general form and its parameters

The cosine form $y = a\cos(b(x - c)) + d$ describes the same family; choosing sine or cosine just changes which feature is easiest to anchor.

Amplitude and midline from the extremes

Period from the parameter b

The period and $b$ are reciprocally linked through $2\pi$: period $= \frac{2\pi}{|b|}$, so $b = \frac{2\pi}{\text{period}}$. A common error is to read $b$ itself as the period; it is not, because $b$ is a frequency-like factor that compresses or stretches the standard $2\pi$ cycle.

Choosing sine or cosine

A point worth stating once is that sine and cosine models of the same graph are equally valid; you pick whichever feature you can anchor cleanly. Use cosine when you know where a maximum occurs (cosine peaks at $x = c$); use sine when you know where the graph crosses its midline going upward (sine does this at $x = c$). Anchoring to the right feature makes the phase shift $c$ fall out immediately, which is far faster than solving for it. This flexibility is exactly what the modelling questions in Topic 3.7 reward.

A second point worth keeping in view is the order in which to read the four parameters off a graph. Amplitude and vertical shift come first, from the maximum and minimum, because they need no choice of sine or cosine. The period comes next, from the peak-to-peak (or trough-to-trough) distance, which sets $b$. Only the phase shift $c$ depends on the sine-or-cosine decision, so leaving it until last keeps the work clean. Following this fixed order, amplitude and midline, then period, then phase, turns every "find the equation of this sinusoid" question into the same short routine, regardless of how the graph is drawn or which features the problem happens to label. The same routine runs in reverse when a question gives you an equation and asks for the graph: read $a$, $d$, $b$ and $c$ in turn and plot the midline, the swing, the cycle length and the starting landmark.

Try this

Q1. What is the amplitude and midline of $y = -4\cos(x) + 1$? [1 point]

• Cue. Amplitude $|-4| = 4$; midline $y = 1$. The negative sign flips the wave but does not change the amplitude.

Q2. A sinusoid has period $\frac{\pi}{2}$. What is $b$? [1 point]

• Cue. $b = \frac{2\pi}{\pi/2} = 4$.