Amplitude, Period, and Phase Shift

Definition A sinusoidal function is a function of the form

where $A$, $B$, $C$ and $D$ are real constants with $A \ne 0$ and $B \ne 0$, and the input $B(x - C)$ of sine or cosine is measured in radians. When $B > 0$ the graph has amplitude $\lvert A \rvert$, period $\frac{2\pi}{B}$, horizontal shift $C$ and midline $y = D$, and the range is $[D - \lvert A \rvert, D + \lvert A \rvert]$. For the sine form with $A > 0$, one cycle starts on the midline at $x = C$, heading up, and ends at $x = C + \frac{2\pi}{B}$.

Definition The amplitude of $y = A\sin\bigl(B(x - C)\bigr) + D$, or of the matching cosine function, is $\lvert A \rvert$. It equals $\frac{M - m}{2}$, where $M = D + \lvert A \rvert$ is the maximum value and $m = D - \lvert A \rvert$ is the minimum value. Multiplying by $A$ stretches the basic wave vertically by the factor $\lvert A \rvert$ (a compression when $\lvert A \rvert < 1$) and also reflects it when $A < 0$.

Definition The midline of a sinusoidal function is the line $y = D$. It is the horizontal line halfway between the maximum value $M$ and the minimum value $m$, so

Definition For $B \ne 0$ the period of a sinusoidal function is

The reason is that as $x$ moves across an interval of length $\frac{2\pi}{\lvert B \rvert}$, the input $B(x - C)$ moves across an interval of length $2\pi$, which is exactly one cycle of sine or cosine.

Definition The frequency of a sinusoidal function is the number of cycles per unit of $x$. It is the reciprocal of the period,

The number $\lvert B \rvert$ itself is called the angular frequency. It measures how many radians the input advances per unit of $x$.

Definition The graph of $y = A\sin\bigl(B(x - C)\bigr) + D$ is the graph of $y = A\sin(Bx) + D$ shifted $C$ units to the right when $C > 0$, and $\lvert C \rvert$ units to the left when $C < 0$. The number $C$ is also called the phase shift. The same holds for cosine. When the formula is written as $A\sin(Bx - c) + D$, factor out $B$ first,

so the shift is $\frac{c}{B}$, not $c$.

Definition In $y = A\sin\bigl(B(x - C)\bigr) + D$ and in the matching cosine function, the sign of each constant has a fixed meaning.

• If $A < 0$, the graph is the graph with the positive constant $-A$ in place of $A$, reflected across the midline $y = D$. The amplitude is still $\lvert A \rvert$.
• If $B < 0$, the function can be rewritten with the positive constant $-B$, as shown below. The period is $\frac{2\pi}{\lvert B \rvert}$ either way.
• If $C > 0$, the shift is to the right, and if $C < 0$, it is to the left. For example $\sin\left(x + \frac{\pi}{4}\right)$ has $C = -\frac{\pi}{4}$.
• If $D > 0$, the midline lies above the $x$-axis, and if $D < 0$, it lies below.

A negative $B$ can always be made positive because sine is odd and cosine is even,

Definition Fitting a sinusoidal model to a quantity that rises and falls regularly means choosing the constants in

so that the model has the same maximum value $M$, minimum value $m$ and period $P$ as the data, and peaks at a time $x_0$ where the data peak. With $A > 0$ and $B > 0$ this gives

The period is the time between two consecutive peaks, or twice the time from a peak to the next low point. A sine model works too, with $C$ at a time where the data cross the midline going up.