Vertical Shift (Trig)
Vertical shift in a trigonometric function moves the entire graph up or down by adding or subtracting a constant, setting the midline.
Formula
\text{midline: } y = D \text{ for } y = A\sin(Bx + C) + D
Definition
Vertical shift moves the entire sine or cosine wave up or down: adding a number shifts it up, subtracting shifts it down. For $y = A\sin(Bx + C) + D$, the vertical shift is $D$, also the midline $y = D$, and the graph oscillates between $D - |A|$ (minimum) and $D + |A|$ (maximum) without affecting amplitude, period, or phase shift. In signal terms, $D$ represents the DC component (zero-frequency term): in a Fourier series it is the constant term $a_0/2$, the mean value of the function over one period, and its Fourier transform acquires a Dirac delta at $\omega = 0$ with coefficient $D$.
Example
$y = \sin(x) + 3$ is the same wave as $y = \sin(x)$ moved up $3$ units, oscillating between $2$ and $4$ instead of $-1$ and $1$; for $y = -2\cos(x) + 5$, the midline is $5$, amplitude $2$, max $7$, min $3$. Temperature models often use vertical shifts, such as $T(t) = 15\sin(2\pi t/365 - 1.4) + 55$ degrees Fahrenheit, where $55^\circ$F is the annual average and $15^\circ$F is the seasonal swing.
Key Insight
The vertical shift sets the "sea level" of the wave, instead of averaging at $y = 0$, the wave averages at $y = D$, and it is easy to read off: it is the average of the maximum and minimum values, $D = (\max + \min)/2$. The vertical shift separates the "steady state" from the "oscillatory" component of a signal; in control systems, the steady-state offset and the oscillatory response are analyzed separately, with the DC gain handling the vertical shift and the AC gain handling amplitude and phase.