Vertical Shift (Trig)
TrigonometryVertical 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 to the function shifts it up; subtracting shifts it down.
Example
$y = \sin(x) + 3$ is the same wave as $y = \sin(x)$ but moved up $3$ units. It oscillates between $2$ and $4$ instead of between $-1$ and $1$.
Key Insight
The vertical shift sets the "sea level" of the wave. Instead of averaging at $y = 0$, the wave averages at $y = D$, the vertical shift value.
Definition
For $y = A\sin(Bx + C) + D$, the vertical shift is $D$, which is also the midline $y = D$. The graph oscillates $D + |A|$ (maximum) to $D - |A|$ (minimum). Vertical shift does not affect the amplitude, period, or phase shift.
Example
$y = -2\cos(x) + 5$: midline $= 5$, amplitude $= 2$, max $= 7$, min $= 3$. The negative sign on $A$ reflects the graph over $y = 5$, but the midline stays at $5$.
Key Insight
The vertical shift is easy to read: it is the average of the maximum and minimum values of the graph. Midline $y = (\max + \min)/2 = D$.
Definition
The vertical shift $D$ in $y = A\sin(Bx + C) + D$ represents the DC component (zero-frequency term) of the signal. In the Fourier series, this corresponds to the constant term $a_0/2$, the mean value of the function over one period. Shifting vertically by $D$ translates the function in the range, and its Fourier transform acquires a Dirac delta at $\omega = 0$ with coefficient $D$.
Example
Temperature models often use vertical shifts: $T(t) = 15\sin(2\pi t/365 - 1.4) + 55$ degrees Fahrenheit, where $55^\circ$F is the annual average (vertical shift) and $15^\circ$F is the amplitude of seasonal variation.
Key Insight
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 the amplitude and phase of oscillations.