Graphing Sine
TrigonometryGraphing sine involves plotting the sinusoidal wave y = A*sin(Bx + C) + D, identifying its amplitude, period, phase shift, and midline.
Formula
y = A\sin(Bx + C) + D
Definition
Graphing sine means drawing the wave shape of $y = \sin(x)$ or a transformed version of it. The basic curve starts at $(0, 0)$, rises to $1$, comes back to $0$, dips to $-1$, and returns to $0$, all over a distance of $2\pi$.
Example
Key points for one cycle of $y = \sin(x)$: $(0, 0)$, $(\pi/2, 1)$, $(\pi, 0)$, $(3\pi/2, -1)$, $(2\pi, 0)$. Connect these smoothly to get the sine wave.
Key Insight
The five key points (start, max, midline-crossing, min, end) are enough to sketch any sine wave. Find them for the transformed version and connect them smoothly.
Definition
To graph $y = A\sin(Bx + C) + D$: find amplitude $|A|$, period $2\pi/|B|$, phase shift $-C/B$, and midline $D$. Locate the five key points of one cycle: starting $x = -C/B$, then increment by $\text{period}/4$ for each successive key point.
Example
Graph $y = 2\sin(\pi x - \pi) + 1$: $A = 2$, $B = \pi$, $C = -\pi$, $D = 1$. Period $= 2\pi/\pi = 2$. Phase shift $= \pi/\pi = 1$. Midline $y = 1$. Start: $x = 1$, $(1,1)$. Max: $(1.5, 3)$. Mid: $(2,1)$. Min: $(2.5,-1)$. End: $(3,1)$.
Key Insight
Transformations happen in a specific order: amplitude (vertical stretch), period (horizontal compress), phase shift (horizontal translate), vertical shift (vertical translate). Identifying each transformation separately avoids errors.
Definition
The function $y = A\sin(Bx + C) + D$ is the general real sinusoid. Using phasor notation, it can be written as $\text{Im}(Ae^{i(Bx+C)}) + D$ or equivalently as the imaginary part of a complex exponential. In signal processing, the general sinusoid is represented as a phasor $Ae^{iC}$ at angular frequency $B$, capturing both amplitude and phase in a single complex number.
Example
In AC circuits, voltage $v(t) = 120\sqrt{2}\sin(120\pi t - \pi/6)$ V represents $120$V RMS at $60$ Hz with a phase lag of $30^\circ$. The phasor is $120\sqrt{2}e^{-i\pi/6}$, and all circuit calculations use complex arithmetic on phasors.
Key Insight
The general sinusoid $y = A\sin(Bx + C) + D$ spans a $4$-dimensional parameter space $(A, B, C, D)$. The set of sinusoids at fixed $B$ forms a $3$-dimensional affine subspace. This structure is why Fourier analysis decomposes functions into a basis of sinusoids at fixed frequencies, each characterized by its amplitude and phase.