Graphing Sine
Graphing 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 $y = A\sin(Bx + C) + D$: the basic curve starts at $(0, 0)$, rises to $1$, returns to $0$, dips to $-1$, and returns to $0$ over a distance of $2\pi$. To graph the general form, find amplitude $|A|$, period $2\pi/|B|$, phase shift $-C/B$, and midline $D$, then locate the five key points of one cycle starting at $x = -C/B$ and incrementing by $\text{period}/4$. In phasor notation, the general sinusoid can be written as $\text{Im}(Ae^{i(Bx+C)}) + D$, capturing amplitude and phase in a single complex number used throughout signal processing.
Example
Key points for one cycle of $y = \sin(x)$: $(0, 0)$, $(\pi/2, 1)$, $(\pi, 0)$, $(3\pi/2, -1)$, $(2\pi, 0)$. For $y = 2\sin(\pi x - \pi) + 1$: $A = 2$, period $= 2$, phase shift $= 1$, midline $= 1$, giving key points $(1,1)$, $(1.5, 3)$, $(2,1)$, $(2.5,-1)$, $(3,1)$. 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 $30^\circ$ phase lag, and its phasor $120\sqrt{2}e^{-i\pi/6}$ lets all circuit calculations use complex arithmetic.
Key Insight
The five key points (start, max, midline-crossing, min, end) are enough to sketch any sine wave; transformations happen in a specific order, amplitude, then period, then phase shift, then vertical shift, and identifying each separately avoids errors. The general sinusoid spans a $4$-dimensional parameter space $(A, B, C, D)$, and sinusoids at a fixed frequency $B$ form a $3$-dimensional affine subspace, which is why Fourier analysis decomposes functions into a basis of sinusoids at fixed frequencies, each characterized by amplitude and phase.