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The period is the distance along the x-axis to complete one cycle of the graph. For $y = A\sin(Bx)$ or $y = A\cos(Bx)$, the period is $\frac{2\pi}{|B|}$.

Amplitude corresponds to the maximum deviation from the graph's central axis. For the functions $y = A\sin(Bx)$ or $y = A\cos(Bx)$, the amplitude is $|A|$.

Frequency is the number of cycles the graph completes in an interval of $2\pi$. For $y = A\sin(Bx)$ or $y = A\cos(Bx)$, the frequency is $|B|/(2\pi)$.

The cotangent graph has undefined values at $x = n\pi$, where $n$ is an integer, leading to vertical asymptotes. The period of the cotangent function is also $\pi$.

The tangent graph has undefined values at $x = (2n + 1)\frac{\pi}{2}$, where $n$ is an integer, resulting in vertical asymptotes. The period of the tangent function is $\pi$.

Phase shift denotes the horizontal displacement of the graph. For $y = A\sin(B(x - C))$ or $y = A\cos(B(x - C))$, the phase shift is $C$ units to the right if $C > 0$ and to the left if $C < 0$.

Vertical shift refers to the upward or downward translation of the graph. For $y = A\sin(Bx) + D$ or $y = A\cos(Bx) + D$, the graph is shifted $D$ units up for $D > 0$ or down for $D < 0$.

A reflection over the x-axis changes the sign of the amplitude. For $y = -A\sin(Bx)$ or $y = -A\cos(Bx)$, the graph is reflected over the x-axis.

Sine graphs are symmetric about the y-axis when they are not phase shifted. This means $f(-x) = -f(x)$ for all $x$ in the graph of $y = \sin(x)$.

To find the equation from a graph, identify the amplitude (A), period ($2\pi/B$), phase shift (C), and vertical shift (D), then use the sine or cosine model $y = A\sin(B(x-C)) + D$ or $y = A\cos(B(x-C)) + D$.

Complex transformations like $y = A\sin(B(x - C)) + D$ involve stretching/compressing vertically by $A$, horizontally by $B$, shifting horizontally by $C$, and vertically by $D$. Analyze these separately for clarity.