Geometry and Trigonometry - Graphs of Trigonometric Functions

General Sine and Cosine Transformations: The functions $y = a \sin(b(x - c)) + d$ and $y = a \cos(b(x - c)) + d$ represent the standard wave models. Visually, these functions create a smooth, repeating 'wave' shape. The cosine graph starts at its maximum value when the phase shift is zero, while the sine graph starts at the midline and moves upwards.

Amplitude ($|a|$): The amplitude is the vertical stretch of the graph, representing half the distance between the maximum and minimum points. Visually, it is the height of the wave crest above the horizontal midline $y = d$. If $a < 0$, the graph is reflected across the midline.

Period and Horizontal Scale ($b$): The period is the horizontal length of one complete cycle. It is calculated as $\frac{2\pi}{b}$ for radians or $\frac{360^{\circ}}{b}$ for degrees. Visually, a larger $b$ value 'compresses' the graph horizontally, meaning more cycles appear within the same distance.

Phase Shift ($c$): The phase shift is the horizontal translation of the graph. A positive $c$ moves the graph to the right, and a negative $c$ moves it to the left. For a sine wave $y = \sin(x - c)$, the point that usually starts at $(0,0)$ is shifted to $(c,0)$.

Vertical Shift and Midline ($d$): The constant $d$ shifts the entire graph up or down. Visually, this creates a new horizontal 'principal axis' or midline at $y = d$, around which the wave oscillates. The maximum value is $d + |a|$ and the minimum value is $d - |a|$.

The Tangent Function $y = a \tan(b(x - c)) + d$: Unlike sine and cosine, the tangent graph is not a continuous wave but a series of separate branches. It has a period of $\frac{\pi}{b}$ (or $\frac{180^{\circ}}{b}$) and features vertical asymptotes where the function is undefined (e.g., at $x = \frac{\pi}{2}$ for $y = \tan(x)$). Visually, it always increases between asymptotes and passes through the midline at the center of each period.

Domain and Range: For sine and cosine functions, the domain is all real numbers $x \in \mathbb{R}$, and the range is restricted to $[d - |a|, d + |a|]$. For the tangent function, the range is all real numbers $y \in \mathbb{R}$, but the domain excludes values where the vertical asymptotes occur.