Geometry and Trigonometry - Graphs of Trigonometric Functions

General Sinusoidal Model: The functions $y = a \\sin(b(x - c)) + d$ and $y = a \\cos(b(x - c)) + d$ represent periodic waves. Visually, the sine graph starts at the midline $(0, d)$ and moves upwards (if $a > 0$), while the cosine graph starts at its maximum value $(0, a + d)$. Both functions oscillate infinitely between a maximum and a minimum value.

Amplitude ($|a|$): This is the vertical stretch factor. Visually, it represents half the vertical distance between the peak (maximum) and the trough (minimum), or the distance from the horizontal midline to either the max or min. A negative $a$ indicates a vertical reflection across the midline.

Period and Horizontal Frequency ($b$): The period is the horizontal length of one complete cycle. It is calculated as $P = \\frac{2\\pi}{|b|}$ for sine and cosine. Visually, as $|b|$ increases, the graph undergoes horizontal compression, meaning the waves appear more frequently over a fixed interval.

Phase Shift ($c$): This is the horizontal translation of the function. In the expression $(x - c)$, a positive $c$ value shifts the graph to the right, while a negative $c$ value shifts it to the left. For a cosine graph, the phase shift often identifies the $x$-coordinate of the first maximum point.

Midline and Vertical Shift ($d$): The constant $d$ shifts the graph vertically. The horizontal line $y = d$ is called the midline or principal axis. The graph is perfectly symmetrical above and below this line. The maximum value is $d + |a|$ and the minimum is $d - |a|$.

The Tangent Function: The graph of $y = a \\tan(b(x - c)) + d$ differs significantly from sine and cosine. It does not have an amplitude because the range is $(-\\infty, \\infty)$. Visually, it consists of a series of repeating 'S-curves' separated by vertical asymptotes where the function is undefined, occurring at $x = \\frac{\\pi}{2b}$ for the basic function $y = \\tan(bx)$.

Domain and Range: For $y = a \\sin(bx)$ or $y = a \\cos(bx)$, the domain is all real numbers $(\\mathbb{R})$ and the range is $[d - |a|, d + |a|]$. For $y = \\tan(x)$, the domain excludes values where $\\cos(x) = 0$ (the asymptotes), and the range is all real numbers.

Transformations of Trig Graphs: To map $f(x)$ to $g(x) = a f(b(x - c)) + d$, apply transformations in this order: 1. Horizontal stretch/compression by factor $\\frac{1}{b}$, 2. Vertical stretch by factor $a$, 3. Reflection (if $a$ or $b$ are negative), 4. Horizontal translation by $c$, 5. Vertical translation by $d$.