$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$

A vertical line $x = a$ where the function approaches infinity or negative infinity as $x$ approaches $a$.

The smallest positive value $P$ such that $f(x + P) = f(x)$ for all $x$ in the domain of $f$.

A horizontal translation of a function's graph.

A transformation that stretches or compresses a graph vertically.

The set of all possible output values (y-values) of a function.

A transformation that flips the graph of a function over the x-axis, changing the sign of the y-values.

A circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system.

Functions that relate the angles of a triangle to the ratios of its sides.

The set of all possible input values (x-values) for which the function is defined.

On the unit circle, $\tan(\theta)$ represents the slope of the terminal ray. It's the ratio of the y-coordinate (sine) to the x-coordinate (cosine).

Tangent has vertical asymptotes where $\cos(\theta) = 0$, because $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Division by zero is undefined.

The slope of the terminal ray on the unit circle repeats every $\pi$ radians (half-rotation), unlike sine and cosine which repeat every $2\pi$ radians.

Quadrant I: Positive (both sine and cosine are positive). Quadrant II: Negative (sine is positive, cosine is negative). Quadrant III: Positive (both sine and cosine are negative). Quadrant IV: Negative (sine is negative, cosine is positive).

As $x$ approaches an asymptote, $\tan(x)$ approaches either positive infinity or negative infinity. The function becomes infinitely steep.

A negative 'a' value reflects the graph of $y = \tan(x)$ over the x-axis.

Changing 'b' in $y = \tan(bx)$ alters the period of the tangent function. A larger 'b' compresses the graph horizontally, decreasing the period, while a smaller 'b' stretches the graph horizontally, increasing the period.

The range of the standard tangent function, $y = \tan(x)$, is all real numbers, or $(-\infty, \infty)$.

Phase shift is the horizontal translation of a tangent function. In the equation $y = a \tan(b(x - c)) + d$, 'c' represents the phase shift. A positive 'c' shifts the graph to the right, while a negative 'c' shifts it to the left.

The 'd' value represents the vertical shift of the tangent function. A positive 'd' shifts the graph upward, while a negative 'd' shifts it downward.

The steepness indicates the rate of change of the function. Near the asymptotes, the function changes very rapidly.

The period is the distance between two consecutive vertical asymptotes.

The graph of $y = -\tan(x)$ is a reflection of $y = \tan(x)$ over the x-axis.

Compare the location of the asymptotes to the standard $y = \tan(x)$ graph. A horizontal shift in the asymptotes indicates a phase shift.

It moves the entire graph up or down, changing the y-coordinates of all points on the graph.

The graph is vertically stretched, making it steeper compared to the graph of $y = \tan(x)$.

The graph is horizontally compressed, decreasing the period and bringing the asymptotes closer together.

It indicates that the tangent function is an odd function, meaning $\tan(-x) = -\tan(x)$.

The period is $\pi$ (the distance between the asymptotes).

The function will decrease from left to right between asymptotes, instead of increasing.