How do you recognize the graph of y = tan(x)?

The graph of \( y = \tan(x) \) has repeating vertical asymptotes and passes through the origin with a wave-like pattern.

The graph of \( y = \tan(x) \) is unique and can be recognised by its distinct features. It has vertical asymptotes (lines the graph approaches but never touches) at \( x = \frac{\pi}{2} + k\pi \) where \( k \) is any integer. These asymptotes occur because the tangent function is undefined at these points. Between each pair of asymptotes, the graph has a wave-like pattern that passes through the origin (0,0) and repeats every \( \pi \) units along the x-axis.

The graph rises steeply from negative infinity to positive infinity as it approaches each asymptote from the left and falls steeply from positive infinity to negative infinity as it approaches from the right. This creates a series of repeating 'S' shapes. The period of the tangent function is \( \pi \), meaning the pattern repeats every \( \pi \) units.

Additionally, the graph of \( y = \tan(x) \) is symmetric about the origin, which means it has rotational symmetry of 180 degrees around the point (0,0). This symmetry is a result of the tangent function being an odd function, satisfying \( \tan(-x) = -\tan(x) \).

Understanding these characteristics will help you identify the graph of \( y = \tan(x) \) easily among other trigonometric functions.