The tangent function

On the unit circle, the terminal ray for angle θ passes through the point $(\cos\theta, \sin\theta)$. The slope of that ray is rise over run: $\sin\theta/\cos\theta$. That quotient is the definition of tangent:

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

Reading tangent as a slope explains its whole personality at once. A nearly flat ray (θ near 0) has slope near 0. As the ray tilts toward vertical (θ near π/2), the slope blows up without bound. A vertical ray has no slope at all, and tangent has no value there. Everything else in this topic is that picture, written out.

Rotate the terminal ray by a half turn. The point $(\cos\theta, \sin\theta)$ moves to the opposite point $(-\cos\theta, -\sin\theta)$: both coordinates flip sign. But the ray through the origin and the opposite point is the same line, so it has the same slope:

$$\tan(\theta + \pi) = \frac{-\sin\theta}{-\cos\theta} = \tan\theta$$

The two sign flips cancel in the quotient. Sine and cosine each need a full turn, $2\pi$, to repeat; tangent repeats every half turn. So the period of $\tan(bx)$ is $\pi/|b|$, not $2\pi/|b|$. Using the sine formula on a tangent function is the single most common period error in this topic.

Because tangent divides by cosine, it is undefined where $\cos\theta = 0$: at $\theta = \pi/2, 3\pi/2, -\pi/2$, every odd multiple of $\pi/2$. At those inputs the graph has vertical asymptotes. Its zeros come from the numerator: $\sin\theta = 0$ at every multiple of $\pi$.

Between consecutive asymptotes lives one complete branch, and every branch looks the same: it climbs from unboundedly negative values, crosses zero exactly once at its center, and climbs off to unboundedly positive values. On each branch tangent is always increasing. There is no peak, no trough, no turning around. Approaching an asymptote from the left, $\tan\theta \to +\infty$; entering the next branch just to the right, it resumes from $-\infty$.

1. Asymptotes: where cosine is zero (odd multiples of π/2).
2. Zeros: where sine is zero (multiples of π).
3. Range: all real numbers, on every single branch.

Amplitude is a bounded-wave measurement: half the distance between a maximum and a minimum. Tangent has neither, its outputs run through every real number, so tangent has no amplitude. In $y = a\tan(bx)$ the coefficient a is a vertical stretch that makes the branches steeper or shallower, but the range stays all reals no matter what a is.

The b does exactly one job: period $= \pi/|b|$. So $y = 3\tan(2x)$ has period $\pi/2$, asymptotes where $2x$ is an odd multiple of $\pi/2$ (that is, $x = \pi/4 + k\pi/2$), zeros at multiples of $\pi/2$, and no maximum value of 3 or anything else. If a question about a tangent graph offers you an amplitude, that is the trap.