Trigonometric Functions

Graphing Tangent & Transformations

Graphing the tangent function and transformations of trigonometric functions.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Graphing Sine & Cosine

Why Tangent's Graph Looks Nothing Like a Wave

Sine and cosine graph as smooth, unbroken waves. Before reading on, think about tan(x) = sin(x)/cos(x) — since it's a ratio, what should happen to its graph at any x-value where cos(x) equals 0?

Wherever cos(x) = 0, tangent is undefined — dividing by 0 — producing a vertical asymptote at every one of those x-values, exactly the same asymptote reasoning from Algebra II's rational functions, just applied to a ratio of trig functions instead of polynomials. Between each pair of asymptotes, the graph shoots from one infinity to the other, giving tangent its distinctive repeating, broken shape.

Tangent's period is also different from sine and cosine's — it's π, not 2π. This follows from checking tan(x + π): since a half revolution flips the sign of both sine and cosine (sin(x + π) = −sin(x), and cos(x + π) = −cos(x)), their ratio is completely unaffected by both signs flipping together: tan(x + π) = (−sin x)/(−cos x) = sin(x)/cos(x) = tan(x). Tangent genuinely repeats twice as often as sine or cosine.

Worked Example — Graphing Tangent and Locating Its Asymptotes

Graph y = tan(x) and identify its vertical asymptotes. Asymptotes occur wherever cos(x) = 0: x = π/2 + kπ, for any integer k. Between consecutive asymptotes, the graph rises from −∞ to +∞, crossing 0 at each multiple of π.

Worked Example — Finding the Period of a Transformed Tangent

Find the period and asymptotes of y = tan(2x). Tangent's period formula is π/|b| (not 2π/|b|, since tangent's own natural period is already π): period = π/2. Asymptotes occur where 2x = π/2 + kπ, so x = π/4 + kπ/2.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

Locate tangent's asymptotes first, before sketching anything else — they anchor the entire repeating pattern, and the rest of the curve fills in predictably between them.

Common Mistakes

Using sine and cosine's period formula, 2π/|b|, for tangent instead of its own formula, π/|b|.
Tangent's natural period is π, not 2π, since it repeats twice as often — always use π/|b| for a transformed tangent function's period.
Trying to find tangent's amplitude, treating it the same way as sine or cosine.
Tangent has no amplitude — its graph extends to positive and negative infinity between each pair of asymptotes, with no maximum or minimum height to measure.

Key Takeaways

Tangent has a vertical asymptote wherever cosine equals 0, since tangent is sine divided by cosine.
Tangent's period is π, since both sine and cosine flip sign together after a half revolution, leaving their ratio unchanged.
Tangent has no amplitude, unlike sine and cosine.

Summary

Tangent's asymptotes and shorter period both follow directly from its definition as a ratio of sine and cosine. The final lesson in this unit asks how to reverse these trig functions — finding an angle from a known ratio.

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