Tangent and cotangent graphs

Tangent and cotangent graphs

Explore the unbounded periodic graphs of tangent and cotangent using labeled curves, exact inputs and transformed equations. This precalculus practice emphasizes what the asymptotes mean, how zeros alternate with excluded inputs, and why these functions need a different graphical description from sine and cosine.

Questions cover positive periods, horizontal transformations, vertical scaling, branch direction and reconstruction of a model from asymptotes and a point. A geometric beam context connects a tangent formula to a finite physical displacement, while error-analysis items test the difference between an undefined input and a large output. Single-select and multiselect tasks check complete sets of claims.

Prerequisites are unit-circle ratios, their denominator restrictions, and amplitude-and-period work for sine and cosine. Radians are used throughout. Statements involving an integer parameter describe every repeated branch, not just the visible portion of a sketch. No calculus is needed: near-asymptote behavior is understood through numerator and denominator signs.

The explanations keep restrictions explicit and distinguish a scale factor from an amplitude. Practice progresses toward graph reconstruction and domain-sensitive edge cases. Each item stands independently, and the graph questions use readable axes and dashed asymptote guides. Successful recognition is a starting point for drawing the branches independently on paper.