Lesson 3: Graphing Tangent and Cotangent
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Sketch the basic graphs of y = tan x and y = cot x
- Identify vertical asymptotes, period (π), domain, and range
- Graph y = A tan(Bx − C) + D with transformations
- Compare and contrast tangent and cotangent graphs
The Basic Tangent Graph
Since tan x = sin x / cos x, the tangent function is undefined wherever cos x = 0 (at odd multiples of π/2).
Key properties of y = tan x:
Domain: all x ≠ π/2 + nπ | Range: (−∞, ∞) | Period: π
Vertical asymptotes at x = ±π/2, ±3π/2, ...
Passes through origin (0, 0). Increasing on each branch.
Key points on (−π/2, π/2): (−π/4, −1), (0, 0), (π/4, 1)
The Basic Cotangent Graph
Key properties of y = cot x:
Domain: all x ≠ nπ | Range: (−∞, ∞) | Period: π
Vertical asymptotes at x = 0, ±π, ±2π, ...
Decreasing on each branch. Key points on (0, π): (π/4, 1), (π/2, 0), (3π/4, −1)
Cotangent is decreasing where tangent is increasing, and its asymptotes are where tangent crosses zero.
Transformations of Tangent
For y = A tan(Bx − C) + D:
- |A| is the vertical stretch (no amplitude for tan since range is all reals)
- Period = π/|B| (note: π, not 2π)
- Phase shift = C/B
- D = vertical shift
Example 1: Transformed Tangent
Graph y = 2 tan(x/2). Find the period and asymptotes.
Period: π/(1/2) = 2π
Asymptotes: x/2 = ±π/2, so x = ±π. More generally, x = π + 2nπ.
The graph is stretched vertically by 2 and horizontally by 2.
Example 2: Phase-Shifted Tangent
Find the period and phase shift of y = tan(2x − π).
Period: π/2. Phase shift: π/2 right. Asymptotes shift to x = π/2 ± π/4 = π/4 and 3π/4.
Graphing Strategy for Tangent/Cotangent
- Find the period (π/|B|) and phase shift (C/B)
- Locate two consecutive vertical asymptotes
- Find the midpoint between asymptotes (the x-intercept for tan, or where the function equals D)
- Find the quarter points (where the function equals A + D and −A + D)
- Sketch the curve passing through these three points between the asymptotes
Check Your Understanding
1. What is the period of y = tan(3x)?
2. Where are the asymptotes of y = cot(x)?
3. Find the phase shift of y = tan(x + π/6).
Key Takeaways
- Tangent and cotangent have period π (not 2π).
- Both have vertical asymptotes and range (−∞, ∞).
- Tangent is increasing; cotangent is decreasing on each branch.
- For transformations, period = π/|B|.