Lesson 4: Graphing Secant and Cosecant

Estimated time: 30-35 minutes

Learning Objectives

By the end of this lesson, you will be able to:

Graph y = sec x and y = csc x using the reciprocal relationship
Identify vertical asymptotes, domain, range, and period
Use the cosine/sine graph as a guide to sketch secant/cosecant
Graph transformed secant and cosecant functions

Graphing Secant Using Cosine

Since sec x = 1/cos x, the secant graph is related to cosine:

Wherever cos x = 0, sec x has a vertical asymptote
Wherever cos x = 1, sec x = 1 (local minimum)
Wherever cos x = −1, sec x = −1 (local maximum)
The secant graph consists of U-shaped branches opening up and down

y = sec x: Domain: x ≠ π/2 + nπ | Range: (−∞, −1] ∪ [1, ∞) | Period: 2π

Strategy: First sketch y = cos x lightly. Draw vertical asymptotes where cos x = 0. Then draw U-shaped curves that touch the cosine curve at its max/min points and curve away toward the asymptotes.

Graphing Cosecant Using Sine

Similarly, csc x = 1/sin x. Asymptotes occur where sin x = 0.

y = csc x: Domain: x ≠ nπ | Range: (−∞, −1] ∪ [1, ∞) | Period: 2π

Example 1: Key Features of csc x

Asymptotes at x = 0, ±π, ±2π, ...

At x = π/2: sin = 1, so csc = 1 (local min of upper branch)

At x = 3π/2: sin = −1, so csc = −1 (local max of lower branch)

Transformations

Example 2: Transformed Cosecant

Graph y = 2 csc(x − π/4). Identify period, phase shift, and asymptotes.

Period: 2π (B = 1). Phase shift: π/4 right.

First graph y = 2 sin(x − π/4) as a guide. The sine graph shifts right by π/4.

Asymptotes where sin(x − π/4) = 0: x − π/4 = nπ, so x = π/4 + nπ.

The U-shaped branches touch the sine guide curve at its peaks (±2) and curve away.

Example 3: Vertical Shift

Graph y = sec(x) + 2. The entire graph shifts up by 2. The U-branches now open from y = 3 upward and from y = 1 downward. Asymptote positions remain the same.

Comparing All Six Trig Graphs

Function	Period	Range	Asymptotes
sin, cos	2π	[−1, 1]	None
tan, cot	π	(−∞, ∞)	Yes
sec, csc	2π	(−∞,−1]∪[1,∞)	Yes

Check Your Understanding

1. Where are the vertical asymptotes of y = csc x?

At x = nπ (where sin x = 0).

2. What is the range of y = 3 sec x?

(−∞, −3] ∪ [3, ∞).

3. To graph y = csc(2x), what guide function would you sketch first?

y = sin(2x). Then place asymptotes where sin(2x) = 0 and draw U-curves.

Key Takeaways

Graph sec by first sketching cos as a guide; graph csc by first sketching sin.
Asymptotes appear where the reciprocal function equals zero.
Sec and csc have period 2π and range (−∞, −1] ∪ [1, ∞).
The same transformation rules (A, B, C, D) apply to sec and csc.