Lesson 4.10

Graphing Secant

Secant is the reciprocal of cosine: . By graphing first as a guide, we can construct the secant graph with its characteristic U-shaped branches.

Introduction

Past Knowledge

You know and can graph cosine with its key features.

Today's Goal

Use the cosine graph as a guide to sketch the secant graph.

Future Success

Understanding reciprocal graphs is key for verifying identities and solving advanced equations.

Key Concepts

Building Secant from Cosine

Draw cosine as a dashed guide curve.
Mark asymptotes where (at ).
Draw U-shapes: upward U at cosine's maxima, downward U at cosine's minima.

Blue = sec(x) · Gray dashed = cos(x) guide · Red = asymptotes

Feature	Value
Period
Asymptotes
Range
No zeros	never equals 0

Worked Examples

Basic

Evaluating Secant

Question: Find .

Final Answer: . This is the vertex of an upward U-branch.

Intermediate

Identifying Asymptotes

Question: Why is undefined?

— division by zero → vertical asymptote.

Final Answer: Undefined because .

Advanced

Sketching Strategy

Question: Describe how to sketch one period of .

1. Draw as a dashed guide from to .

2. Draw vertical asymptotes at and .

3. At and (cos max), draw an upward U touching .

4. At (cos min), draw a downward U touching .

Final Answer: Two upward U-branches and one downward U-branch per period, guided by cosine.

Common Pitfalls

Connecting the U-Branches

Each U-branch is a separate piece. Do not draw a continuous curve through the asymptotes — the function is undefined there, and the branches never cross the horizontal band between and .

Real-Life Applications

Structural Engineering

The secant function appears in formulas for forces on curved structures. When analyzing the forces on a suspension cable, engineers encounter in the tension formula — as the cable angle approaches horizontal (), the tension increases without bound, mirroring secant's asymptotic behavior.