Lesson 4.8

The Tangent Graph

Unlike sine and cosine, tangent has no amplitude — it shoots to . Its graph features vertical asymptotes and a period of , not .

Introduction

Past Knowledge

and is undefined when .

Today's Goal

Graph and locate its vertical asymptotes and zeros.

Future Success

Tangent and cotangent are the basis for secant and cosecant graphs.

Key Concepts

Key Features of

Feature	Value
Period
Asymptotes	(where )
Zeros	(where )
Domain	All real numbers except
Range

Red dashed = asymptotes · Green dots = zeros

The S-Shape

Each branch of the tangent curve is an S-shape that rises from to , passing through a zero at the midpoint between adjacent asymptotes. The curve is always increasing.

Worked Examples

Basic

Finding Asymptotes

Question: List the asymptotes of in .

Step 1: Asymptotes occur at .

Step 2: In :

Final Answer:

Intermediate

Modified Tangent Period

Question: Find the period and asymptotes of .

Step 1: For tangent, period

Step 2: Asymptotes:

Final Answer: Period . Asymptotes at .

Advanced

Tangent Value from the Graph

Question: Why does ? Confirm from the graph.

Step 1: is the midpoint between (a zero) and (an asymptote).

Step 2: On the graph, the S-curve passes through . Algebraically: .

Final Answer: Confirmed:

Common Pitfalls

Using for the Tangent Period

The period of tangent is , not . The formula is , not .

Real-Life Applications

Road Grade & Slope

Road engineers describe slope as a percentage “grade” — a 6% grade means the road rises 6 feet for every 100 feet horizontally. Mathematically, grade = . As the angle approaches , the tangent (and thus the grade) approaches infinity — a vertical cliff!