Lesson 6.3

Vertical Asymptotes

Where the graph explodes to infinity. Understanding the invisible walls that functions cannot cross.

Introduction

Where the graph explodes to infinity. Understanding the invisible walls that functions cannot cross.

Past Knowledge

In Lesson 6.2, we found "problem numbers" where the denominator was zero. In this lesson, we see what the graph actually DOES at those numbers.

Today's Goal

A Vertical Asymptote (VA) is a vertical line . As the graph gets closer and closer to this line, the y-values shoot up to or crash down to .

Future Success

Think of it as a force field. The graph can get infinitely close, but it can never touch or cross a Vertical Asymptote.

Key Concepts

How to Find Them

Step 1: Simplify FIRST!

You MUST factor and cancel common terms before finding asymptotes. (If they cancel, they are Holes, not VAs—see Lesson 6.5).

Step 2: Set Denominator = 0

Whatever factors are LEFT in the bottom will create Vertical Asymptotes.

Infinite Behavior

Odd Multiplicity (Example:

)

"Opposite Directions"

[Graph goes UP on one side, DOWN on other]

Like a Volcanic Eruption vs Waterfall

Even Multiplicity (Example:

)

"Same Direction"

[Both sides go UP or both go DOWN]

Like a Chimney or a Pit

Worked Examples

Level: Basic

Example 1: Finding Equations

Find the vertical asymptotes of .

Is it simplified? Yes. Nothing cancels.

Set Bottom = 0

Answer

Two Vertical Asymptotes:

(The y-axis)

Level: Intermediate

Example 2: Hidden Asymptotes

Find VAs for .

Factor Denom:

Set factors to zero:

Equations:

and

Level: Advanced

Example 3: Behavior at the Line

Describe the behavior near the asymptote for .

VA is at

.
The factor is squared (

), so Multiplicity = 2 (Even).

Interpretation: Since square numbers are always positive, the denominator is always positive (but tiny) near 3.

Conclusion

on BOTH sides of

Common Pitfalls

Saying "The Asymptote is 5":
It's a LINE, not a number. You MUST write "". Writing just "5" is wrong because that could mean .
Confusing Holes and Asymptotes:
If is in the top AND bottom, it cancels out. That's a Hole. It is NOT an asymptote. Only simplify first!

Real-Life Applications

Black Holes

In relativity, the gravitational force near a black hole is modeled by equations with vertical asymptotes at the "Event Horizon" (Schwarzschild radius).

As you approach , the time dilation approaches infinity. To an outside observer, time literally stops at the asymptote.