Lesson 6.4

Horizontal Asymptotes

The End Behavior of fractions. Who wins the tug-of-war between the numerator and the denominator?

1

Introduction

Prerequisite Connection: In Lesson 4.3, we learned that for polynomials like , the graph goes to . But what if we divide by ?

Today's Increment: A Horizontal Asymptote (HA) describes what happens when gets HUGE (like a billion). It's the long-term trend of the graph.

Visual: Imagine a plane leveling off after takeoff. That flat cruising altitude is the Horizontal Asymptote.

2

Explanation of Key Concepts

The Degree Comparison Test

Compare the degree of the Numerator (Top) vs the Denominator (Bottom).

BOBO
Bigger On BOttom
y = 0
Example:
BOTN
Bigger On Top
None
Example:
EATS DC
Exponents Are The Same
Divide Coefficients
Example:

Why This Works

When is 1,000,000, only the highest powers matter.
In , the and become meaningless dust.

It becomes essentially . The 's cancel out, and you are left with .

3

Worked Examples

Level: Basic

Example 1: Bigger On Bottom

Find the horizontal asymptote of .

Step 1: Identify Degrees
Numerator Degree: 1 ()
Denominator Degree: 2 ()
Step 2: Compare
1 < 2. The bottom wins. (BOBO)
Answer
(The x-axis)
Level: Intermediate

Example 2: Balanced Powers

Find the HA of .

Both Top and Bottom are Degree 2. It's a tie.
Ratio =
6
2
= 3
Answer
Level: Advanced

Example 3: No Limit

Analyze the end behavior of .

Top Degree: 3. Bottom Degree: 1.
Top is bigger. The numerator grows way faster than the denominator.
Result: The graph goes to . It does NOT level off.
Answer
No Horizontal Asymptote.
(Hint: It might have a Slant Asymptote! See Lesson 6.6)
4

Common Pitfalls

  • Mixing up X and Y:

    Vertical Asymptotes are (Domain).
    Horizontal Asymptotes are (Range).

  • Can graphs cross Horizontal Asymptotes?

    YES! They can cross in the middle. The asymptote only cares about the "Ends" (infinity). Vertical asymptotes are brick walls; Horizontal asymptotes are just suggestions until the end.

5

Real-World Application

Pharmacology: Saturation Concentration

The concentration of a drug in the bloodstream over time is often modeled by .

As time goes on (), the degrees are equal (1 vs 1). The concentration levels off at . This is the maximum saturation level—you can't absorb any more than this, no matter how much time passes.

6

Practice Quiz

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