Lesson 6.4

Horizontal Asymptotes

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

Introduction

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

Past Knowledge

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

Today's Goal

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

Future Success

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

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 .

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)

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.

Real-Life Applications

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.

Practice Quiz

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