Lesson 6.6

Slant Asymptotes

When the numerator is exactly "one degree stronger" than the denominator, the graph doesn't level off—it shoots away in a straight diagonal line.

1

Introduction

Prerequisite Connection: In Lesson 5.1, we mastered Long Division. In Lesson 6.4, we learned that if the Top Degree > Bottom Degree, there is no Horizontal Asymptote.

Today's Increment: If the top degree is exactly one greater (e.g., or ), the Asymptote is a diagonal line . We call this a Slant (or Oblique) Asymptote.

2

Explanation of Key Concepts

The Division Strategy

To find the equation of the Slant Asymptote, simply perform the division.

The "Ignore the Remainder" Rule

As , the fraction shrinks to zero.
So, the graph behaves exactly like the Quotient part.
.
3

Worked Examples

Level: Basic

Example 1: Using Synthetic Division

Find the slant asymptote of .

Degrees check: Top = 2, Bottom = 1. Difference is 1. Yes, it has a Slant Asymptote.
-1
1
4
-5
↓
-1
-3
1
3
-8 (R)
Quotient: . Remainder: -8.
Answer
Asymptote Equation:
We discard the remainder (-8).
Level: Intermediate

Example 2: Complex Divisor

Find the SA for .

Degrees: Top 3, Bottom 2. The difference is 1, so yes, SA exists.
Divisor is (not linear), so we must use Long Division.
Quotient only concerns the first two terms:

Quick Trick: You can stop dividing once you reach the constant term!
Answer
Level: Advanced

Example 3: Checking the Rule

Does have a Slant Asymptote?

Top Degree: 4
Bottom Degree: 2
Difference = .
Since the difference is NOT 1, it is not line-like (linear).
It is actually a (Parabolic) Asymptote, but that is generally outside the scope of this course.
Answer
No Slant Asymptote.
4

Common Pitfalls

  • Including the Remainder:

    The equation should just be . Do not write . The whole point is that the fractional part goes away at infinity.

  • Using Synthetic Division incorrectly:

    Remember Synthetic Division only works easily for divisors like . If you have or , you might be safer sticking to Long Division to avoid errors.

5

Real-World Application

Economics: Average Cost

The cost to produce items is often .

The Average Unit Cost is .

As gets huge, the term vanishes (fixed costs are spread out), and your per-unit cost approaches the line .

6

Practice Quiz

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