How To Identify the 3 Types of Non-Vertical Asymptotes
Feb 23, 2017 · 2 min read
Non-Vertical (Horizontal and Slant/Oblique Asymptotes) are all about recognizing if a function is TOP-HEAVY, BOTTOM-HEAVY, OR BALANCED based on the degrees of x.
What I mean by “top-heavy” is that there is a higher degree of x in the numerator than in the denominator.
1. Top-Heavy
- What does it mean?
- there is a higher degree of x in the numerator than in the denominator
- ex: (x-5)³/(x²-4)
- What type of Non-Vertical Asymptote does it make?
- SLANT ASYMPTOTE (as long as the numerator is only 1 degree higher than the denominator)
- How do you find the equation?
- Do Long Division of the top divided by the bottom
- Ignore the remainder — this is not part of the equation
2. Bottom-Heavy
- What does it mean?
- there is a lower degree of x in the numerator than in the denominator
- ex: (x-5)²/(x³-4)
- What type of Non-Vertical Asymptote does it make?
- HORIZONTAL ASYMPTOTE
- How do you find the equation?
- The equation is always y=0 because you’re dividing by infinitely bigger and bigger numbersthat make the y value closer and closer to 0
3. Balanced
- What does it mean?
- there is an equal degree of x in the numerator than in the denominator
- ex: (x-5)²/(x²-4)
- What type of Non-Vertical Asymptote does it make?
- HORIZONTAL ASYMPTOTE
- How do you find the equation?
- The equation is going to be a ratio of the coefficients in front of the largest degrees of x
- ex: (3x³ — 4x² + x — 1) / (-2x³+8) would have a horizontal asymptote at y = -3/2
