Asymptote #1
VA = Zero of Denominator
Set denominator = 0. But cancel common factors first — those become Holes, not VAs.
Asymptote #2
HA: Compare Degrees
n < m: y = 0 | n = m: y = a/b | n > m: No HA (Slant!)
Hole
Cancel → Plug In
Factor and cancel the common factor. Plug its zero into the simplified function to get y-coordinate.
Slant
Divide → Ignore Remainder
Degree of numerator = degree of denominator + 1. Do polynomial long division; the quotient (no remainder) is the slant asymptote.
X-Intercept
Numerator = 0
Set the simplified numerator = 0. Exclude values that cause holes or VAs.
Y-Intercept
Plug x = 0
Substitute x = 0 into the original function. Only exists if x = 0 is in the domain.
Trap ⚠️
Hole ≠ Asymptote
Most common mistake! A cancelled factor = Hole. An un-cancelled denominator zero = VA. Never mix these up.
Trap ⚠️
HA Can Be Crossed
The graph can cross a horizontal asymptote in the middle — it only describes end behavior (x→±∞).