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Rational Function Asymptotes
AP Precalculus · Topics 1.7–1.9
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Learning Objectives
Identify vertical asymptotes from zeros of denominator (LO 1.7.A)
Determine horizontal asymptotes by comparing degrees (LO 1.8.A)
Find slant asymptotes when deg(N) = deg(D) + 1
Recognize holes from common factors
Asymptote Rules
Vertical:
denominator = 0
Horizontal:
deg(N) < deg(D) → y = 0
deg(N) = deg(D) → y = aₙ/bₙ
deg(N) > deg(D) → none
Slant:
deg(N) = deg(D)+1
Tags
rational function
vertical asymptote
horizontal asymptote
slant asymptote
AP Precalculus
VA
x=0
HA
y=0
Slant
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Preset Functions
1/x
1/(x²−1)
(x+1)/(x−2)
x²/(x−1)
(x²−4)/(x−2)
(2x²+1)/(x²−4)
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8
Display
Vert. Asymptotes
Horiz/Slant
Grid
Function
f(x) = 1/x
Asymptote Summary
VA: x=0 · HA: y=0
Quick Quiz
What is the horizontal asymptote of f(x) = (3x² + 1)/(x² − 4)?
y = 0
y = 3
y = −4
No HA