Slant asymptotes & sketching
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Flip to reveal answersWhen does a rational function have a slant (oblique) asymptote?
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All 8 Flashcards — Slant asymptotes & sketching
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Question
When does a rational function have a slant (oblique) asymptote?
Answer
When the numerator's degree is exactly one more than the denominator's.
Question
How do you find the slant asymptote?
Answer
Divide top by bottom; the quotient line y = mx + c is the asymptote (the remainder term → 0).
Question
Slant asymptote of (x² + 1)/(x − 1)?
Answer
Divide: x + 1 + 2/(x − 1), so y = x + 1.
Question
Steps to sketch a rational function?
Answer
x-intercepts (top = 0), y-intercept (x = 0), vertical asymptotes (bottom = 0), horizontal/slant asymptote, then fit the branches.
Question
Can a function have both a vertical and a slant asymptote?
Answer
Yes — e.g. (x² + 1)/(x − 1) has vertical x = 1 and slant y = x + 1.
Question
Slant asymptote of (2x² − x + 1)/(x + 1)?
Answer
y = 2x − 3 (the quotient of the division).
Question
Does a curve ever cross its slant asymptote?
Answer
It can cross it (asymptotes are about behaviour as x → ±∞), unlike never crossing a vertical one.
Question
What do you draw first when sketching?
Answer
The asymptotes as dashed lines, then the intercepts.
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Full study notes for Slant asymptotes & sketching
Topic 2.13 hub
Rational functions (HL only)
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