Top one degree bigger → a slant line: If the top's degree is exactly one more than the bottom's, the curve follows a slant (oblique) line for large x. Divide to find it:
f(x) = (quotient mx + c) + remainder/(bottom). The line y = mx + c is the asymptote.
IB-style question — find the slant asymptote
Find the asymptotes of f(x) = (x² + 1)/(x − 1).
Step by step
- Vertical: denominator = 0.
- Divide x² + 1 by (x − 1): quotient x + 1, remainder 2.
- As x → ±∞ the last term → 0, so the curve hugs the line.
Final answer
Vertical asymptote x = 1; oblique asymptote y = x + 1.
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Asymptotes first, then fit the curve: Recipe: x-intercepts (top = 0), y-intercept (put x = 0), vertical asymptotes (bottom = 0), and the horizontal/slant asymptote (degrees). Draw the asymptotes dashed, then fit the curve to them.
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IB-style question — sketch features
For f(x) = (x² + 1)/(x − 1), state the intercepts.
Step by step
- x-intercepts: top = 0 ⇒ x² + 1 = 0 has no real solution.
- y-intercept: put x = 0.
Final answer
No x-intercept; y-intercept (0, −1).