Key Idea: A rational function is one polynomial divided by another. To sketch it, you find its asymptotes and intercepts first. It is mostly Paper 2 with the GDC, but the algebra is on Paper 1 too.
| Case | Asymptote |
|---|---|
| Bottom = 0 (top not also 0) | Vertical line there |
| Top degree < bottom degree | y = 0 |
| Equal degrees | y = ratio of leading coefficients |
| Top degree = bottom + 1 | Slant: divide, the quotient is the line |
Slant asymptotes & sketching
- Asymptotes — Draw them as dashed lines first.
- Intercepts — x-intercepts: top = 0. y-intercept: x = 0.
- Branches — Fit the curve to the asymptotes.
Asymptotes first, then intercepts, then the curve.
Asymptotes of (x + 2)/(x² − 9)? x = 3, x = −3 and y = 0.
Slant asymptote of (2x² − x + 1)/(x + 1)? y = 2x − 3.
Top and bottom are both 0 at one x. What is there? A hole, not a vertical asymptote.
Why do you drop the remainder for a slant asymptote? It vanishes for large x.