Complete Study of a Rational Function — The 8 Phases #
| # | Phase | What you find |
|---|---|---|
| 1 | Domain | Excluded values |
| 2 | Symmetry | Even, odd, or none |
| 3 | Intercepts | Points on X and Y axes |
| 4 | Sign | Positive and negative regions |
| 5 | Asymptotes | Limiting lines |
| 6 | First derivative | Increasing/decreasing, max/min |
| 7 | Second derivative | Concavity, inflection points |
| 8 | Graph | Complete curve |
Worked Example: \(f(x) = \frac{x^2 - 1}{x^2 - 4}\) #
- Domain: \(x \neq \pm 2\)
- Symmetry: even (symmetric about Y-axis)
- Intercepts: Y: \((0, 1/4)\), X: \((\pm 1, 0)\)
- Vertical asymptotes: \(x = \pm 2\)
- Horizontal asymptote: \(y = 1\)
- First derivative: \(f’(x) = \frac{-6x}{(x^2-4)^2}\) → maximum at \((0, 1/4)\)
Conclusion #
Following the 8 phases in order gives you a complete picture. The key is not to skip any step.
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