Studying Rational Functions with Probable Graph

Studying Rational Functions

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A rational function \(y = \frac{N(x)}{D(x)}\). Simplified study phases:

1. Domain: \(D(x) \neq 0\)
2. Y-intercept: set \(x = 0\)
3. X-intercepts: solve \(N(x) = 0\)
4. Sign: sign table for \(f(x) > 0\)
5. Vertical asymptotes: where \(D(x) = 0\)
6. Horizontal asymptote: compare degrees of N and D
7. Sketch the graph combining all information

Worked Example: \(y = \frac{2x}{x-1}\)

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Domain: \(x \neq 1\). Passes through origin. Vertical asymptote: \(x = 1\). Horizontal asymptote: \(y = 2\).

Conclusion

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Following these phases in order lets you sketch a probable graph without computing infinitely many points.